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The document provides a comprehensive overview of geometrical optics, including foundational principles such as Fermat's principle, the laws of reflection and refraction, and various types of optical devices like lenses and prisms. It elaborates on the mathematical relationships governing ray behavior as light interacts with different media and covers applications of these principles in devices such as dispersing and reflecting prisms. Key definitions related to optical systems, including focal points and stops, are also explained to illustrate how they influence image formation.

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1 LENS SOLO HERMELIN Updated: 27.10.07http://www.solohermelin.com
2 Table of Content (continue( SOLO OPTICS Plane-Parallel Plate The Three Laws of Geometrical Optics Fermat’s Principle (1657) Prisms Lens Definitions Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law Derivation of Lens Makers’ Formula First Order, Paraxial or Gaussian Optics Ray Tracing Matrix Formulation References
3 SOLO The Three Laws of Geometrical Optics 1. Law of Rectilinear Propagation In an uniform homogeneous medium the propagation of an optical disturbance is in straight lines. . Law of Reflection An optical disturbance reflected by a surface has the property that the incident ray, the surface normal, and the reflected ray all lie in a plane, and the angle between the incident ray and the surface normal is equal to the angle between the reflected ray and the surface normal: . Law of Refraction An optical disturbance moving from a medium of refractive index n1 into a medium of refractive index n2 will have its incident ray, the surface normal between the media , and the reflected ray in a plane, and the relationship between angle between the incident ray and the surface normal θi and the angle between the reflected ray and the surface normal θt given by Snell’s Law: ti nn θθ sinsin 21 ⋅=⋅ ri θθ = “The branch of optics that addresses the limiting case λ0 → 0, is known as Geometrical Optics, since in this approximation the optical laws may be formulated in the language of geometry.” Max Born & Emil Wolf, “Principles of Optics”, 6th Ed., Ch. 3 Foundation of Geometrical Optics
4 SOLO Foundation of Geometrical Optics Fermat’s Principle (1657) The Principle of Fermat (principle of the shortest optical path( asserts that the optical length of an actual ray between any two points is shorter than the optical ray of any other curve that joints these two points and which is in a certai neighborhood of it. An other formulation of the Fermat’s Principle requires only Stationarity (instead of minimal length). ∫ 2 1 P P dsn An other form of the Fermat’s Principle is: Princple of Least Time The path following by a ray in going from one point in space to another is the path that makes the time of transit of the associated wave stationary (usually a minimum). The idea that the light travels in the shortest path was first put forward by Hero of Alexandria in his work “Catoptrics”, cc 100B.C.-150 A.C. Hero showed by a geometrical method that the actual path taken by a ray of light reflected from plane mirror is shorter than any other reflected path that might be drawn between the source and point of observation.
5 SOLO 1. The optical path is reflected at the boundary between two regions ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   In this case we have and21 nn = ( ) ( ) ( ) 0ˆˆ 21 21 =⋅−=⋅        − rdssrd sd rd sd rd rayray   We can write the previous equation as: i.e. is normal to , i.e. to the boundary where the reflection occurs. 21 ˆˆ ss − rd  ( ) 0ˆˆˆ 2121 =−×− ssn REFLECTION & REFRACTION Reflection Laws Development Using Fermat Principle This is equivalent with: ri θθ = Incident ray and Reflected ray are in the same plane normal to the boundary.&
6 SOLO 2. The optical path passes between two regions with different refractive indexes n1 to n2. (continue – 1) ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   where is on the boundary between the two regions andrd  ( ) ( ) sd rd s sd rd s rayray 2 :ˆ, 1 :ˆ 21  == Therefore is normal to . 2211 ˆˆ snsn − rd  Since can be in any direction on the boundary between the two regions is parallel to the unit vector normal to the boundary surface, and we have rd  2211 ˆˆ snsn − 21 ˆ −n ( ) 0ˆˆˆ 221121 =−×− snsnn We recovered the Snell’s Law from Geometrical Optics REFLECTION & REFRACTION Refraction Laws Development Using Fermat Principle ti nn θθ sinsin 21 = Incident ray and Refracted ray are in the same plane normal to the boundary. &
7 SOLO Plane-Parallel Plate A single ray traverses a glass plate with parallel surfaces and emerges parallel to its original direction but with a lateral displacement d. Optics ( ) ( )irriri lld φφφφφφ cossincossinsin −=−= r t l φcos =       −= r i ri td φ φ φφ cos cos sinsin ir nn φφ sinsin 0 =Snell’s Law       −= n n td r i i 0 cos cos 1sin φ φ φ For small anglesi φ       −≈ n n td i 0 1φ
8 SOLO Plane-Parallel Plate (continue – 1( Two rays traverse a glass plate with parallel surfaces and emerge parallel to their original direction but with a lateral displacement l. Optics ( ) ( )irriri lld φφφφφφ cossincossinsin −=−= r t l φcos =       −= r i ri td φ φ φφ cos cos sinsin ir nn φφ sinsin 0 =Snell’s Law       −= n n td r i i 0 cos cos 1sin φ φ φ       −== r i i n n t d l φ φ φ cos cos 1 sin 0 For small anglesi φ       −≈ n n tl 0 1
9 SOLO Prisms Type of prisms: A prism is an optical device that refract, reflect or disperse light into its spectral components. They are also used to polarize light by prisms from birefringent media. Optics - Prisms 2. Reflective 1. Dispersive 3. Polarizing
10 OpticsSOLO Dispersive Prisms ( ) ( )2211 itti θθθθδ −+−= 21 it θθα += αθθδ −+= 21 ti 202 sinsin ti nn θθ =Snell’s Law 10 ≈n ( ) ( )[ ]1 1 2 1 2 sinsinsinsin tit nn θαθθ −== −− ( )[ ] ( )[ ]11 21 11 1 2 sincossin1sinsinsincoscossinsin ttttt nn θαθαθαθαθ −−=−= −− Snell’s Law 110 sinsin ti nn θθ = 11 sin 1 sin it n θθ = ( )[ ]1 2/1 1 221 2 sincossinsinsin iit n θαθαθ −−= − ( )[ ] αθαθαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n The ray deviation angle is 10 ≈n
11 OpticsSOLO Prisms ( )[ ] αθαθαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n
12 OpticsSOLO Prisms ( )[ ] αθαθαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n αθθδ −+= 21 ti Let find the angle θi1 for which the deviation angle δ is minimal; i.e. δm. This happens when  01 0 11 2 1 =−+= ii t i d d d d d d θ α θ θ θ δ Taking the differentials of Snell’s Law equations 22 sinsin ti n θθ = 11 sinsin ti n θθ = 2222 coscos iitt dnd θθθθ = 1111 coscos ttii dnd θθθθ = Dividing the equations  1 2 1 2 1 1 2 1 2 1 cos cos cos cos −− = i t i t t i t i d d d d θ θ θ θ θ θ θ θ 2 22 1 22 2 2 2 2 1 2 2 2 1 2 2 2 1 2 sin sin /sin1 /sin1 sin1 sin1 sin1 sin1 t i t i i t t i n n n n θ θ θ θ θ θ θ θ − − = − − = − − = − − 1 1 2 −= i t d d θ θ 21 it θθα += 1 2 1 −= i t d d θ θ 2 2 1 2 2 2 1 2 cos cos cos cos i t t i θ θ θ θ = 21 ti θθ = 1≠n
13 OpticsSOLO Prisms ( )[ ] αθαθαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n We found that if the angle θi1 = θt2 the deviation angle δ is minimal; i.e. δm. Using the Snell’s Law equations 22 sinsin ti n θθ = 11 sinsin ti n θθ = 21 ti θθ = 21 it θθ = This means that the ray for which the deviation angle δ is minimum passes through the prism parallel to it’s base. Find the angle θi1 for which the deviation angle δ is minimal; i.e. δm (continue – 1(.
14 OpticsSOLO Prisms ( )[ ] αθαθαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n Using the Snell’s Law 11 sinsin ti n θθ = 21 it θθ = This equation is used for determining the refractive index of transparent substances. 21 it θθα += αθθδ −+= 21 ti 21 ti θθ = mδδ = 2/1 αθ =t αθδ −= 12 im ( ) 2/1 αδθ += mi ( )[ ] 2/sin 2/sin α αδ + = m n Find the angle θi1 for which the deviation angle δ is minimal; i.e. δm (continue – 2(.
15 OpticsSOLO Prisms The refractive index of transparent substances varies with the wavelength λ. ( )[ ]{ } αθαθλαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n
16 OpticsSOLO http://physics.nad.ru/Physics/English/index.htm Prisms Color λ0 (nm( υ [THz] Red Orange Yellow Green Blue Violet 780 - 622 622 - 597 597 - 577 577 - 492 492 - 455 455 - 390 384 – 482 482 – 503 503 – 520 520 – 610 610 – 659 659 - 769 1 nm = 10-9 m, 1 THz = 1012 Hz ( )[ ]{ } αθαθλαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n In 1672 Newton wrote “A New Theory about Light and Colors” in which he said that the white light consisted of a mixture of various colors and the diffraction was color dependent. Isaac Newton 1542 - 1727
17 SOLO Dispersing Prisms Pellin-Broca Prism Abbe Prism Ernst Karl Abbe 1840-1905 At Pellin-Broca Prism an incident ray of wavelength λ passes the prism at a dispersing angle of 90°. Because the dispersing angle is a function of wavelength the ray at other wavelengths exit at different angles. By rotating the prism around an axis normal to the page different rays will exit at the 90°. At Abbe Prism the dispersing angle is 60°. Optics - Prisms
18 SOLO Dispersing Prisms (continue – 1( Amici Prism Optics - Prisms
19 SOLO Reflecting Prisms BED∠−=  180δ  360=∠+∠+∠+ ABEBEDADEα 1 90 i ABE θ+=∠  2 90 t ADE θ+=∠   3609090 12 =++∠+++ it BED θθα 12180 itBED θθα −−−=∠  αθθδ ++=∠−= 21180 tiBED The bottom of the prism is a reflecting mirror Since the ray BC is reflected to CD DCGBCF ∠=∠ Also CGDBFC ∠=∠ CDGFBC ∠=∠ FBCt ∠−=  901θ CDGi ∠−=  902θ 21 it θθ = 202 sinsin ti nn θθ =Snell’s Law Snell’s Law 110 sinsin ti nn θθ = 21 ti θθ = αθδ += 1 2 i CDGFBC ∆∆ ~ Optics - Prisms
20 SOLO Reflecting Prisms Porro Prism Porro-Abbe Prism Schmidt-Pechan Prism Penta Prism Optics - Prisms Roof Penta Prism
21 SOLO Reflecting Prisms Abbe-Koenig Prism Dove Prism Amici-roof Prism Optics - Prisms
22 SOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Nicol Prism The Nicol Prism is made up from two prisms of calcite cemented with Canada balsam. The ordinary ray can be made to totally reflect off the prism boundary, leving only the extraordinary ray.. Polarizing Prisms Optics - Prisms
23 SOLO Polarizing Prisms A Glan-Foucault prism deflects polarized light transmitting the s-polarized component. The optical axis of the prism material is perpendicular to the plane of the diagram. A Glan-Taylor prism reflects polarized light at an internal air-gap, transmitting only the p-polarized component. The optical axes are vertical in the plane of the diagram. A Glan-Thompson prism deflects the p-polarized ordinary ray whilst transmitting the s-polarized extraordinary ray. The two halves of the prism are joined with Optical cement, and the crystal axis are perpendicular to the plane of the diagram. Optics - Prisms
24 OpticsSOLO Lens Definitions Optical Axis: the common axis of symmetry of an optical system; a line that connects all centers of curvature of the optical surfaces. Lateral Magnification: the ratio between the size of an image measured perpendicular to the optical axis and the size of the conjugate object. Longitudinal Magnification: the ratio between the lengthof an image measured along the optical axis and the length of the conjugate object. First (Front( Focal Point: the point on the optical axis on the left of the optical system (FFP( to which parallel rays on it’s right converge. Second (Back( Focal Point: the point on the optical axis on the right of the optical system (BFP( to which parallel rays on it’s left converge.
25 OpticsSOLO Definitions (continue – 2( Aperture Stop (AS(: the physical diameter which limits the size of the cone of radiation which the optical system will accept from an axial point on the object. Field Stop (FS(: the physical diameter which limits the angular field of view of an optical system. The Field Stop limit the size of the object that can be seen by the optical system in order to control the quality of the image. A.S. F.S. IΣ Aperture and Field Stops Image plane Hecht "Optics"
26 OpticsSOLO Definitions (continue – 2( Entrance Pupil: the image of the Aperture Stop as seen from the object through the (EnP( elements preceding the Aperture Stop. Exit Pupil: the image of the Aperture Stop as seen from an axial point on the (ExP( image plane. Entrance pupil Exit pupil A.S. IΣ xpE npE Chief Ray Entrance and Exit pupils Image plane Marginal Ray Hecht "Optics" Entrance pupilExit pupil A.S. I Σ xpE npE Chief Ray Image plane A front Aperture Stop Hecht "Optics" Chief Ray: an object Ray passing through the center of the aperture stop and (CR( appearing to pass through the centers of entrance and exit pupils. Marginal Ray: an object Ray passing through the edge of the aperture stop. (MR(
27 OpticsSOLO Definitions (continue – 2( Entrance pupil Exit pupil A.S. IΣ Chief Ray Marginal Ray Exp Enp Image plane Hecht "Optics"
28 OpticsSOLO Definitions (continue – 1( Principal Planes: the two planes defined by the intersection of the parallel incident rays entering an optical system with the rays converging to the focal points after passing through the optical system. Principal Points: the intersection of the principal planes with the optical axes. Nodal Points: two axial points of an optical system, so located that an oblique ray directed toward the first appears to emerge from the second, parallel to the original direction. For systems in air, the Nodal Points coincide with the Principal Points. Cardinal Points: the Focal Points, Principal Points and the Nodal Points.
29 OpticsSOLO Definitions (continue – 3( Relative Aperture (f# (: the ratio between the effective focal length (EFL( f to Entrance Pupil diameter D. Numerical Aperture (NA(: sine of the half cone angle u of the image forming ray bundles multiplied by the final index n of the optical system. If the object is at infinity and assuming n = 1 (air(: Dff /:# = unNA sin: ⋅= # 1 2 1 2 1 sin ff D uNA =      ==
30 OpticsSOLO Perfect Imaging System • All rays originating at one object point reconverge to one image point after passing through the optical system. • All of the objects points lying on one plane normal to the optical axis are imaging onto one plane normal to the axis. • The image is geometrically similar to the object.
31 OpticsSOLO Lens Convention of Signs 1. All Figures are drawn with the light traveling from left to right. 2. All object distances are considered positive when they are measured to the left of the vertex and negative when they are measured to the right. 3. All image distances are considered positive when they are measured to the right of the vertex and negative when they are measured to the left. 4. Both focal length are positive for a converging system and negative for a diverging system. 5. Object and Image dimensions are positive when measured upward from the axis and negative when measured downward. 6. All convex surfaces are taken as having a positive radius, and all concave surfaces are taken as having a negative radius.
32 OpticsSOLO Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle Karl Friederich Gauss 1777-1855 The optical path connecting points M, T, M’ is ''lnlnpathOptical ⋅+⋅= Applying cosine theorem in triangles MTC and M’TC we obtain: ( ) ( )[ ] 2/122 cos2 βRsRRsRl +−++= ( ) ( )[ ] 2/122 cos'2'' βRsRRsRl −+−+= ( ) ( )[ ] ( ) ( )[ ] 2/122 2/122 cos'2''cos2 ββ RsRRsRnRsRRsRnpathOptical −+−+⋅++−++⋅= Therefore According to Fermat’s Principle when the point T moves on the spherical surface we must have ( ) 0= βd pathOpticald ( ) ( ) ( ) 0 ' sin''sin = −⋅ − +⋅ = l RsRn l RsRn d pathOpticald ββ β from which we obtain       ⋅ − ⋅ =+ l sn l sn Rl n l n ' ''1 ' ' For small α and β we have ''& slsl ≈≈ and we obtain R nn s n s n − =+ ' ' ' Gaussian Formula for a Single Spherical Surface
33 OpticsSOLO Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law Apply Snell’s Law: 'sin'sin φφ nn = If the incident and refracted rays MT and TM’ are paraxial the angles and are small and we can write Snell’s Law: φ 'φ From the Figure βαφ += γβφ −=' ''φφ nn = ( ) ( ) ( ) βγαγββα nnnnnn −=+⇒−=+ ''' For paraxial rays α, β, γ are small angles, therefore '/// shrhsh ≈≈≈ γβα ( ) r h nn s h n s h n −=+ ' ' ' or ( ) r nn s n s n − =+ ' ' ' Gaussian Formula for a Single Spherical Surface Karl Friederich Gauss 1777-1855 Willebrord van Roijen Snell 1580-1626 ( ) φ φφ φφ φ ≈+++=   O` 53 !5!3 sin
34 OpticsSOLO Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law for s → ∞ the incoming rays are parallel to optical axis and they will refract passing trough a common point called the focus F’. ( ) r nn s n s n − =+ ' ' ' ( ) r nn f nn − =+ ∞ ' ' ' r nn n f − = ' ' ' for s’ → ∞ the refracting rays are parallel to optical axis and therefore the incoming rays passes trough a common point called the focus F. ( ) r nnn f n − = ∞ + '' r nn n f − = ' '' n n f f =
35 OpticsSOLO Derivation of Lens Makers’ Formula We have a lens made of two spherical surfaces of radiuses r1 and r2 and a refractive index n’, separating two media having refraction indices n a and n”. Ray MT1 is refracted by the first spherical surface (if no second surface exists) to T1M’. ( ) 111 ' ' ' r nn s n s n − =+ 11111 ''& sMAsTA == Ray T1T2 is refracted by the second spherical surface to T2M”. 2222 ""&'' sMAsMA == ( ) 222 '" " " ' ' r nn s n s n − =+ Assuming negligible lens thickness we have , and since M’ is a virtual object for the second surface (negative sign) we have 21 '' ss ≈ 21 '' ss −≈ ( ) 221 '" " " ' ' r nn s n s n − =+−
36 OpticsSOLO Derivation of Lens Makers’ Formula (continue – 1) ( ) 111 ' ' ' r nn s n s n − =+ Add those equations ( ) 221 '" " " ' ' r nn s n s n − =+− ( ) ( ) 2121 '"' " " r nn r nn s n s n − + − =+ The focal lengths are defined by tacking s1 → ∞ to obtain f” and s”2 → ∞ to obtain f ( ) ( ) f n r nn r nn f n = − + − = 212 '"' " " Let define s1 as s and s”2 as s” to obtain ( ) ( ) 21 '"' " " r nn r nn s n s n − + − =+ ( ) ( ) f n r nn r nn f n = − + − = 21 '"' " "
37 OpticsSOLO Derivation of Lens Makers’ Formula (continue – 2) If the media on both sides of the lens is the same n = n”.       −      −=+ 21 11 1 ' " 11 rrn n ss       −      −== 21 11 1 '1 " 1 rrn n ff Therefore " 11 " 11 ffss ==+ Lens Makers’ Formula
38 OpticsSOLO First Order, Paraxial or Gaussian Optics In 1841 Gauss gave an exposition in “Dioptrische Untersuchungen” for thin lenses, for the rays arriving at shallow angles with respect to Optical axis (paraxial). Karl Friederich Gauss 1777-1855 Derivation of Lens Formula From the similarity of the triangles and using the convention: ( ) '' ' '~' f y s yy TAFTSQ = −+ ⇒∆∆ Lens Formula in Gaussian form ( ) ( ) f y s yy FASQTS '' ~ − = −+ ⇒∆∆ ( ) 0' >− y Sum of the equations: ( ) ( ) ( ) ' ' ' '' f y f y s yy s yy + − = −+ + −+ since f = f’ fss 1 ' 11 =+ ( ) φ φφ φφ φ ≈+++=   O` 53 !5!3 sin
39 OpticsSOLO First Order, Paraxial or Gaussian Optics (continue – 1) Gauss explanation can be extended to the first order approximation to any optical system. Karl Friederich Gauss 1777-1855 Lens Formula in Gaussian form fss 1 ' 11 =+ s – object distance (from the first principal point to the object). s’ – image distance (from the second principal point to the image). f – EFL (distance between a focal point to the closest principal plane). 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis y
40 OpticsSOLO Derivation of Lens Formula (continue) From the similarity of the triangles and using the convention: ( ) f y x y FASQMF ' ~ − =⇒∆∆ Lens Formula in Newton’s form ( ) f y x y QMFTAF = − ⇒∆∆ ' ' '''~' ( ) 0' >− y Multiplication of the equations: ( ) ( ) 2 ' ' ' f yy xx yy −⋅ = ⋅ −⋅ or 2 ' fxx =⋅ Isaac Newton 1643-1727 First Order, Paraxial or Gaussian Optics (continue – 2) Published by Newton in “Opticks” 1710 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis y
41 OpticsSOLO Derivation of Lens Formula (continue) First Order, Paraxial or Gaussian Optics (continue – 3) Lateral or Transverse Magnification f x x f s s h h mT ''' −=−=−== Quantity (+) sign (-) sign s real object virtual object s’ real image virtual image f converging lens diverging lens h erect object inverted object h’ erect image inverted image mT erect image inverted image 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis y
42 OpticsSOLO Derivation of Lens Formula (Summary) If the media on both sides of the lens is the same n = n”.       −      −=+ 21 11 1 ' " 11 rrn n ss       −      −== 21 11 1 '1 " 1 rrn n ff Therefore " 11 " 11 ffss ==+ Lens Makers’ Formula f x x f s s h h mT ''' −=−=−== Gauss’ Lens Formula Magnification
43 OpticsSOLO
44 OpticsSOLO
45 OpticsSOLO Ray Tracing F C O I Object Virtua l Image Convex Mirror R/2 R/2 R FC O I Object Real Image Concave Mirror Ray Tracing is a graphically implementation of paralax ray analysis. The construction doesn’t take into consideration the nonideal behavior, or aberration of real lens. The image of an off-axis point can be located by the intersection of any two of the following three rays: 1. A ray parallel to the axis that is reflected through F’. 2. A ray through F that is reflected parallel to the axis. 3. A ray through the center C of the lens that remains undeviated and undisplaced (for thin lens).
46 OpticsSOLO
47 OpticsSOLO Matrix Formulation The Matrix Formulation of the Ray Tracing method for the paraxial assumption was proposed at the beginning of nineteen-thirties by T.Smith. Assuming a paraxial ray entering at some input plane of an optical system at the distance r1 from the symmetry axis and with a slope r1’ and exiting at some output plane at the distance r2 from the symmetry axis and with a slope r2’, than the following linear (matrix) relation applies:       =            =      ''' 1 1 1 1 2 2 r r M r r DC BA r r       = DC BA Mwhere ray transfer matrix When the media to the left of the input plane and to the right of the output plane have the same refractive index, we have: 1det =⋅−⋅= CBDAM
48 OpticsSOLO Matrix Formulation (continue -1) Uniform Optical Medium In an Uniform Optical Medium of length d no change in ray angles occurs: '' ' 12 112 rr rdrr = +=       = 10 1 d M Medium Optical Uniform Planar Interface Between Two Different Media 12 rr = '' 1 2 1 2 12 r n n r rr = = Apply Snell’s Law: 2211 sinsin φφ nn = paraxial assumption: φφφφ ≈=⇒≈ tan'sin r From Snell’s Law: '' 1 2 1 2 r n n r =       = 21 /0 01 nn M Interface Planar 1det 2 1 ≠= n n M Interface Planar 1det = Medium Optical Uniform M The focal length of this system is infinite and it has not specific principal planes.
49 OpticsSOLO Matrix Formulation (continue -2) A Parallel-Sided Slab of refractive index n bounded on both sides with media of refractive index n1 = 1 We have three regions: • on the right of the slab (exit of ray):             =      '/0 01 ' 3 3 124 4 r r nnr r • in the slab:             =      '10 1 ' 2 2 3 3 r rd r r • on the left of the slab (entrance of ray):             =      '/0 01 ' 1 1 212 2 r r nnr r Therefore:                         =      '/0 01 10 1 /0 01 ' 1 1 21124 4 r r nn d nnr r             =                  = 21 21 122112 /0 /1 /0 01 /0 01 10 1 /0 01 nn nnd nnnn d nn M mediaentranceslabmediaexit Slab Sided Parallel        = 10 /1 21 nnd M Slab Sided Parallel 1det = Slab Sided Parallel M
50 OpticsSOLO Matrix Formulation (continue -3) Spherical Interface Between Two Different Media 12 rr = Apply Snell’s Law: rnin sinsin 21 = paraxial assumption: rrii ≈≈ sin&sin From Snell’s Law: rnin 21 = ( )           − =           −= 2 1 2 1 2 1 12 21 0101 n n n D n n Rn nnM Interface Spherical 1det 2 1 ≠= n n M Interface Spherical 12 11 ' ' φ φ += += rr ri From the Figure: ( ) ( )122111 '' φφ +=+ rnrn 111 / Rr=φ ( ) 12 121 2 11 2 ' ' Rn rnn n rn r − += ( ) 1 12 11 112 2 12 ' ' n rn Rn rnn r rr + − = = ( ) 1 12 1 : R nn D − =where: Power of the surface If R1 is given in meters D1 gives diopters
51 OpticsSOLO Matrix Formulation (continue -4) Thick Lens We have three regions: • on the right of the slab (exit of ray):                 − =      ' 01 ' 3 3 1 2 1 2 4 4 r r n n n D r r • in the slab:             =      '10 1 ' 2 2 3 3 r rd r r • on the left of the slab (entrance of ray):                 − =      ' 01 ' 1 1 2 1 2 1 2 2 r r n n n D r r Therefore:                   − −           − =                −                − =      ' 101 ' 01 10 1 01 ' 1 1 2 1 2 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 4 4 r r n n n D n n d n D d n n n D r r n n n D d n n n D r r               −      − + − − = 2 2 21 21 1 21 2 1 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick ( ) 2 21 2 R nn D − = ( ) 1 12 1 : R nn D − =               −      − + −− = − 2 1 21 21 1 21 2 1 2 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick 1det = Lens Thick M or 21 DD ⇔
52 OpticsSOLO Matrix Formulation (continue -5) Thick Lens (continue -1) Let use the second Figure where Ray 2 is parallel to Symmetry Axis of the Optical System that is refracted trough the Second Focal Point.                         −      − + − − =           '1 1 ' 1 1 2 2 21 21 1 21 2 1 2 1 4 4 r r n D d nn DD d n DD n n d n D d r r We found: 2141 /'&0' frrr −==Ray 2: By substituting Ray2 parameters we obtain: 1 2 1 21 21 1 21 4 1 ' r f r nn DD d n DD r −=      − + −= 1 21 21 1 21 2 −       − + = nn DD d n DD f frrr /'&0' 414 −==Ray 1: We found:                         −      − + −− =           '1 1 ' 4 4 2 1 21 21 1 21 2 1 2 2 1 1 r r n D d nn DD d n DD n n d n D d r r 4 1 4 21 21 1 21 1 1 ' r f r nn DD d n DD r −=      − + = 2 1 21 21 1 21 1 f nn DD d n DD f −=      − + = −
53 OpticsSOLO Matrix Formulation (continue -6) Thin Lens For thick lens we found               −      − + − − = 2 2 21 21 1 21 2 1 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick       − + = 21 21 1 211 nn DD d n DD f For thin lens we can assume d = 0 and obtain           − = 1 1 01 f M Lens Thin 1 211 n DD f + = ( ) 2 21 2 R nn D − = ( ) 1 12 1 : R nn D − =       −      −= + = 211 2 1 21 11 1 1 RRn n n DD f
54 OpticsSOLO Matrix Formulation (continue -7) Thin Lens (continue – 1) For a biconvex lens we have R2 negative         +      −= 211 2 11 1 1 RRn n f For a biconcave lens we have R1 negative         +      −−= 211 2 11 1 1 RRn n f           − = 1 1 01 f M Lens Thin
55 OpticsSOLO Matrix Formulation (continue -8) A Length of Uniform Medium Plus a Thin Lens           −− =                − == + f d f d d f MMM Medium Uniform Lens Thin Lens Thin Medium Uniform 1 1 1 10 1 1 1 01 Combination of Two Thin Lenses             +−−−+−− −+− =           −−          −− == 21 21 2 2 2 1 1 1 21 2 21 1 21 21 2 2 1 1 1 1 22 2 1 11 1 1 1 1 1 1 1 1122 ff dd f d f d f d ff d ff f dd dd f d f d f d f d f d MMMMM dMedium Uniform fLens Thin dMedium Uniform fLens Thin Lenses Thin Two The Focal Length of the Combination of Two Thin Lenses is: 21 2 21 111 ff d fff −+= Return to Chromatic Aberration
56 OpticsSOLO Real Imaging Systems – Aberrations Departures from the idealized conditions of Gaussian Optics in a real Optical System are called Aberrations Monochromatic Aberrations Chromatic Aberrations • Monochromatic Aberrations Departures from the first order theory are embodied in the five primary aberrations 1. Spherical Aberrations 2. Coma 3. Astigmatism 4. Field Curvature 5. Distortion This classification was done in 1857 by Philipp Ludwig von Seidel (1821 – 1896) • Chromatic Aberrations 1. Axial Chromatic Aberration 2. Lateral Chromatic Aberration
57 OpticsSOLO Real Imaging Systems – Aberrations (continue – 5)
58 OpticsSOLO Real Imaging Systems – Aberrations (continue – 5)
59 OpticsSOLO Real Imaging Systems – Aberrations (continue – 5)
60 OpticsSOLO Real Imaging Systems – Aberrations
61 OpticsSOLO Real Imaging Systems – Aberrations
62 OpticsSOLO Real Imaging Systems – Aberrations (continue – 1) Seidel Aberrations Consider a spherical surface of radius R, with an object P0 and the image P0’ on the Optical Axis. The Chief Ray is P0 V0 P0’ and a General Ray P0 Q P0’. The Wave Aberration is defined as the difference in the optical path lengths between a General Ray and the Chief Ray. ( ) [ ] [ ] ( ) ( )snsnQPnQPnPVPQPPrW +−+=−= '''''' 00000000 On-Axis Point Object The aperture stop AS, entrance pupil EnP, and exit pupil ExP are located at the refracting surface.
63 OpticsSOLO Real Imaging Systems – Aberrations (continue – 2) Seidel Aberrations (continue – 1)         −−=−−= 2 2 22 11 R r RrRRz Define: ( ) 2 2 11 2 2 R r xxf R r x −=+= −= ( ) ( ) 2/1 1 2 1 ' − += xxf ( ) ( ) 2/3 1 4 1 " − +−= xxf ( ) ( ) 2/5 1 8 3 '" − +−= xxf Develop f (x) in a Taylor series ( ) ( ) ( ) ( ) ( ) ++++= 0"' 6 0" 2 0' 1 0 32 f x f x f x fxf 1 168 11 32 <++−+=+ x xx xx  Rr R r R r R r R r Rz <+++=         −−= 5 6 3 42 2 2 1682 11 On Axis Point Object From the Figure: ( ) 222 rzRR +−= 02 22 =+− rRzz
64 OpticsSOLO Real Imaging Systems – Aberrations (continue – 3) Seidel Aberrations (continue – 2) From the Figure: ( )[ ] [ ] ( )[ ] ( ) 2/1 2 2/12 2 2/12222/122 0 212 2 22       − +=+−= ++−=+−= −= z s sR sszsR rsszzrszQP rzRz ( ) ( )       + − − − +−≈ <++−+=+   2 4 2 2 1 168 11 2 1 1 32 z s sR z s sR s x xx xx ( ) ( )         +      + − −      + − +−= +≈  2 3 42 4 2 3 42 2 82 822 1 82 1 3 42 R r R r s sR R r R r s sR s R r R r z ( )[ ] +               −+      −+      −+−≈+−= 4 2 2 22/122 0 11 8 111 8 111 2 1 r sRssRR r sR srszQP ( )[ ] +               −+      −+      −+≈+−= 4 2 2 22/122 0 1 ' 1 '8 11 ' 1 8 11 ' 1 2 1 ''' r RssRsR r Rs srzsPQ In the same way: On Axis Point Object
65 OpticsSOLO Real Imaging Systems – Aberrations (continue – 4) Seidel Aberrations (continue – 3) +               −+      −+      −+−≈ 4 2 2 2 0 11 8 111 8 111 2 1 r sRssRR r sR sQP +               −+      −+      −+≈ 4 2 2 2 0 1 ' 1 '8 11 ' 1 8 11 ' 1 2 1 '' r RssRsR r Rs sPQ Therefore: ( ) ( ) ( ) 4 22 2 42 000 11 ' 11 ' ' 8 1 82 ' ' ' '''' r sRs n sRs n R rr R nn s n s n snsnQPnQPnrW               −−      −−      +      − −−= +−+= Since P0’ is the Gaussian image of P0 we have ( ) R nn s n s n − = − + ' ' ' and: ( ) 44 22 0 11 ' 11 ' ' 8 1 rar sRs n sRs n rW S =               −−      −−= On Axis Point Object
66 OpticsSOLO Real Imaging Systems – Aberrations (continue – 5) Seidel Aberrations (continue – 4) Off-Axis Point Object Consider the spherical surface of radius R, with an object P and its Gaussian image P’ outside the Optical Axis. The aperture stop AS, entrance pupil EnP, and xit pupil ExP are located at the refracting surface. Using ''~ 00 CPPCPP ∆∆ the transverse magnification ( ) ( ) s n s n nn s s n s n nn s Rs Rs h h Mt − − +− − − − = +− − = − = ' ' ' ' ' ' ' '' ( )sn sn nn s s nn nn s s nn Mt − = −+− +−− = ' ' ' ' ' ' ' '
67 OpticsSOLO Real Imaging Systems – Aberrations (continue – 6) Seidel Aberrations (continue – 5) Off-Axis Point Object The Wave Aberration is defined as the difference n the optical path lengths between the General Ray and the Undeviated Ray. ( ) [ ] [ ] [ ] [ ]{ } [ ] [ ]{ } ( )4 0 4 0 '''' '' VVVQa PVPPPVPVPPQP PVPPQPQW S −= −−−= −= For the approximately similar triangles VV0C and CP0’P’ we have: CP CV PP VV ''' 0 0 0 0 ≈ '' ' '' ' 0 0 0 0 hbh Rs R PP CP CV VV = − =≈ Rs R b − = ' :               −−      −−= 22 11 ' 11 ' ' 8 1 sRs n sRs n aS
68 OpticsSOLO Real Imaging Systems – Aberrations (continue – 7) Seidel Aberrations (continue – 6) Off-Axis Point Object Wave Aberration. ( ) [ ] [ ] ( )4 0 4 '' VVVQaPVPPQPQW S −=−= Define the polar coordinate (r,θ) of the projection of Q in the plane of exit pupil, with V0 at the origin. θθ cos'2'cos2 222 0 2 0 2 2 hbrhbrVVrVVrVQ ++=++= '0 hbVV = ( ) [ ] [ ] ( ) ( )[ ]442222 4 0 4 'cos'2' '' hbhbrhbra VVVQaPVPPQPQW S S −++= −=−= θ ( ) ( )θθθθ cos'4'2cos'4cos'4';, 33222222234 rhbrhbrhbrhbrahrW S ++++=
69 OpticsSOLO Real Imaging Systems – Aberrations (continue – 8) Seidel Aberrations (continue – 7) eneral Optical Systems ( ) θθθθ cos''cos'cos'';, 33222222234 rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp ++++= A General Optical Systems has more than on Reflecting or Refracting surface. The image of one surface acts as an bject for the next surface, therefore the aberration is additive. We must address the aberration in the plane of the exit pupil, since the rays follow straight lines from the plane of the exit pupil. The general Wave Aberration Function is: 1. Spherical Aberrations CoefficientSpC 2. Coma CoefficientCoC 3. Astigmatism CoefficientAs C 4. Field Curvature CoefficientFC C 5. Distortion CoefficientDi C where:
70 OpticsSOLO Real Imaging Systems – Aberrations (continue – 7) Seidel Aberrations (continue – 6) ( ) ( )θθθθ cos'4'2cos'4cos'4';, 33222222234 rhbrhbrhbrhbrahrW S ++++=
71 OpticsSOLO Real Imaging Systems – Aberrations (continue – 7) Seidel Aberrations (continue – 6) ( ) θθθθ cos''cos'cos'';, 32222234 rhCrhCrhCrhCrChrW DiFCAsCoSp ++++=
72 OpticsSOLO Real Imaging Systems – Aberrations (continue – 9) Seidel Aberrations (continue – 8) nWPP TR /= Assume that P’ is the image of P. The point PT is on the Exit Pupil (Exp) and on the True Wave Front (TWF) that propagates toward P’. This True Wave Front is not a sphere because of the Aberration. Without the aberration the wave front would be the Reference Sphere (RS) with radius PRP. W (x’,y’;h’) - wave aberration n - lens refraction index L’ - distance between Exp and Image plane ά - angle between the normals to the TWF and RS at PT. Assume that P’R and P’T are two points on RS and TWF, respectively, and on a ray close to PRPT ray, converging to P’, the image of P. lPPPP TRTR ∆+=''
73 OpticsSOLO Real Imaging Systems – Aberrations (continue – 9) Seidel Aberrations (continue – 8) ( ) ' ';',' ' ' ' x hyxW n L x ∂ ∂ =∆ ( ) ' ';',' ' ' ' y hyxW n L y ∂ ∂ =∆ θ θ sin' cos' ry rx = = ( ) nhyxWPP TR /';','= lPPPP TRTR ∆+='' α= ∆ ∆ = ∆ − = ∂ ∂ →∆→∆ r l r PPPP x W n r TRTR r 00 lim '' lim 1 x W n L Lr ∂ ∂ ==∆ ' 'α
74 OpticsSOLO Real Imaging Systems – Aberrations (continue – 1) 1. Spherical Aberrations ( ) ( ) ( )';',''' ';, 222 4 hyxWyxC rChrW SpSp SpSp =+= =θ ( ) ' ' ' 4 ' ';',' ' ' ' 2 xrC n L x hyxW n L x Sp = ∂ =∆ ( ) ' ' ' 4 ' ';',' ' ' ' 2 yrC n L y hyxW n L y Sp = ∂ =∆ To Update ( ) ( )[ ] 32/122 ' ' 4'' rC n L yxr Sp=∆+∆=∆ Consider only the Spherical Wave Aberration Function The Spherical Wave Aberration is a Circle in the Image Plane
75 OpticsSOLO Real Imaging Systems – Aberrations (continue – 2) 2. Coma Assume an object point outside the Optical Axis. Meridional (Tangential) plane is the plane defined by the object point and the Optical Axis. Sagittal plane is the plane normal to Meridional plane that contains the Chief Ray passing through the Object point.
76 OpticsSOLO Real Imaging Systems – Aberrations (continue – 2) 2. Coma Consider only the Coma Wave Aberration Function ( ) ( ) ''''cos'';, 223 xyxhCrhChrW CoCoCo +== θθ ( ) ( ) ( ) ( )θθ 2cos2 ' '' cos21 ' '' ''3 ' '' ' ';',' ' ' ' 2222 +=+=+= ∂ =∆ r n Lh Cr n Lh Cyx n Lh C x hyxW n L x CoCoCo ( ) ( ) θ2sin ' '' ''2 ' '' ' ';',' ' ' ' 2 r n Lh Cyx n Lh C y hyxW n L y CoCo == ∂ =∆ 1 ' '' ' 2 ' '' ' 2 2 2 2 =             ∆ +             − ∆ r n Lh C y r n Lh C x CoCo ( )( ) ( ) ( )222 '2' rRyrRx CoCo =∆+−∆ ( ) 2 ' '' : r n Lh CrR CoCo =
77 OpticsSOLO Real Imaging Systems – Aberrations (continue – 2) 2. Coma We obtained 2 ' '' : MAXCoS r n Lh CC = ( )( ) ( ) ( )222 '2' rRyrRx CoCo =∆+−∆ ( ) MAXCoCo rrr n Lh CrR ≤≤= 0 ' '' : 2 Define: 1 2 3 4 P Image Plane O SC SC ST CC 3= Coma Blur Spot Shape Tangential Coma Sagittal Coma  30 'h 'x 'y
78 OpticsSOLO Real Imaging Systems – Aberrations (continue – 2) Graphical Explanation of Coma Blur
79 OpticsSOLO Real Imaging Systems – Aberrations (continue – 2) phical Explanation of Coma Blur (continue – 1)
81 OpticsSOLO Real Imaging Systems – Aberrations (continue – 3) 3. Astigmatism
82 OpticsSOLO Real Imaging Systems – Aberrations (continue – 3) 3. Astigmatism
83 OpticsSOLO Real Imaging Systems – Aberrations (continue – 3) 3. Astigmatism
84 OpticsSOLO Real Imaging Systems – Aberrations (continue – 4) Field Curvature
85 OpticsSOLO Real Imaging Systems – Aberrations (continue – 5) 5. Distortion
86 OpticsSOLO Real Imaging Systems – Aberrations (continue – 8) Seidel Aberrations (continue – 7) hin Lens Aberrations ( ) 2222234 'cos'cos'';, rhCrhCrhCrChrW FCAsCoSp +++= θθθ ven a thin lens formed by two faces with radiuses r1 and r2 h centers C1 and C2. PP0 is object, P”P”0 is the Gaussian ge formed by the first surface, ’0 is the image of virtual object ”0 of the second surface. ( ) ( ) ( ) ( )       ++ − + +−++ −− −= qpnq n n pnn n n fnn CSp 14 1 2 123 1132 1 22 3 3 ( )       − + ++= q n n pn sfn CCo 1 1 12 '4 1 2 ( )2 '2/1 sfCAs −= ( ) ( )2 '4/1 sfnnCFC +−= where: f s OA C11 r F” F ''f ''s 2 r 1=n n h "h D 0P P 0'P 0"P "P 'P 'h 's CR AS EnP ExP r ( )θ,rQ OC2 1=n ( ) [ ] [ ]0000 '', OPPQPPrW −=θ Coddington shape factor: Coddington position factor: ss ss p − + = ' ' 12 12 rr rr q − + = From: we find:
87 OpticsSOLO Real Imaging Systems – Aberrations (continue – 8) Seidel Aberrations (continue – 7) oddington Position Factor 2R 1R f 1C 2FO 1F 2 C 2n 1n s 's '2 sfs == 2R 1 R f 1 C 2F1F 2 C 2n 1 n s 's fss =∞= ', 2R 1R f 1 C 2F1F 2 C 2n 1 n s 's fss <> ',0 2R 1 R f 1 C 2F1F 2 C 2n 1 n s 's ∞== ', sfs 2R 1Rf 1 C2F1 F 2 C 2n 1 n s 's 0', << sfs CRCR 2 R 1 R f1C 2 F 1F 2 C2n 1 n s 's 0'0 <<> sfs 2 R 1 R f1 C 2F1F 2C2n 1n s 's fss =∞= ', 1=p 2 R1R f 1C 2F1F 2C 2n 1 n s 's ∞== ', sfs 1>p 2 R1R f 1C 2F1F 2C 2n 1 n s 's 0',0 ><< ssf 0=p 2 R1R f 1C 2FO 1F 2C 2n 1 n s 's '2 sfs == 1−=p1−<p ss ss p − + = ' ' ss ss p − + = ' ' ' 111 ssf += ' 2 11 2 s f s f p −=−=
88 OpticsSOLO Coddington Position Factor f f2f2− f− 0 Figure Object Location Image Location Image Properties Shape Factor Infinity Principal focus 'ss fs 2> fsf 2'<< fs 2= fs 2'= fsf 2<< fs 2'> 's 's s s fs = ∞='s s s 's fs < fs <' Real, inverted small p = -1 Real, inverted smaller -1 < p <0 Real, inverted same size p = 0 Real, inverted larger 0 < p <1 No image p = 1 Virtual, erect larger p>1 's 's 0<s fs <' p < -1 Imaginary, inverted small
89 OpticsSOLO Real Imaging Systems – Aberrations (continue – 8) Seidel Aberrations (continue – 7) oddington Shape Factor 1 02 1 −= < ∞= q R R 2 R 1 R 2 C 2 n 1 n Plano Convex 2 n 1 0,0 21 21 −< > << q RR RR 1 C 2 C 1 n 1 R 2 R Positive Meniscus 2 R 1 R f 1 C 2 F1 F 2 C 2 n 1 n 0 0,0 21 21 = = <> q RR RR Equi Convex 2 R 1 R 1C2n 1n Plano Convex 1 0 2 1 = ∞= > q R R 2 R1 R f 1 C 2 F 2 C 2 n 1 n 1 0,0 21 21 > < >> q RR RR Positive Meniscus 12 12 RR RR q − + = 2 R 1 R f 2F1 F 2C 2n 1 n 1C Negative Meniscus 1 0,0 21 21 −< > >> q RR RR 1 0, 21 −= >∞= q RR Plano Concave 2 R 1 R f 2 F1 F 2 C 2n 1 n 2 R1 R f 1 C 2F1 F 2C 2 n 1 n 0 0,0 21 21 = = >< q RR RR Equi Concave 2 R 1 R f 1F 2F 1C 2 n 1 n 1 ,0 21 = ∞=< q RR Plano Concave Negative Meniscus 1 0,0 21 21 > < << q RR RR 2 R 1 R f 2 F1 F 2C 2n 1 n 1C
90 REFLECTION & REFRACTIONSOLO http://freepages.genealogy.rootsweb.com/~coddingtons/15763.htm History of Reflection & Refraction Reverent Henry Coddington (1799 – 1845) English mathematician and cleric. He wrote an Elementary Treatise on Optics (1823, 1st Ed., 1825, 2nd Ed.). The book was displayed the interest on Geometrical Optics, but hinted to the acceptance of the Wave Theory. Coddington wrote “A System of Optics” in two parts: 1. “A Treatise of Reflection and Refraction of Light” (1829), containing a thorough investigation of reflection and refraction. 2. “A Treatise on Eye and on Optical Instruments” (1630), where he explained the theory of construction of various kinds of telescopes and microscopes. He recommended the ue of the grooved sphere lens, first described by David Brewster in 1820 and inuse today as the “Coddington lens”. Coddington introduced for lens: Coddington Shape Factor: Coddington Position Factor: 12 12 rr rr q − + = ss ss p − + = ' ' Coddington Lens http://www.eyeantiques.com/MicroscopesAndTelescopes/Coddington%20microscope_thick_wood.htm
91 OpticsSOLO Real Imaging Systems – Aberrations (continue – 8) Seidel Aberrations (continue – 7) hin Lens Spherical Aberrations ( ) 4 rCrW SpSA = ven a thin lens and object O on the ical Axis (OA). A paraxial ray will cross OA at point I, at a distance s’p from lens. A general ray, that reaches the lens distance r from OA, will cross OA at nt E, at a distance s’r. ( ) ( ) ( ) ( )       ++ − + +−++ −− −= qpnq n n pnn n n fnn CSp 14 1 2 123 1132 1 22 3 3 where: Define: 2 R 1 R 1 C IO 2C Paraxial focal plane2 n 1n s ps' E rs' Long. SA Lat. SA φ Paraxial Ray General Ray 'φ r rp ssSALongAberrationSphericalalLongitudin ''. −== ( ) rrp srssSALatAberrationSphericalLateral '/''. −== We have:
92 OpticsSOLO Real Imaging Systems – Aberrations (continue – 1) 12 12 RR RR qK − + == ( ) ( ) ( ) ( ) ( )       ++ − + +−++ −− −= qpnq n n pnn n n fnn r rWSp 14 1 2 123 1132 22 3 3 4 Thin Lens Spherical Aberrations (continue – 1)
93 OpticsSOLO Real Imaging Systems – Aberrations (continue – 1) Thin Lens Spherical Aberrations (continue – 3) 2 R 1 R 1 C IO 2C Paraxial focal plane2 n 1 n s ps' E rs' Long. SA Lat. SA φ Paraxial Ray General Ray 'φ r 12 12 RR RR q − + = F.A. Jenkins & H.E. White, “Fundamentals of Optics”, 4th Ed., McGraw-Hill, 1976, pg. 157 Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm In Figure we can see a comparison of the Seidel Third Order Theory with the ray tracing.
94 OpticsSOLO Real Imaging Systems – Aberrations (continue – 1) We can see that the Thin Lens Spherical Aberration WSp is a parabolic function of the Coddington Shape Factor q, with the vertex at (qmin,WSp min) ( ) ( ) ( ) ( )       ++ − + +−++ −− −= qpnq n n pnn n n fnn r WSp 14 1 2 123 1132 22 3 3 4 Thin Lens Spherical Aberrations (continue -2) The minimum Spherical Aberration for a given Coddington Position Factor p is obtained by: ( ) ( ) 014 1 2 2 132 3 4 =    ++ − + − −= ∂ ∂ pnq n n fnn r q W p Sp 1 1 2 2 min + − −= n n pq         + −      − −= 2 2 3 4 min 2132 p n n n n f r WSp The minimum Spherical Aberration is zero for ( ) ( ) 1 1 2 2 2 > − + = n nn p
95 OpticsSOLO Real Imaging Systems – Aberrations (continue – 1) In order to obtain the radii of the lens for a given focal length f and given Shape Factor and Position Factor we can perform the following: Thin Lens Spherical Aberrations (continue – 3) Those relations were given by Coddington. ' 2 11 2 s f s f p −=−= p f s p f s − = + = 1 2 '& 1 2 ( ) fRR n ss 111 1 ' 11 21 =      −−=+ ( ) ( ) 1 12 & 1 12 21 − − = + − = q nf R q nf R 12 12 RR RR q − + = 12 1 12 2 2 1& 2 1 RR R q RR R q − =− − =+ ( ) ( )12 21 1 RRn RR f −− = 2 R 1 R 1 C IO 2 C Paraxial focal plane2 n 1 n s ps' E rs' Long. SA Lat. SA φ Paraxial Ray General Ray 'φ r
96 OpticsSOLO Real Imaging Systems – Aberrations (continue – 8) Seidel Aberrations (continue – 7) hin Lens Coma ( ) ( ) ( ) ( )     − + ++ + = +== q n n pn sfn xyxh xyxhCrhChrW CoCoCo 1 1 12 '4 '''' ''''cos'';, 2 22 223 θθ or thin lens the coma factor is given by: where:we find: ( ) 2 22 2 1 1 12 4 ''' : MAXMAXCoS rq n n pn fn h r n sh CC     − + ++== 1 2 3 4 P Image Plane O SC SC ST CC 3= Coma Blur Spot Shape Tangential Coma Sagittal Coma  30 'h 'x 'y ( )( ) ( ) ( )222 '2' rRyrRx CoCo =∆+−∆ ( ) MAXCoCo rrr n sh CrR ≤≤= 0 '' : 2 Define: ( ) ( ) ( ) ( )θθ 2cos2 '' cos21 '' ''3 '' ' ';','' ' 2222 +=+=+= ∂ =∆ r n sh Cr n sh Cyx n sh C x hyxW n s x CoCoCo ( ) ( ) θ2sin '' ''2 '' ' ';','' ' 2 r n sh Cyx n sh C y hyxW n s y CoCo == ∂ =∆
97 OpticsSOLO Real Imaging Systems – Aberrations (continue – 8) Seidel Aberrations (continue – 7) hin Lens F.A. Jenkins & H.E. White, “Fundamentals of Optics”, 4th Ed., McGraw-Hill, 1976, pg. 165 Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm, y = 2 cm ( ) 2 22 1 1 12 4 ' : MAXS rq n n pn fn h C       − + ++= oma is linear in q ( ) ( ) ( ) p n nn qCS 1 112 0 + −+ −=⇐= n Figure 800.00 =⇐= qCS The Spherical Aberration is arabolic in q ( ) ( ) ( ) ( )       ++ − + +−++ −− −= qpnq n n pnn n n fnn CSp 14 1 2 123 1132 1 22 3 3 1 1 2 2 min + − −= n n pq         + −      − −= 2 2 3min 2132 1 p n n n n f CSp In Figure 714.0min =q
98 OpticsSOLO Real Imaging Systems – Aberrations (continue – 5)
99 OpticsSOLO Real Imaging Systems – Aberrations (continue – 5)
100 SOLO Optics Chromatic Aberration Chromatic Aberrations arise in Polychromatic IR Systems because the material index n is actually a function of frequency. Rays at different frequencies will traverse an optical system along different paths.
101 SOLO Optics Chromatic Aberration
102 SOLO Optics Chester Moor Hall (1704 – 1771) designed in secrecy the achromatic lens. He experienced with different kinds of glass until he found in 1729 a combination of convex component formed from crown glass with a concave component formed from flint glass, but he didn’t request for a patent. http://microscopy.fsu.edu/optics/timeline/people/dollond.html In 1750 John Dollond learned from George Bass on Hall achromatic lens and designed his own lenses, build some telescopes and urged by his son Peter (1739 – 1820) applied for a patent. Born & Wolf,”Principles of Optics”, 5th Ed.,p.176 Chromatic Aberration In 1733 he built several telescopes with apertures of 2.5” and 20”. To keep secrecy Hall ordered the two components from different opticians in London, but they subcontract the same glass grinder named George Bass, who, on finding that both Lenses were from the same customer and had one radius in common, placed them in contact and saw that the image is free of color. The other London opticians objected and took the case to court, bringing Moore-Hall as a witness. The court agree that Moore- Hall was the inventor, but the judge Lord Camden, ruled in favor of Dollond saying:”It is not the person who locked up his invention in the scritoire that ought to profit by a patent for such invention, but he who brought it forth for the benefit of the public”
103 SOLO Optics Chromatic Aberration Every piece of glass will separate white light into a spectrum given the appropriate angle. This is called dispersion. Some types of glasses such as flint glasses have a high level of dispersion and are great for making prisms. Crown glass produces less dispersion for light entering the same angle as flint, and is much more suited for lenses. Chromatic aberration occurs when the shorter wavelength light (blue) is bent more than the longer wavelength (red). So a lens that suffers from chromatic aberration will have a different focal length for each color To make an achromat, two lenses are put together to work as a group called a doublet. A positive (convex) lens made of high quality crown glass is combined with a weaker negative (concave) lens that is made of flint glass. The result is that the positive lens controls the focal length of the doublet, while the negative lens is the aberration control. The negative lens is of much weaker strength than the positive, but has higher dispersion. This brings the blue and the red light back together (B). However, the green light remains uncorrected (A), producing a secondary spectrum consisting of the green and blue-red rays. The distance between the green focal point and the blue-red focal point indicates the quality of the achromat. Typically, most achromats yield about 75 to 80 % of their numerical aperture with practical resolution
104 SOLO Optics Chromatic Aberration In addition, to the correction for the chromatic aberration the achromat is corrected for spherical aberration, but just for green light. The Illustration shows how the green light is corrected to a single focal length (A), while the blue-red (purple) is still uncorrected with respect to spherical aberration. This illustrates the fact that spherical aberration has to be corrected for each color, called spherochromatism. The effect of the blue and red spherochromatism failure is minimized by the fact that human perception of the blue and red color is very weak with respect to green, especially in dim light. So the color halos will be hardly noticeable. However, in photomicroscopy, the film is much more sensitive to blue light, which would produce a fuzzy image. So achromats that are used for photography will have a green filter placed in the optical path.
105 SOLO Optics Chromatic Aberration As the optician's understanding of optical aberrations improved they were able to engineer achromats with shorter and shorter secondary spectrums. They were able to do this by using special types of glass call flourite. If the two spectra are brought very close together the lens is said to be a semi-apochromat or flour. However, to finally get the two spectra to merge, a third optical element is needed. The resulting triplet is called an apochromat. These lenses are at the pinnacle of the optical family, and their quality and price reflect that. The apochromat lenses are corrected for chromatic aberration in all three colors of light and corrected for spherical aberration in red and blue. Unlike the achromat the green light has the least amount of correction, though it is still very good. The beauty of the apochromat is that virtually the entire numerical aperture is corrected, resulting in a resolution that achieves what is theoretically possible as predicted by Abbe equation.
106 SOLO Optics Chromatic Aberration With two lenses (n1, f1), (n2,f2) separated by a distance d we found 2121 111 ff d fff −+= Let use ( ) ( ) 222111 1/1&1/1 ρρ −=−= nfnf We have ( ) ( ) ( ) ( ) 22112211 1111 1 ρρρρ −−−−+−= nndnn f nF – blue index produced by hydrogen wavelength 486.1 nm. nC – red index produced by hydrogen wavelength 656.3 nm. nd – yellow index produced by helium wavelength 587.6 nm. Assume that for two colors red and blue we have fR = fB ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 22112211 22112211 1111 1111 1 ρρρρ ρρρρ −−−−+−= −−−−+−= FFFF CCCC nndnn nndnn f
107 SOLO Optics Chromatic Aberration Let analyze the case d = 0 (the two lenses are in contact) nd – yellow index produced by helium wavelength 587.6 nm. We have ( ) ( ) ( ) ( ) 22112211 1111 1 ρρρρ −+−=−+−= FFCC nnnn f ( ) ( ) ( ) ( )1 1 1 1 1 2 1 2 2 1 − − −= − − −= F F C C n n n n ρ ρ ( ) ( )CF CF nn nn 11 22 2 1 − − −= ρ ρ For the yellow light (roughly the midway between the blue and red extremes) the compound lens will have the focus fY: ( ) ( ) YY f d f d Y nn f 21 /1 22 /1 11 11 1 ρρ −+−= ( ) ( ) Y Y d d f f n n 1 2 1 2 2 1 1 1 − − = ρ ρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1/ 1/ 1 1 111 222 2 1 11 22 1 2 −− −− −= − − − − −= dCF dCF d d CF CF Y Y nnn nnn n n nn nn f f
108 SOLO Optics Chromatic Aberration ( ) ( ) ( ) ( )1/ 1/ 111 222 1 2 −− −− −= dCF dCF Y Y nnn nnn f f The quantities are called Dispersive Powers of the two materials forming the lenses. ( ) ( ) ( ) ( )1 & 1 2 22 1 11 − − − − d CF d CF n nn n nn Their inverses are called V-numbers or Abbe numbers. ( ) ( ) ( ) ( )CF d CF d nn n V nn n V 22 2 2 11 1 1 1 & 1 − − = − − =
109 OpticsSOLO To define glass we need to know more than one index of refraction. In general we choose the indexes of refraction of three colors: nF – blue index produced by hydrogen wavelength 486.1 nm. nC – red index produced by hydrogen wavelength 656.3 nm. nd – yellow index produced by helium wavelength 587.6 nm. Define: nF – nC - mean dispersion CF d nn n v − − = 1 - Abbe’s Number or v value or V-number Crowns: glasses of low dispersion (nF – nC small and V-number above 55) Flints: glasses of high dispersion (nF – nC high and V-number bellow 50) Fraunhofer line color Wavelength (nm) Spectacle Crown C - 1 Extra Dense Flint EDF - 3 F d C Blue Yellow Red 486.1 587.6 656.3 1.5293 1.5230 1.5204 1.7378 1.7200 1.7130V - number 58.8 29.0
110 OpticsSOLO Refractive indices and Abbe’s numbers of various glass materials
111 OpticsSOLO Camera Lenses Hecht, “Optics” Addison Wesley, 4th Ed., 2002, pp.218
112 OpticsSOLO Camera Lenses Born & Wolfe, “Principle of Optics”, Pergamon Press, 5th Ed., pp.236-237
113 SOLO References Lens Design 1. Kingslake, R., “Lens Design Fundamentals”, Academic Press, N.Y., 1978 6. Geary, J. M., “Introduction to Lens Design with Practical ZEMAX Examples”, Willmann-Bell, Inc., 2002 5. Laikin, M., “Lens Design”, Marcel Dekker, N.Y., 1991 2. Malacara, D., Ed., “Optical Shop Testing”, John Wiley & Sons, N.Y., 1978 7. Kidger, M. J., “Fundamental Optical Design”, SPIE Press., 2002 3. Kingslake, R., “Optical System Design”, Academic Press, N.Y., 1983 4. O’Shea, D.,C., “Elements of Modern Optical Design”, John Wiley & Sons, N.Y., 1985
114 SOLO References OPTICS 1. Waldman, G., Wootton, J., “Electro-Optical Systems Performance Modeling”, Artech House, Boston, London, 1993 2. Wolfe, W.L., Zissis, G.J., “The Infrared Handbook”, IRIA Center, Environmental Research Institute of Michigan, Office of Naval Research, 1978 3. “The Infrared & Electro-Optical Systems Handbook”, Vol. 1-7 4. Spiro, I.J., Schlessinger, M., “The Infrared Technology Fundamentals”, Marcel Dekker, Inc., 1989
115 SOLO References [1] M. Born, E. Wolf, “Principle of Optics – Electromagnetic Theory of Propagation, Interference and Diffraction of Light”, 6th Ed., Pergamon Press, 1980, [2] C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996, OPTICS
116 SOLO References Foundation of Geometrical Optics [3] E.Hecht, A. Zajac, “Optics ”, 3th Ed., Addison Wesley Publishing Company, 1997, [4] M.V. Klein, T.E. Furtak, “Optics ”, 2nd Ed., John Wiley & Sons, 1986
117 OPTICSSOLO References Optics Polarization A. Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons, 1984 M. Born, E. Wolf, “Principles of Optics”, Pergamon Press,6th Ed., 1980 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 1979, Ch.8 C.C. Davis, “Lasers and Electro-Optics”, Cambridge University Press, 1996 G.R. Fowles, “Introduction to Modern Optics”,2nd Ed., Dover, 1975, Ch.2 M.V.Klein, T.E. Furtak, “Optics”, 2nd Ed., John Wiley & Sons, 1986 http://en.wikipedia.org/wiki/Polarization W.C.Elmore, M.A. Heald, “Physics of Waves”, Dover Publications, 1969 E. Collett, “Polarization Light in Fiber Optics”, PolaWave Group, 2003 W. Swindell, Ed., “Polarization Light”, Benchmark Papers in Optics, V.1, Dowden, Hutchinson & Ross, Inc., 1975 http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi)
January 4, 2015 118 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

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Lens

  • 1.
  • 2.
    2 Table of Content(continue( SOLO OPTICS Plane-Parallel Plate The Three Laws of Geometrical Optics Fermat’s Principle (1657) Prisms Lens Definitions Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law Derivation of Lens Makers’ Formula First Order, Paraxial or Gaussian Optics Ray Tracing Matrix Formulation References
  • 3.
    3 SOLO The Three Lawsof Geometrical Optics 1. Law of Rectilinear Propagation In an uniform homogeneous medium the propagation of an optical disturbance is in straight lines. . Law of Reflection An optical disturbance reflected by a surface has the property that the incident ray, the surface normal, and the reflected ray all lie in a plane, and the angle between the incident ray and the surface normal is equal to the angle between the reflected ray and the surface normal: . Law of Refraction An optical disturbance moving from a medium of refractive index n1 into a medium of refractive index n2 will have its incident ray, the surface normal between the media , and the reflected ray in a plane, and the relationship between angle between the incident ray and the surface normal θi and the angle between the reflected ray and the surface normal θt given by Snell’s Law: ti nn θθ sinsin 21 ⋅=⋅ ri θθ = “The branch of optics that addresses the limiting case λ0 → 0, is known as Geometrical Optics, since in this approximation the optical laws may be formulated in the language of geometry.” Max Born & Emil Wolf, “Principles of Optics”, 6th Ed., Ch. 3 Foundation of Geometrical Optics
  • 4.
    4 SOLO Foundation ofGeometrical Optics Fermat’s Principle (1657) The Principle of Fermat (principle of the shortest optical path( asserts that the optical length of an actual ray between any two points is shorter than the optical ray of any other curve that joints these two points and which is in a certai neighborhood of it. An other formulation of the Fermat’s Principle requires only Stationarity (instead of minimal length). ∫ 2 1 P P dsn An other form of the Fermat’s Principle is: Princple of Least Time The path following by a ray in going from one point in space to another is the path that makes the time of transit of the associated wave stationary (usually a minimum). The idea that the light travels in the shortest path was first put forward by Hero of Alexandria in his work “Catoptrics”, cc 100B.C.-150 A.C. Hero showed by a geometrical method that the actual path taken by a ray of light reflected from plane mirror is shorter than any other reflected path that might be drawn between the source and point of observation.
  • 5.
    5 SOLO 1. The opticalpath is reflected at the boundary between two regions ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   In this case we have and21 nn = ( ) ( ) ( ) 0ˆˆ 21 21 =⋅−=⋅        − rdssrd sd rd sd rd rayray   We can write the previous equation as: i.e. is normal to , i.e. to the boundary where the reflection occurs. 21 ˆˆ ss − rd  ( ) 0ˆˆˆ 2121 =−×− ssn REFLECTION & REFRACTION Reflection Laws Development Using Fermat Principle This is equivalent with: ri θθ = Incident ray and Reflected ray are in the same plane normal to the boundary.&
  • 6.
    6 SOLO 2. The opticalpath passes between two regions with different refractive indexes n1 to n2. (continue – 1) ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   where is on the boundary between the two regions andrd  ( ) ( ) sd rd s sd rd s rayray 2 :ˆ, 1 :ˆ 21  == Therefore is normal to . 2211 ˆˆ snsn − rd  Since can be in any direction on the boundary between the two regions is parallel to the unit vector normal to the boundary surface, and we have rd  2211 ˆˆ snsn − 21 ˆ −n ( ) 0ˆˆˆ 221121 =−×− snsnn We recovered the Snell’s Law from Geometrical Optics REFLECTION & REFRACTION Refraction Laws Development Using Fermat Principle ti nn θθ sinsin 21 = Incident ray and Refracted ray are in the same plane normal to the boundary. &
  • 7.
    7 SOLO Plane-Parallel Plate A singleray traverses a glass plate with parallel surfaces and emerges parallel to its original direction but with a lateral displacement d. Optics ( ) ( )irriri lld φφφφφφ cossincossinsin −=−= r t l φcos =       −= r i ri td φ φ φφ cos cos sinsin ir nn φφ sinsin 0 =Snell’s Law       −= n n td r i i 0 cos cos 1sin φ φ φ For small anglesi φ       −≈ n n td i 0 1φ
  • 8.
    8 SOLO Plane-Parallel Plate (continue– 1( Two rays traverse a glass plate with parallel surfaces and emerge parallel to their original direction but with a lateral displacement l. Optics ( ) ( )irriri lld φφφφφφ cossincossinsin −=−= r t l φcos =       −= r i ri td φ φ φφ cos cos sinsin ir nn φφ sinsin 0 =Snell’s Law       −= n n td r i i 0 cos cos 1sin φ φ φ       −== r i i n n t d l φ φ φ cos cos 1 sin 0 For small anglesi φ       −≈ n n tl 0 1
  • 9.
    9 SOLO Prisms Type of prisms: Aprism is an optical device that refract, reflect or disperse light into its spectral components. They are also used to polarize light by prisms from birefringent media. Optics - Prisms 2. Reflective 1. Dispersive 3. Polarizing
  • 10.
    10 OpticsSOLO Dispersive Prisms ( )( )2211 itti θθθθδ −+−= 21 it θθα += αθθδ −+= 21 ti 202 sinsin ti nn θθ =Snell’s Law 10 ≈n ( ) ( )[ ]1 1 2 1 2 sinsinsinsin tit nn θαθθ −== −− ( )[ ] ( )[ ]11 21 11 1 2 sincossin1sinsinsincoscossinsin ttttt nn θαθαθαθαθ −−=−= −− Snell’s Law 110 sinsin ti nn θθ = 11 sin 1 sin it n θθ = ( )[ ]1 2/1 1 221 2 sincossinsinsin iit n θαθαθ −−= − ( )[ ] αθαθαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n The ray deviation angle is 10 ≈n
  • 11.
    11 OpticsSOLO Prisms ( )[ ]αθαθαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n
  • 12.
    12 OpticsSOLO Prisms ( )[ ]αθαθαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n αθθδ −+= 21 ti Let find the angle θi1 for which the deviation angle δ is minimal; i.e. δm. This happens when  01 0 11 2 1 =−+= ii t i d d d d d d θ α θ θ θ δ Taking the differentials of Snell’s Law equations 22 sinsin ti n θθ = 11 sinsin ti n θθ = 2222 coscos iitt dnd θθθθ = 1111 coscos ttii dnd θθθθ = Dividing the equations  1 2 1 2 1 1 2 1 2 1 cos cos cos cos −− = i t i t t i t i d d d d θ θ θ θ θ θ θ θ 2 22 1 22 2 2 2 2 1 2 2 2 1 2 2 2 1 2 sin sin /sin1 /sin1 sin1 sin1 sin1 sin1 t i t i i t t i n n n n θ θ θ θ θ θ θ θ − − = − − = − − = − − 1 1 2 −= i t d d θ θ 21 it θθα += 1 2 1 −= i t d d θ θ 2 2 1 2 2 2 1 2 cos cos cos cos i t t i θ θ θ θ = 21 ti θθ = 1≠n
  • 13.
    13 OpticsSOLO Prisms ( )[ ]αθαθαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n We found that if the angle θi1 = θt2 the deviation angle δ is minimal; i.e. δm. Using the Snell’s Law equations 22 sinsin ti n θθ = 11 sinsin ti n θθ = 21 ti θθ = 21 it θθ = This means that the ray for which the deviation angle δ is minimum passes through the prism parallel to it’s base. Find the angle θi1 for which the deviation angle δ is minimal; i.e. δm (continue – 1(.
  • 14.
    14 OpticsSOLO Prisms ( )[ ]αθαθαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n Using the Snell’s Law 11 sinsin ti n θθ = 21 it θθ = This equation is used for determining the refractive index of transparent substances. 21 it θθα += αθθδ −+= 21 ti 21 ti θθ = mδδ = 2/1 αθ =t αθδ −= 12 im ( ) 2/1 αδθ += mi ( )[ ] 2/sin 2/sin α αδ + = m n Find the angle θi1 for which the deviation angle δ is minimal; i.e. δm (continue – 2(.
  • 15.
    15 OpticsSOLO Prisms The refractive indexof transparent substances varies with the wavelength λ. ( )[ ]{ } αθαθλαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n
  • 16.
    16 OpticsSOLO http://physics.nad.ru/Physics/English/index.htm Prisms Color λ0 (nm(υ [THz] Red Orange Yellow Green Blue Violet 780 - 622 622 - 597 597 - 577 577 - 492 492 - 455 455 - 390 384 – 482 482 – 503 503 – 520 520 – 610 610 – 659 659 - 769 1 nm = 10-9 m, 1 THz = 1012 Hz ( )[ ]{ } αθαθλαθδ −−−+= − 1 2/1 1 221 1 sincossinsinsin iii n In 1672 Newton wrote “A New Theory about Light and Colors” in which he said that the white light consisted of a mixture of various colors and the diffraction was color dependent. Isaac Newton 1542 - 1727
  • 17.
    17 SOLO Dispersing Prisms Pellin-Broca Prism AbbePrism Ernst Karl Abbe 1840-1905 At Pellin-Broca Prism an incident ray of wavelength λ passes the prism at a dispersing angle of 90°. Because the dispersing angle is a function of wavelength the ray at other wavelengths exit at different angles. By rotating the prism around an axis normal to the page different rays will exit at the 90°. At Abbe Prism the dispersing angle is 60°. Optics - Prisms
  • 18.
    18 SOLO Dispersing Prisms (continue– 1( Amici Prism Optics - Prisms
  • 19.
    19 SOLO Reflecting Prisms BED∠−= 180δ  360=∠+∠+∠+ ABEBEDADEα 1 90 i ABE θ+=∠  2 90 t ADE θ+=∠   3609090 12 =++∠+++ it BED θθα 12180 itBED θθα −−−=∠  αθθδ ++=∠−= 21180 tiBED The bottom of the prism is a reflecting mirror Since the ray BC is reflected to CD DCGBCF ∠=∠ Also CGDBFC ∠=∠ CDGFBC ∠=∠ FBCt ∠−=  901θ CDGi ∠−=  902θ 21 it θθ = 202 sinsin ti nn θθ =Snell’s Law Snell’s Law 110 sinsin ti nn θθ = 21 ti θθ = αθδ += 1 2 i CDGFBC ∆∆ ~ Optics - Prisms
  • 20.
    20 SOLO Reflecting Prisms Porro PrismPorro-Abbe Prism Schmidt-Pechan Prism Penta Prism Optics - Prisms Roof Penta Prism
  • 21.
    21 SOLO Reflecting Prisms Abbe-Koenig Prism DovePrism Amici-roof Prism Optics - Prisms
  • 22.
    22 SOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Polarization can beachieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Nicol Prism The Nicol Prism is made up from two prisms of calcite cemented with Canada balsam. The ordinary ray can be made to totally reflect off the prism boundary, leving only the extraordinary ray.. Polarizing Prisms Optics - Prisms
  • 23.
    23 SOLO Polarizing Prisms A Glan-Foucaultprism deflects polarized light transmitting the s-polarized component. The optical axis of the prism material is perpendicular to the plane of the diagram. A Glan-Taylor prism reflects polarized light at an internal air-gap, transmitting only the p-polarized component. The optical axes are vertical in the plane of the diagram. A Glan-Thompson prism deflects the p-polarized ordinary ray whilst transmitting the s-polarized extraordinary ray. The two halves of the prism are joined with Optical cement, and the crystal axis are perpendicular to the plane of the diagram. Optics - Prisms
  • 24.
    24 OpticsSOLO Lens Definitions Optical Axis:the common axis of symmetry of an optical system; a line that connects all centers of curvature of the optical surfaces. Lateral Magnification: the ratio between the size of an image measured perpendicular to the optical axis and the size of the conjugate object. Longitudinal Magnification: the ratio between the lengthof an image measured along the optical axis and the length of the conjugate object. First (Front( Focal Point: the point on the optical axis on the left of the optical system (FFP( to which parallel rays on it’s right converge. Second (Back( Focal Point: the point on the optical axis on the right of the optical system (BFP( to which parallel rays on it’s left converge.
  • 25.
    25 OpticsSOLO Definitions (continue –2( Aperture Stop (AS(: the physical diameter which limits the size of the cone of radiation which the optical system will accept from an axial point on the object. Field Stop (FS(: the physical diameter which limits the angular field of view of an optical system. The Field Stop limit the size of the object that can be seen by the optical system in order to control the quality of the image. A.S. F.S. IΣ Aperture and Field Stops Image plane Hecht "Optics"
  • 26.
    26 OpticsSOLO Definitions (continue –2( Entrance Pupil: the image of the Aperture Stop as seen from the object through the (EnP( elements preceding the Aperture Stop. Exit Pupil: the image of the Aperture Stop as seen from an axial point on the (ExP( image plane. Entrance pupil Exit pupil A.S. IΣ xpE npE Chief Ray Entrance and Exit pupils Image plane Marginal Ray Hecht "Optics" Entrance pupilExit pupil A.S. I Σ xpE npE Chief Ray Image plane A front Aperture Stop Hecht "Optics" Chief Ray: an object Ray passing through the center of the aperture stop and (CR( appearing to pass through the centers of entrance and exit pupils. Marginal Ray: an object Ray passing through the edge of the aperture stop. (MR(
  • 27.
    27 OpticsSOLO Definitions (continue –2( Entrance pupil Exit pupil A.S. IΣ Chief Ray Marginal Ray Exp Enp Image plane Hecht "Optics"
  • 28.
    28 OpticsSOLO Definitions (continue –1( Principal Planes: the two planes defined by the intersection of the parallel incident rays entering an optical system with the rays converging to the focal points after passing through the optical system. Principal Points: the intersection of the principal planes with the optical axes. Nodal Points: two axial points of an optical system, so located that an oblique ray directed toward the first appears to emerge from the second, parallel to the original direction. For systems in air, the Nodal Points coincide with the Principal Points. Cardinal Points: the Focal Points, Principal Points and the Nodal Points.
  • 29.
    29 OpticsSOLO Definitions (continue –3( Relative Aperture (f# (: the ratio between the effective focal length (EFL( f to Entrance Pupil diameter D. Numerical Aperture (NA(: sine of the half cone angle u of the image forming ray bundles multiplied by the final index n of the optical system. If the object is at infinity and assuming n = 1 (air(: Dff /:# = unNA sin: ⋅= # 1 2 1 2 1 sin ff D uNA =      ==
  • 30.
    30 OpticsSOLO Perfect Imaging System •All rays originating at one object point reconverge to one image point after passing through the optical system. • All of the objects points lying on one plane normal to the optical axis are imaging onto one plane normal to the axis. • The image is geometrically similar to the object.
  • 31.
    31 OpticsSOLO Lens Convention of Signs 1.All Figures are drawn with the light traveling from left to right. 2. All object distances are considered positive when they are measured to the left of the vertex and negative when they are measured to the right. 3. All image distances are considered positive when they are measured to the right of the vertex and negative when they are measured to the left. 4. Both focal length are positive for a converging system and negative for a diverging system. 5. Object and Image dimensions are positive when measured upward from the axis and negative when measured downward. 6. All convex surfaces are taken as having a positive radius, and all concave surfaces are taken as having a negative radius.
  • 32.
    32 OpticsSOLO Derivation of GaussianFormula for a Single Spherical Surface Lens Using Fermat’s Principle Karl Friederich Gauss 1777-1855 The optical path connecting points M, T, M’ is ''lnlnpathOptical ⋅+⋅= Applying cosine theorem in triangles MTC and M’TC we obtain: ( ) ( )[ ] 2/122 cos2 βRsRRsRl +−++= ( ) ( )[ ] 2/122 cos'2'' βRsRRsRl −+−+= ( ) ( )[ ] ( ) ( )[ ] 2/122 2/122 cos'2''cos2 ββ RsRRsRnRsRRsRnpathOptical −+−+⋅++−++⋅= Therefore According to Fermat’s Principle when the point T moves on the spherical surface we must have ( ) 0= βd pathOpticald ( ) ( ) ( ) 0 ' sin''sin = −⋅ − +⋅ = l RsRn l RsRn d pathOpticald ββ β from which we obtain       ⋅ − ⋅ =+ l sn l sn Rl n l n ' ''1 ' ' For small α and β we have ''& slsl ≈≈ and we obtain R nn s n s n − =+ ' ' ' Gaussian Formula for a Single Spherical Surface
  • 33.
    33 OpticsSOLO Derivation of GaussianFormula for a Single Spherical Surface Lens Using Snell’s Law Apply Snell’s Law: 'sin'sin φφ nn = If the incident and refracted rays MT and TM’ are paraxial the angles and are small and we can write Snell’s Law: φ 'φ From the Figure βαφ += γβφ −=' ''φφ nn = ( ) ( ) ( ) βγαγββα nnnnnn −=+⇒−=+ ''' For paraxial rays α, β, γ are small angles, therefore '/// shrhsh ≈≈≈ γβα ( ) r h nn s h n s h n −=+ ' ' ' or ( ) r nn s n s n − =+ ' ' ' Gaussian Formula for a Single Spherical Surface Karl Friederich Gauss 1777-1855 Willebrord van Roijen Snell 1580-1626 ( ) φ φφ φφ φ ≈+++=   O` 53 !5!3 sin
  • 34.
    34 OpticsSOLO Derivation of GaussianFormula for a Single Spherical Surface Lens Using Snell’s Law for s → ∞ the incoming rays are parallel to optical axis and they will refract passing trough a common point called the focus F’. ( ) r nn s n s n − =+ ' ' ' ( ) r nn f nn − =+ ∞ ' ' ' r nn n f − = ' ' ' for s’ → ∞ the refracting rays are parallel to optical axis and therefore the incoming rays passes trough a common point called the focus F. ( ) r nnn f n − = ∞ + '' r nn n f − = ' '' n n f f =
  • 35.
    35 OpticsSOLO Derivation of LensMakers’ Formula We have a lens made of two spherical surfaces of radiuses r1 and r2 and a refractive index n’, separating two media having refraction indices n a and n”. Ray MT1 is refracted by the first spherical surface (if no second surface exists) to T1M’. ( ) 111 ' ' ' r nn s n s n − =+ 11111 ''& sMAsTA == Ray T1T2 is refracted by the second spherical surface to T2M”. 2222 ""&'' sMAsMA == ( ) 222 '" " " ' ' r nn s n s n − =+ Assuming negligible lens thickness we have , and since M’ is a virtual object for the second surface (negative sign) we have 21 '' ss ≈ 21 '' ss −≈ ( ) 221 '" " " ' ' r nn s n s n − =+−
  • 36.
    36 OpticsSOLO Derivation of LensMakers’ Formula (continue – 1) ( ) 111 ' ' ' r nn s n s n − =+ Add those equations ( ) 221 '" " " ' ' r nn s n s n − =+− ( ) ( ) 2121 '"' " " r nn r nn s n s n − + − =+ The focal lengths are defined by tacking s1 → ∞ to obtain f” and s”2 → ∞ to obtain f ( ) ( ) f n r nn r nn f n = − + − = 212 '"' " " Let define s1 as s and s”2 as s” to obtain ( ) ( ) 21 '"' " " r nn r nn s n s n − + − =+ ( ) ( ) f n r nn r nn f n = − + − = 21 '"' " "
  • 37.
    37 OpticsSOLO Derivation of LensMakers’ Formula (continue – 2) If the media on both sides of the lens is the same n = n”.       −      −=+ 21 11 1 ' " 11 rrn n ss       −      −== 21 11 1 '1 " 1 rrn n ff Therefore " 11 " 11 ffss ==+ Lens Makers’ Formula
  • 38.
    38 OpticsSOLO First Order, Paraxialor Gaussian Optics In 1841 Gauss gave an exposition in “Dioptrische Untersuchungen” for thin lenses, for the rays arriving at shallow angles with respect to Optical axis (paraxial). Karl Friederich Gauss 1777-1855 Derivation of Lens Formula From the similarity of the triangles and using the convention: ( ) '' ' '~' f y s yy TAFTSQ = −+ ⇒∆∆ Lens Formula in Gaussian form ( ) ( ) f y s yy FASQTS '' ~ − = −+ ⇒∆∆ ( ) 0' >− y Sum of the equations: ( ) ( ) ( ) ' ' ' '' f y f y s yy s yy + − = −+ + −+ since f = f’ fss 1 ' 11 =+ ( ) φ φφ φφ φ ≈+++=   O` 53 !5!3 sin
  • 39.
    39 OpticsSOLO First Order, Paraxialor Gaussian Optics (continue – 1) Gauss explanation can be extended to the first order approximation to any optical system. Karl Friederich Gauss 1777-1855 Lens Formula in Gaussian form fss 1 ' 11 =+ s – object distance (from the first principal point to the object). s’ – image distance (from the second principal point to the image). f – EFL (distance between a focal point to the closest principal plane). 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis y
  • 40.
    40 OpticsSOLO Derivation of LensFormula (continue) From the similarity of the triangles and using the convention: ( ) f y x y FASQMF ' ~ − =⇒∆∆ Lens Formula in Newton’s form ( ) f y x y QMFTAF = − ⇒∆∆ ' ' '''~' ( ) 0' >− y Multiplication of the equations: ( ) ( ) 2 ' ' ' f yy xx yy −⋅ = ⋅ −⋅ or 2 ' fxx =⋅ Isaac Newton 1643-1727 First Order, Paraxial or Gaussian Optics (continue – 2) Published by Newton in “Opticks” 1710 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis y
  • 41.
    41 OpticsSOLO Derivation of LensFormula (continue) First Order, Paraxial or Gaussian Optics (continue – 3) Lateral or Transverse Magnification f x x f s s h h mT ''' −=−=−== Quantity (+) sign (-) sign s real object virtual object s’ real image virtual image f converging lens diverging lens h erect object inverted object h’ erect image inverted image mT erect image inverted image 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis y
  • 42.
    42 OpticsSOLO Derivation of LensFormula (Summary) If the media on both sides of the lens is the same n = n”.       −      −=+ 21 11 1 ' " 11 rrn n ss       −      −== 21 11 1 '1 " 1 rrn n ff Therefore " 11 " 11 ffss ==+ Lens Makers’ Formula f x x f s s h h mT ''' −=−=−== Gauss’ Lens Formula Magnification
  • 43.
  • 44.
  • 45.
    45 OpticsSOLO Ray Tracing F C O I ObjectVirtua l Image Convex Mirror R/2 R/2 R FC O I Object Real Image Concave Mirror Ray Tracing is a graphically implementation of paralax ray analysis. The construction doesn’t take into consideration the nonideal behavior, or aberration of real lens. The image of an off-axis point can be located by the intersection of any two of the following three rays: 1. A ray parallel to the axis that is reflected through F’. 2. A ray through F that is reflected parallel to the axis. 3. A ray through the center C of the lens that remains undeviated and undisplaced (for thin lens).
  • 46.
  • 47.
    47 OpticsSOLO Matrix Formulation The MatrixFormulation of the Ray Tracing method for the paraxial assumption was proposed at the beginning of nineteen-thirties by T.Smith. Assuming a paraxial ray entering at some input plane of an optical system at the distance r1 from the symmetry axis and with a slope r1’ and exiting at some output plane at the distance r2 from the symmetry axis and with a slope r2’, than the following linear (matrix) relation applies:       =            =      ''' 1 1 1 1 2 2 r r M r r DC BA r r       = DC BA Mwhere ray transfer matrix When the media to the left of the input plane and to the right of the output plane have the same refractive index, we have: 1det =⋅−⋅= CBDAM
  • 48.
    48 OpticsSOLO Matrix Formulation (continue-1) Uniform Optical Medium In an Uniform Optical Medium of length d no change in ray angles occurs: '' ' 12 112 rr rdrr = +=       = 10 1 d M Medium Optical Uniform Planar Interface Between Two Different Media 12 rr = '' 1 2 1 2 12 r n n r rr = = Apply Snell’s Law: 2211 sinsin φφ nn = paraxial assumption: φφφφ ≈=⇒≈ tan'sin r From Snell’s Law: '' 1 2 1 2 r n n r =       = 21 /0 01 nn M Interface Planar 1det 2 1 ≠= n n M Interface Planar 1det = Medium Optical Uniform M The focal length of this system is infinite and it has not specific principal planes.
  • 49.
    49 OpticsSOLO Matrix Formulation (continue-2) A Parallel-Sided Slab of refractive index n bounded on both sides with media of refractive index n1 = 1 We have three regions: • on the right of the slab (exit of ray):             =      '/0 01 ' 3 3 124 4 r r nnr r • in the slab:             =      '10 1 ' 2 2 3 3 r rd r r • on the left of the slab (entrance of ray):             =      '/0 01 ' 1 1 212 2 r r nnr r Therefore:                         =      '/0 01 10 1 /0 01 ' 1 1 21124 4 r r nn d nnr r             =                  = 21 21 122112 /0 /1 /0 01 /0 01 10 1 /0 01 nn nnd nnnn d nn M mediaentranceslabmediaexit Slab Sided Parallel        = 10 /1 21 nnd M Slab Sided Parallel 1det = Slab Sided Parallel M
  • 50.
    50 OpticsSOLO Matrix Formulation (continue-3) Spherical Interface Between Two Different Media 12 rr = Apply Snell’s Law: rnin sinsin 21 = paraxial assumption: rrii ≈≈ sin&sin From Snell’s Law: rnin 21 = ( )           − =           −= 2 1 2 1 2 1 12 21 0101 n n n D n n Rn nnM Interface Spherical 1det 2 1 ≠= n n M Interface Spherical 12 11 ' ' φ φ += += rr ri From the Figure: ( ) ( )122111 '' φφ +=+ rnrn 111 / Rr=φ ( ) 12 121 2 11 2 ' ' Rn rnn n rn r − += ( ) 1 12 11 112 2 12 ' ' n rn Rn rnn r rr + − = = ( ) 1 12 1 : R nn D − =where: Power of the surface If R1 is given in meters D1 gives diopters
  • 51.
    51 OpticsSOLO Matrix Formulation (continue-4) Thick Lens We have three regions: • on the right of the slab (exit of ray):                 − =      ' 01 ' 3 3 1 2 1 2 4 4 r r n n n D r r • in the slab:             =      '10 1 ' 2 2 3 3 r rd r r • on the left of the slab (entrance of ray):                 − =      ' 01 ' 1 1 2 1 2 1 2 2 r r n n n D r r Therefore:                   − −           − =                −                − =      ' 101 ' 01 10 1 01 ' 1 1 2 1 2 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 4 4 r r n n n D n n d n D d n n n D r r n n n D d n n n D r r               −      − + − − = 2 2 21 21 1 21 2 1 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick ( ) 2 21 2 R nn D − = ( ) 1 12 1 : R nn D − =               −      − + −− = − 2 1 21 21 1 21 2 1 2 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick 1det = Lens Thick M or 21 DD ⇔
  • 52.
    52 OpticsSOLO Matrix Formulation (continue-5) Thick Lens (continue -1) Let use the second Figure where Ray 2 is parallel to Symmetry Axis of the Optical System that is refracted trough the Second Focal Point.                         −      − + − − =           '1 1 ' 1 1 2 2 21 21 1 21 2 1 2 1 4 4 r r n D d nn DD d n DD n n d n D d r r We found: 2141 /'&0' frrr −==Ray 2: By substituting Ray2 parameters we obtain: 1 2 1 21 21 1 21 4 1 ' r f r nn DD d n DD r −=      − + −= 1 21 21 1 21 2 −       − + = nn DD d n DD f frrr /'&0' 414 −==Ray 1: We found:                         −      − + −− =           '1 1 ' 4 4 2 1 21 21 1 21 2 1 2 2 1 1 r r n D d nn DD d n DD n n d n D d r r 4 1 4 21 21 1 21 1 1 ' r f r nn DD d n DD r −=      − + = 2 1 21 21 1 21 1 f nn DD d n DD f −=      − + = −
  • 53.
    53 OpticsSOLO Matrix Formulation (continue-6) Thin Lens For thick lens we found               −      − + − − = 2 2 21 21 1 21 2 1 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick       − + = 21 21 1 211 nn DD d n DD f For thin lens we can assume d = 0 and obtain           − = 1 1 01 f M Lens Thin 1 211 n DD f + = ( ) 2 21 2 R nn D − = ( ) 1 12 1 : R nn D − =       −      −= + = 211 2 1 21 11 1 1 RRn n n DD f
  • 54.
    54 OpticsSOLO Matrix Formulation (continue-7) Thin Lens (continue – 1) For a biconvex lens we have R2 negative         +      −= 211 2 11 1 1 RRn n f For a biconcave lens we have R1 negative         +      −−= 211 2 11 1 1 RRn n f           − = 1 1 01 f M Lens Thin
  • 55.
    55 OpticsSOLO Matrix Formulation (continue-8) A Length of Uniform Medium Plus a Thin Lens           −− =                − == + f d f d d f MMM Medium Uniform Lens Thin Lens Thin Medium Uniform 1 1 1 10 1 1 1 01 Combination of Two Thin Lenses             +−−−+−− −+− =           −−          −− == 21 21 2 2 2 1 1 1 21 2 21 1 21 21 2 2 1 1 1 1 22 2 1 11 1 1 1 1 1 1 1 1122 ff dd f d f d f d ff d ff f dd dd f d f d f d f d f d MMMMM dMedium Uniform fLens Thin dMedium Uniform fLens Thin Lenses Thin Two The Focal Length of the Combination of Two Thin Lenses is: 21 2 21 111 ff d fff −+= Return to Chromatic Aberration
  • 56.
    56 OpticsSOLO Real Imaging Systems– Aberrations Departures from the idealized conditions of Gaussian Optics in a real Optical System are called Aberrations Monochromatic Aberrations Chromatic Aberrations • Monochromatic Aberrations Departures from the first order theory are embodied in the five primary aberrations 1. Spherical Aberrations 2. Coma 3. Astigmatism 4. Field Curvature 5. Distortion This classification was done in 1857 by Philipp Ludwig von Seidel (1821 – 1896) • Chromatic Aberrations 1. Axial Chromatic Aberration 2. Lateral Chromatic Aberration
  • 57.
    57 OpticsSOLO Real Imaging Systems– Aberrations (continue – 5)
  • 58.
    58 OpticsSOLO Real Imaging Systems– Aberrations (continue – 5)
  • 59.
    59 OpticsSOLO Real Imaging Systems– Aberrations (continue – 5)
  • 60.
  • 61.
  • 62.
    62 OpticsSOLO Real Imaging Systems– Aberrations (continue – 1) Seidel Aberrations Consider a spherical surface of radius R, with an object P0 and the image P0’ on the Optical Axis. The Chief Ray is P0 V0 P0’ and a General Ray P0 Q P0’. The Wave Aberration is defined as the difference in the optical path lengths between a General Ray and the Chief Ray. ( ) [ ] [ ] ( ) ( )snsnQPnQPnPVPQPPrW +−+=−= '''''' 00000000 On-Axis Point Object The aperture stop AS, entrance pupil EnP, and exit pupil ExP are located at the refracting surface.
  • 63.
    63 OpticsSOLO Real Imaging Systems– Aberrations (continue – 2) Seidel Aberrations (continue – 1)         −−=−−= 2 2 22 11 R r RrRRz Define: ( ) 2 2 11 2 2 R r xxf R r x −=+= −= ( ) ( ) 2/1 1 2 1 ' − += xxf ( ) ( ) 2/3 1 4 1 " − +−= xxf ( ) ( ) 2/5 1 8 3 '" − +−= xxf Develop f (x) in a Taylor series ( ) ( ) ( ) ( ) ( ) ++++= 0"' 6 0" 2 0' 1 0 32 f x f x f x fxf 1 168 11 32 <++−+=+ x xx xx  Rr R r R r R r R r Rz <+++=         −−= 5 6 3 42 2 2 1682 11 On Axis Point Object From the Figure: ( ) 222 rzRR +−= 02 22 =+− rRzz
  • 64.
    64 OpticsSOLO Real Imaging Systems– Aberrations (continue – 3) Seidel Aberrations (continue – 2) From the Figure: ( )[ ] [ ] ( )[ ] ( ) 2/1 2 2/12 2 2/12222/122 0 212 2 22       − +=+−= ++−=+−= −= z s sR sszsR rsszzrszQP rzRz ( ) ( )       + − − − +−≈ <++−+=+   2 4 2 2 1 168 11 2 1 1 32 z s sR z s sR s x xx xx ( ) ( )         +      + − −      + − +−= +≈  2 3 42 4 2 3 42 2 82 822 1 82 1 3 42 R r R r s sR R r R r s sR s R r R r z ( )[ ] +               −+      −+      −+−≈+−= 4 2 2 22/122 0 11 8 111 8 111 2 1 r sRssRR r sR srszQP ( )[ ] +               −+      −+      −+≈+−= 4 2 2 22/122 0 1 ' 1 '8 11 ' 1 8 11 ' 1 2 1 ''' r RssRsR r Rs srzsPQ In the same way: On Axis Point Object
  • 65.
    65 OpticsSOLO Real Imaging Systems– Aberrations (continue – 4) Seidel Aberrations (continue – 3) +               −+      −+      −+−≈ 4 2 2 2 0 11 8 111 8 111 2 1 r sRssRR r sR sQP +               −+      −+      −+≈ 4 2 2 2 0 1 ' 1 '8 11 ' 1 8 11 ' 1 2 1 '' r RssRsR r Rs sPQ Therefore: ( ) ( ) ( ) 4 22 2 42 000 11 ' 11 ' ' 8 1 82 ' ' ' '''' r sRs n sRs n R rr R nn s n s n snsnQPnQPnrW               −−      −−      +      − −−= +−+= Since P0’ is the Gaussian image of P0 we have ( ) R nn s n s n − = − + ' ' ' and: ( ) 44 22 0 11 ' 11 ' ' 8 1 rar sRs n sRs n rW S =               −−      −−= On Axis Point Object
  • 66.
    66 OpticsSOLO Real Imaging Systems– Aberrations (continue – 5) Seidel Aberrations (continue – 4) Off-Axis Point Object Consider the spherical surface of radius R, with an object P and its Gaussian image P’ outside the Optical Axis. The aperture stop AS, entrance pupil EnP, and xit pupil ExP are located at the refracting surface. Using ''~ 00 CPPCPP ∆∆ the transverse magnification ( ) ( ) s n s n nn s s n s n nn s Rs Rs h h Mt − − +− − − − = +− − = − = ' ' ' ' ' ' ' '' ( )sn sn nn s s nn nn s s nn Mt − = −+− +−− = ' ' ' ' ' ' ' '
  • 67.
    67 OpticsSOLO Real Imaging Systems– Aberrations (continue – 6) Seidel Aberrations (continue – 5) Off-Axis Point Object The Wave Aberration is defined as the difference n the optical path lengths between the General Ray and the Undeviated Ray. ( ) [ ] [ ] [ ] [ ]{ } [ ] [ ]{ } ( )4 0 4 0 '''' '' VVVQa PVPPPVPVPPQP PVPPQPQW S −= −−−= −= For the approximately similar triangles VV0C and CP0’P’ we have: CP CV PP VV ''' 0 0 0 0 ≈ '' ' '' ' 0 0 0 0 hbh Rs R PP CP CV VV = − =≈ Rs R b − = ' :               −−      −−= 22 11 ' 11 ' ' 8 1 sRs n sRs n aS
  • 68.
    68 OpticsSOLO Real Imaging Systems– Aberrations (continue – 7) Seidel Aberrations (continue – 6) Off-Axis Point Object Wave Aberration. ( ) [ ] [ ] ( )4 0 4 '' VVVQaPVPPQPQW S −=−= Define the polar coordinate (r,θ) of the projection of Q in the plane of exit pupil, with V0 at the origin. θθ cos'2'cos2 222 0 2 0 2 2 hbrhbrVVrVVrVQ ++=++= '0 hbVV = ( ) [ ] [ ] ( ) ( )[ ]442222 4 0 4 'cos'2' '' hbhbrhbra VVVQaPVPPQPQW S S −++= −=−= θ ( ) ( )θθθθ cos'4'2cos'4cos'4';, 33222222234 rhbrhbrhbrhbrahrW S ++++=
  • 69.
    69 OpticsSOLO Real Imaging Systems– Aberrations (continue – 8) Seidel Aberrations (continue – 7) eneral Optical Systems ( ) θθθθ cos''cos'cos'';, 33222222234 rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp ++++= A General Optical Systems has more than on Reflecting or Refracting surface. The image of one surface acts as an bject for the next surface, therefore the aberration is additive. We must address the aberration in the plane of the exit pupil, since the rays follow straight lines from the plane of the exit pupil. The general Wave Aberration Function is: 1. Spherical Aberrations CoefficientSpC 2. Coma CoefficientCoC 3. Astigmatism CoefficientAs C 4. Field Curvature CoefficientFC C 5. Distortion CoefficientDi C where:
  • 70.
    70 OpticsSOLO Real Imaging Systems– Aberrations (continue – 7) Seidel Aberrations (continue – 6) ( ) ( )θθθθ cos'4'2cos'4cos'4';, 33222222234 rhbrhbrhbrhbrahrW S ++++=
  • 71.
    71 OpticsSOLO Real Imaging Systems– Aberrations (continue – 7) Seidel Aberrations (continue – 6) ( ) θθθθ cos''cos'cos'';, 32222234 rhCrhCrhCrhCrChrW DiFCAsCoSp ++++=
  • 72.
    72 OpticsSOLO Real Imaging Systems– Aberrations (continue – 9) Seidel Aberrations (continue – 8) nWPP TR /= Assume that P’ is the image of P. The point PT is on the Exit Pupil (Exp) and on the True Wave Front (TWF) that propagates toward P’. This True Wave Front is not a sphere because of the Aberration. Without the aberration the wave front would be the Reference Sphere (RS) with radius PRP. W (x’,y’;h’) - wave aberration n - lens refraction index L’ - distance between Exp and Image plane ά - angle between the normals to the TWF and RS at PT. Assume that P’R and P’T are two points on RS and TWF, respectively, and on a ray close to PRPT ray, converging to P’, the image of P. lPPPP TRTR ∆+=''
  • 73.
    73 OpticsSOLO Real Imaging Systems– Aberrations (continue – 9) Seidel Aberrations (continue – 8) ( ) ' ';',' ' ' ' x hyxW n L x ∂ ∂ =∆ ( ) ' ';',' ' ' ' y hyxW n L y ∂ ∂ =∆ θ θ sin' cos' ry rx = = ( ) nhyxWPP TR /';','= lPPPP TRTR ∆+='' α= ∆ ∆ = ∆ − = ∂ ∂ →∆→∆ r l r PPPP x W n r TRTR r 00 lim '' lim 1 x W n L Lr ∂ ∂ ==∆ ' 'α
  • 74.
    74 OpticsSOLO Real Imaging Systems– Aberrations (continue – 1) 1. Spherical Aberrations ( ) ( ) ( )';',''' ';, 222 4 hyxWyxC rChrW SpSp SpSp =+= =θ ( ) ' ' ' 4 ' ';',' ' ' ' 2 xrC n L x hyxW n L x Sp = ∂ =∆ ( ) ' ' ' 4 ' ';',' ' ' ' 2 yrC n L y hyxW n L y Sp = ∂ =∆ To Update ( ) ( )[ ] 32/122 ' ' 4'' rC n L yxr Sp=∆+∆=∆ Consider only the Spherical Wave Aberration Function The Spherical Wave Aberration is a Circle in the Image Plane
  • 75.
    75 OpticsSOLO Real Imaging Systems– Aberrations (continue – 2) 2. Coma Assume an object point outside the Optical Axis. Meridional (Tangential) plane is the plane defined by the object point and the Optical Axis. Sagittal plane is the plane normal to Meridional plane that contains the Chief Ray passing through the Object point.
  • 76.
    76 OpticsSOLO Real Imaging Systems– Aberrations (continue – 2) 2. Coma Consider only the Coma Wave Aberration Function ( ) ( ) ''''cos'';, 223 xyxhCrhChrW CoCoCo +== θθ ( ) ( ) ( ) ( )θθ 2cos2 ' '' cos21 ' '' ''3 ' '' ' ';',' ' ' ' 2222 +=+=+= ∂ =∆ r n Lh Cr n Lh Cyx n Lh C x hyxW n L x CoCoCo ( ) ( ) θ2sin ' '' ''2 ' '' ' ';',' ' ' ' 2 r n Lh Cyx n Lh C y hyxW n L y CoCo == ∂ =∆ 1 ' '' ' 2 ' '' ' 2 2 2 2 =             ∆ +             − ∆ r n Lh C y r n Lh C x CoCo ( )( ) ( ) ( )222 '2' rRyrRx CoCo =∆+−∆ ( ) 2 ' '' : r n Lh CrR CoCo =
  • 77.
    77 OpticsSOLO Real Imaging Systems– Aberrations (continue – 2) 2. Coma We obtained 2 ' '' : MAXCoS r n Lh CC = ( )( ) ( ) ( )222 '2' rRyrRx CoCo =∆+−∆ ( ) MAXCoCo rrr n Lh CrR ≤≤= 0 ' '' : 2 Define: 1 2 3 4 P Image Plane O SC SC ST CC 3= Coma Blur Spot Shape Tangential Coma Sagittal Coma  30 'h 'x 'y
  • 78.
    78 OpticsSOLO Real Imaging Systems– Aberrations (continue – 2) Graphical Explanation of Coma Blur
  • 79.
    79 OpticsSOLO Real Imaging Systems– Aberrations (continue – 2) phical Explanation of Coma Blur (continue – 1)
  • 80.
    81 OpticsSOLO Real Imaging Systems– Aberrations (continue – 3) 3. Astigmatism
  • 81.
    82 OpticsSOLO Real Imaging Systems– Aberrations (continue – 3) 3. Astigmatism
  • 82.
    83 OpticsSOLO Real Imaging Systems– Aberrations (continue – 3) 3. Astigmatism
  • 83.
    84 OpticsSOLO Real Imaging Systems– Aberrations (continue – 4) Field Curvature
  • 84.
    85 OpticsSOLO Real Imaging Systems– Aberrations (continue – 5) 5. Distortion
  • 85.
    86 OpticsSOLO Real Imaging Systems– Aberrations (continue – 8) Seidel Aberrations (continue – 7) hin Lens Aberrations ( ) 2222234 'cos'cos'';, rhCrhCrhCrChrW FCAsCoSp +++= θθθ ven a thin lens formed by two faces with radiuses r1 and r2 h centers C1 and C2. PP0 is object, P”P”0 is the Gaussian ge formed by the first surface, ’0 is the image of virtual object ”0 of the second surface. ( ) ( ) ( ) ( )       ++ − + +−++ −− −= qpnq n n pnn n n fnn CSp 14 1 2 123 1132 1 22 3 3 ( )       − + ++= q n n pn sfn CCo 1 1 12 '4 1 2 ( )2 '2/1 sfCAs −= ( ) ( )2 '4/1 sfnnCFC +−= where: f s OA C11 r F” F ''f ''s 2 r 1=n n h "h D 0P P 0'P 0"P "P 'P 'h 's CR AS EnP ExP r ( )θ,rQ OC2 1=n ( ) [ ] [ ]0000 '', OPPQPPrW −=θ Coddington shape factor: Coddington position factor: ss ss p − + = ' ' 12 12 rr rr q − + = From: we find:
  • 86.
    87 OpticsSOLO Real Imaging Systems– Aberrations (continue – 8) Seidel Aberrations (continue – 7) oddington Position Factor 2R 1R f 1C 2FO 1F 2 C 2n 1n s 's '2 sfs == 2R 1 R f 1 C 2F1F 2 C 2n 1 n s 's fss =∞= ', 2R 1R f 1 C 2F1F 2 C 2n 1 n s 's fss <> ',0 2R 1 R f 1 C 2F1F 2 C 2n 1 n s 's ∞== ', sfs 2R 1Rf 1 C2F1 F 2 C 2n 1 n s 's 0', << sfs CRCR 2 R 1 R f1C 2 F 1F 2 C2n 1 n s 's 0'0 <<> sfs 2 R 1 R f1 C 2F1F 2C2n 1n s 's fss =∞= ', 1=p 2 R1R f 1C 2F1F 2C 2n 1 n s 's ∞== ', sfs 1>p 2 R1R f 1C 2F1F 2C 2n 1 n s 's 0',0 ><< ssf 0=p 2 R1R f 1C 2FO 1F 2C 2n 1 n s 's '2 sfs == 1−=p1−<p ss ss p − + = ' ' ss ss p − + = ' ' ' 111 ssf += ' 2 11 2 s f s f p −=−=
  • 87.
    88 OpticsSOLO Coddington Position Factor ff2f2− f− 0 Figure Object Location Image Location Image Properties Shape Factor Infinity Principal focus 'ss fs 2> fsf 2'<< fs 2= fs 2'= fsf 2<< fs 2'> 's 's s s fs = ∞='s s s 's fs < fs <' Real, inverted small p = -1 Real, inverted smaller -1 < p <0 Real, inverted same size p = 0 Real, inverted larger 0 < p <1 No image p = 1 Virtual, erect larger p>1 's 's 0<s fs <' p < -1 Imaginary, inverted small
  • 88.
    89 OpticsSOLO Real Imaging Systems– Aberrations (continue – 8) Seidel Aberrations (continue – 7) oddington Shape Factor 1 02 1 −= < ∞= q R R 2 R 1 R 2 C 2 n 1 n Plano Convex 2 n 1 0,0 21 21 −< > << q RR RR 1 C 2 C 1 n 1 R 2 R Positive Meniscus 2 R 1 R f 1 C 2 F1 F 2 C 2 n 1 n 0 0,0 21 21 = = <> q RR RR Equi Convex 2 R 1 R 1C2n 1n Plano Convex 1 0 2 1 = ∞= > q R R 2 R1 R f 1 C 2 F 2 C 2 n 1 n 1 0,0 21 21 > < >> q RR RR Positive Meniscus 12 12 RR RR q − + = 2 R 1 R f 2F1 F 2C 2n 1 n 1C Negative Meniscus 1 0,0 21 21 −< > >> q RR RR 1 0, 21 −= >∞= q RR Plano Concave 2 R 1 R f 2 F1 F 2 C 2n 1 n 2 R1 R f 1 C 2F1 F 2C 2 n 1 n 0 0,0 21 21 = = >< q RR RR Equi Concave 2 R 1 R f 1F 2F 1C 2 n 1 n 1 ,0 21 = ∞=< q RR Plano Concave Negative Meniscus 1 0,0 21 21 > < << q RR RR 2 R 1 R f 2 F1 F 2C 2n 1 n 1C
  • 89.
    90 REFLECTION & REFRACTIONSOLO http://freepages.genealogy.rootsweb.com/~coddingtons/15763.htm Historyof Reflection & Refraction Reverent Henry Coddington (1799 – 1845) English mathematician and cleric. He wrote an Elementary Treatise on Optics (1823, 1st Ed., 1825, 2nd Ed.). The book was displayed the interest on Geometrical Optics, but hinted to the acceptance of the Wave Theory. Coddington wrote “A System of Optics” in two parts: 1. “A Treatise of Reflection and Refraction of Light” (1829), containing a thorough investigation of reflection and refraction. 2. “A Treatise on Eye and on Optical Instruments” (1630), where he explained the theory of construction of various kinds of telescopes and microscopes. He recommended the ue of the grooved sphere lens, first described by David Brewster in 1820 and inuse today as the “Coddington lens”. Coddington introduced for lens: Coddington Shape Factor: Coddington Position Factor: 12 12 rr rr q − + = ss ss p − + = ' ' Coddington Lens http://www.eyeantiques.com/MicroscopesAndTelescopes/Coddington%20microscope_thick_wood.htm
  • 90.
    91 OpticsSOLO Real Imaging Systems– Aberrations (continue – 8) Seidel Aberrations (continue – 7) hin Lens Spherical Aberrations ( ) 4 rCrW SpSA = ven a thin lens and object O on the ical Axis (OA). A paraxial ray will cross OA at point I, at a distance s’p from lens. A general ray, that reaches the lens distance r from OA, will cross OA at nt E, at a distance s’r. ( ) ( ) ( ) ( )       ++ − + +−++ −− −= qpnq n n pnn n n fnn CSp 14 1 2 123 1132 1 22 3 3 where: Define: 2 R 1 R 1 C IO 2C Paraxial focal plane2 n 1n s ps' E rs' Long. SA Lat. SA φ Paraxial Ray General Ray 'φ r rp ssSALongAberrationSphericalalLongitudin ''. −== ( ) rrp srssSALatAberrationSphericalLateral '/''. −== We have:
  • 91.
    92 OpticsSOLO Real Imaging Systems– Aberrations (continue – 1) 12 12 RR RR qK − + == ( ) ( ) ( ) ( ) ( )       ++ − + +−++ −− −= qpnq n n pnn n n fnn r rWSp 14 1 2 123 1132 22 3 3 4 Thin Lens Spherical Aberrations (continue – 1)
  • 92.
    93 OpticsSOLO Real Imaging Systems– Aberrations (continue – 1) Thin Lens Spherical Aberrations (continue – 3) 2 R 1 R 1 C IO 2C Paraxial focal plane2 n 1 n s ps' E rs' Long. SA Lat. SA φ Paraxial Ray General Ray 'φ r 12 12 RR RR q − + = F.A. Jenkins & H.E. White, “Fundamentals of Optics”, 4th Ed., McGraw-Hill, 1976, pg. 157 Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm In Figure we can see a comparison of the Seidel Third Order Theory with the ray tracing.
  • 93.
    94 OpticsSOLO Real Imaging Systems– Aberrations (continue – 1) We can see that the Thin Lens Spherical Aberration WSp is a parabolic function of the Coddington Shape Factor q, with the vertex at (qmin,WSp min) ( ) ( ) ( ) ( )       ++ − + +−++ −− −= qpnq n n pnn n n fnn r WSp 14 1 2 123 1132 22 3 3 4 Thin Lens Spherical Aberrations (continue -2) The minimum Spherical Aberration for a given Coddington Position Factor p is obtained by: ( ) ( ) 014 1 2 2 132 3 4 =    ++ − + − −= ∂ ∂ pnq n n fnn r q W p Sp 1 1 2 2 min + − −= n n pq         + −      − −= 2 2 3 4 min 2132 p n n n n f r WSp The minimum Spherical Aberration is zero for ( ) ( ) 1 1 2 2 2 > − + = n nn p
  • 94.
    95 OpticsSOLO Real Imaging Systems– Aberrations (continue – 1) In order to obtain the radii of the lens for a given focal length f and given Shape Factor and Position Factor we can perform the following: Thin Lens Spherical Aberrations (continue – 3) Those relations were given by Coddington. ' 2 11 2 s f s f p −=−= p f s p f s − = + = 1 2 '& 1 2 ( ) fRR n ss 111 1 ' 11 21 =      −−=+ ( ) ( ) 1 12 & 1 12 21 − − = + − = q nf R q nf R 12 12 RR RR q − + = 12 1 12 2 2 1& 2 1 RR R q RR R q − =− − =+ ( ) ( )12 21 1 RRn RR f −− = 2 R 1 R 1 C IO 2 C Paraxial focal plane2 n 1 n s ps' E rs' Long. SA Lat. SA φ Paraxial Ray General Ray 'φ r
  • 95.
    96 OpticsSOLO Real Imaging Systems– Aberrations (continue – 8) Seidel Aberrations (continue – 7) hin Lens Coma ( ) ( ) ( ) ( )     − + ++ + = +== q n n pn sfn xyxh xyxhCrhChrW CoCoCo 1 1 12 '4 '''' ''''cos'';, 2 22 223 θθ or thin lens the coma factor is given by: where:we find: ( ) 2 22 2 1 1 12 4 ''' : MAXMAXCoS rq n n pn fn h r n sh CC     − + ++== 1 2 3 4 P Image Plane O SC SC ST CC 3= Coma Blur Spot Shape Tangential Coma Sagittal Coma  30 'h 'x 'y ( )( ) ( ) ( )222 '2' rRyrRx CoCo =∆+−∆ ( ) MAXCoCo rrr n sh CrR ≤≤= 0 '' : 2 Define: ( ) ( ) ( ) ( )θθ 2cos2 '' cos21 '' ''3 '' ' ';','' ' 2222 +=+=+= ∂ =∆ r n sh Cr n sh Cyx n sh C x hyxW n s x CoCoCo ( ) ( ) θ2sin '' ''2 '' ' ';','' ' 2 r n sh Cyx n sh C y hyxW n s y CoCo == ∂ =∆
  • 96.
    97 OpticsSOLO Real Imaging Systems– Aberrations (continue – 8) Seidel Aberrations (continue – 7) hin Lens F.A. Jenkins & H.E. White, “Fundamentals of Optics”, 4th Ed., McGraw-Hill, 1976, pg. 165 Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm, y = 2 cm ( ) 2 22 1 1 12 4 ' : MAXS rq n n pn fn h C       − + ++= oma is linear in q ( ) ( ) ( ) p n nn qCS 1 112 0 + −+ −=⇐= n Figure 800.00 =⇐= qCS The Spherical Aberration is arabolic in q ( ) ( ) ( ) ( )       ++ − + +−++ −− −= qpnq n n pnn n n fnn CSp 14 1 2 123 1132 1 22 3 3 1 1 2 2 min + − −= n n pq         + −      − −= 2 2 3min 2132 1 p n n n n f CSp In Figure 714.0min =q
  • 97.
    98 OpticsSOLO Real Imaging Systems– Aberrations (continue – 5)
  • 98.
    99 OpticsSOLO Real Imaging Systems– Aberrations (continue – 5)
  • 99.
    100 SOLO Optics Chromatic Aberration ChromaticAberrations arise in Polychromatic IR Systems because the material index n is actually a function of frequency. Rays at different frequencies will traverse an optical system along different paths.
  • 100.
  • 101.
    102 SOLO Optics Chester MoorHall (1704 – 1771) designed in secrecy the achromatic lens. He experienced with different kinds of glass until he found in 1729 a combination of convex component formed from crown glass with a concave component formed from flint glass, but he didn’t request for a patent. http://microscopy.fsu.edu/optics/timeline/people/dollond.html In 1750 John Dollond learned from George Bass on Hall achromatic lens and designed his own lenses, build some telescopes and urged by his son Peter (1739 – 1820) applied for a patent. Born & Wolf,”Principles of Optics”, 5th Ed.,p.176 Chromatic Aberration In 1733 he built several telescopes with apertures of 2.5” and 20”. To keep secrecy Hall ordered the two components from different opticians in London, but they subcontract the same glass grinder named George Bass, who, on finding that both Lenses were from the same customer and had one radius in common, placed them in contact and saw that the image is free of color. The other London opticians objected and took the case to court, bringing Moore-Hall as a witness. The court agree that Moore- Hall was the inventor, but the judge Lord Camden, ruled in favor of Dollond saying:”It is not the person who locked up his invention in the scritoire that ought to profit by a patent for such invention, but he who brought it forth for the benefit of the public”
  • 102.
    103 SOLO Optics Chromatic Aberration Everypiece of glass will separate white light into a spectrum given the appropriate angle. This is called dispersion. Some types of glasses such as flint glasses have a high level of dispersion and are great for making prisms. Crown glass produces less dispersion for light entering the same angle as flint, and is much more suited for lenses. Chromatic aberration occurs when the shorter wavelength light (blue) is bent more than the longer wavelength (red). So a lens that suffers from chromatic aberration will have a different focal length for each color To make an achromat, two lenses are put together to work as a group called a doublet. A positive (convex) lens made of high quality crown glass is combined with a weaker negative (concave) lens that is made of flint glass. The result is that the positive lens controls the focal length of the doublet, while the negative lens is the aberration control. The negative lens is of much weaker strength than the positive, but has higher dispersion. This brings the blue and the red light back together (B). However, the green light remains uncorrected (A), producing a secondary spectrum consisting of the green and blue-red rays. The distance between the green focal point and the blue-red focal point indicates the quality of the achromat. Typically, most achromats yield about 75 to 80 % of their numerical aperture with practical resolution
  • 103.
    104 SOLO Optics Chromatic Aberration Inaddition, to the correction for the chromatic aberration the achromat is corrected for spherical aberration, but just for green light. The Illustration shows how the green light is corrected to a single focal length (A), while the blue-red (purple) is still uncorrected with respect to spherical aberration. This illustrates the fact that spherical aberration has to be corrected for each color, called spherochromatism. The effect of the blue and red spherochromatism failure is minimized by the fact that human perception of the blue and red color is very weak with respect to green, especially in dim light. So the color halos will be hardly noticeable. However, in photomicroscopy, the film is much more sensitive to blue light, which would produce a fuzzy image. So achromats that are used for photography will have a green filter placed in the optical path.
  • 104.
    105 SOLO Optics Chromatic Aberration Asthe optician's understanding of optical aberrations improved they were able to engineer achromats with shorter and shorter secondary spectrums. They were able to do this by using special types of glass call flourite. If the two spectra are brought very close together the lens is said to be a semi-apochromat or flour. However, to finally get the two spectra to merge, a third optical element is needed. The resulting triplet is called an apochromat. These lenses are at the pinnacle of the optical family, and their quality and price reflect that. The apochromat lenses are corrected for chromatic aberration in all three colors of light and corrected for spherical aberration in red and blue. Unlike the achromat the green light has the least amount of correction, though it is still very good. The beauty of the apochromat is that virtually the entire numerical aperture is corrected, resulting in a resolution that achieves what is theoretically possible as predicted by Abbe equation.
  • 105.
    106 SOLO Optics Chromatic Aberration Withtwo lenses (n1, f1), (n2,f2) separated by a distance d we found 2121 111 ff d fff −+= Let use ( ) ( ) 222111 1/1&1/1 ρρ −=−= nfnf We have ( ) ( ) ( ) ( ) 22112211 1111 1 ρρρρ −−−−+−= nndnn f nF – blue index produced by hydrogen wavelength 486.1 nm. nC – red index produced by hydrogen wavelength 656.3 nm. nd – yellow index produced by helium wavelength 587.6 nm. Assume that for two colors red and blue we have fR = fB ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 22112211 22112211 1111 1111 1 ρρρρ ρρρρ −−−−+−= −−−−+−= FFFF CCCC nndnn nndnn f
  • 106.
    107 SOLO Optics Chromatic Aberration Letanalyze the case d = 0 (the two lenses are in contact) nd – yellow index produced by helium wavelength 587.6 nm. We have ( ) ( ) ( ) ( ) 22112211 1111 1 ρρρρ −+−=−+−= FFCC nnnn f ( ) ( ) ( ) ( )1 1 1 1 1 2 1 2 2 1 − − −= − − −= F F C C n n n n ρ ρ ( ) ( )CF CF nn nn 11 22 2 1 − − −= ρ ρ For the yellow light (roughly the midway between the blue and red extremes) the compound lens will have the focus fY: ( ) ( ) YY f d f d Y nn f 21 /1 22 /1 11 11 1 ρρ −+−= ( ) ( ) Y Y d d f f n n 1 2 1 2 2 1 1 1 − − = ρ ρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1/ 1/ 1 1 111 222 2 1 11 22 1 2 −− −− −= − − − − −= dCF dCF d d CF CF Y Y nnn nnn n n nn nn f f
  • 107.
    108 SOLO Optics Chromatic Aberration () ( ) ( ) ( )1/ 1/ 111 222 1 2 −− −− −= dCF dCF Y Y nnn nnn f f The quantities are called Dispersive Powers of the two materials forming the lenses. ( ) ( ) ( ) ( )1 & 1 2 22 1 11 − − − − d CF d CF n nn n nn Their inverses are called V-numbers or Abbe numbers. ( ) ( ) ( ) ( )CF d CF d nn n V nn n V 22 2 2 11 1 1 1 & 1 − − = − − =
  • 108.
    109 OpticsSOLO To define glasswe need to know more than one index of refraction. In general we choose the indexes of refraction of three colors: nF – blue index produced by hydrogen wavelength 486.1 nm. nC – red index produced by hydrogen wavelength 656.3 nm. nd – yellow index produced by helium wavelength 587.6 nm. Define: nF – nC - mean dispersion CF d nn n v − − = 1 - Abbe’s Number or v value or V-number Crowns: glasses of low dispersion (nF – nC small and V-number above 55) Flints: glasses of high dispersion (nF – nC high and V-number bellow 50) Fraunhofer line color Wavelength (nm) Spectacle Crown C - 1 Extra Dense Flint EDF - 3 F d C Blue Yellow Red 486.1 587.6 656.3 1.5293 1.5230 1.5204 1.7378 1.7200 1.7130V - number 58.8 29.0
  • 109.
    110 OpticsSOLO Refractive indices andAbbe’s numbers of various glass materials
  • 110.
  • 111.
    112 OpticsSOLO Camera Lenses Born & Wolfe,“Principle of Optics”, Pergamon Press, 5th Ed., pp.236-237
  • 112.
    113 SOLO References Lens Design 1. Kingslake,R., “Lens Design Fundamentals”, Academic Press, N.Y., 1978 6. Geary, J. M., “Introduction to Lens Design with Practical ZEMAX Examples”, Willmann-Bell, Inc., 2002 5. Laikin, M., “Lens Design”, Marcel Dekker, N.Y., 1991 2. Malacara, D., Ed., “Optical Shop Testing”, John Wiley & Sons, N.Y., 1978 7. Kidger, M. J., “Fundamental Optical Design”, SPIE Press., 2002 3. Kingslake, R., “Optical System Design”, Academic Press, N.Y., 1983 4. O’Shea, D.,C., “Elements of Modern Optical Design”, John Wiley & Sons, N.Y., 1985
  • 113.
    114 SOLO References OPTICS 1. Waldman, G.,Wootton, J., “Electro-Optical Systems Performance Modeling”, Artech House, Boston, London, 1993 2. Wolfe, W.L., Zissis, G.J., “The Infrared Handbook”, IRIA Center, Environmental Research Institute of Michigan, Office of Naval Research, 1978 3. “The Infrared & Electro-Optical Systems Handbook”, Vol. 1-7 4. Spiro, I.J., Schlessinger, M., “The Infrared Technology Fundamentals”, Marcel Dekker, Inc., 1989
  • 114.
    115 SOLO References [1] M. Born,E. Wolf, “Principle of Optics – Electromagnetic Theory of Propagation, Interference and Diffraction of Light”, 6th Ed., Pergamon Press, 1980, [2] C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996, OPTICS
  • 115.
    116 SOLO References Foundation of GeometricalOptics [3] E.Hecht, A. Zajac, “Optics ”, 3th Ed., Addison Wesley Publishing Company, 1997, [4] M.V. Klein, T.E. Furtak, “Optics ”, 2nd Ed., John Wiley & Sons, 1986
  • 116.
    117 OPTICSSOLO References Optics Polarization A.Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons, 1984 M. Born, E. Wolf, “Principles of Optics”, Pergamon Press,6th Ed., 1980 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 1979, Ch.8 C.C. Davis, “Lasers and Electro-Optics”, Cambridge University Press, 1996 G.R. Fowles, “Introduction to Modern Optics”,2nd Ed., Dover, 1975, Ch.2 M.V.Klein, T.E. Furtak, “Optics”, 2nd Ed., John Wiley & Sons, 1986 http://en.wikipedia.org/wiki/Polarization W.C.Elmore, M.A. Heald, “Physics of Waves”, Dover Publications, 1969 E. Collett, “Polarization Light in Fiber Optics”, PolaWave Group, 2003 W. Swindell, Ed., “Polarization Light”, Benchmark Papers in Optics, V.1, Dowden, Hutchinson & Ross, Inc., 1975 http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi)
  • 117.
    January 4, 2015118 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

Editor's Notes

  • #87 This can be found in V.N. Mahajan, ”Aberration Theory Made Simple”, SPIE, 1991, pg.23
  • #88 This can be found in V.N. Mahajan, ”Aberration Theory Made Simple”, SPIE, 1991, pg.23
  • #90 This can be found in V.N. Mahajan, ”Aberration Theory Made Simple”, SPIE, 1991, pg.23
  • #92 This can be found in V.N. Mahajan, ”Aberration Theory Made Simple”, SPIE, 1991, pg.23 F.A. Jenkins &amp; H.E. White, “Fundamentals of Optics”,4th Ed., McGraw-Hill, 1976, pp.157-160 M.V. Klein &amp; T. Furtak, “Optics”, 2nd Ed., John Wiley &amp; Sons, 1986, pp.243-245 Jurge R. Meyer-Arendt, “Introduction to Classical &amp; Modern Optics”, 3th Ed., Prentice-Hall Inc., 1989, pp.108-116
  • #94 F.A. Jenkins &amp; H.E. White, “Fundamentals of Optics”,4th Ed., McGraw-Hill, 1976, pg.159
  • #96 F.A. Jenkins &amp; H.E. White, “Fundamentals of Optics”,4th Ed., McGraw-Hill, 1976, pg.159
  • #97 This can be found in V.N. Mahajan, ”Aberration Theory Made Simple”, SPIE, 1991, pg.23 M.V. Klein &amp; T. Furtak, “Optics”, 2nd Ed., John Wiley &amp; Sons, 1986, pg.245 F.A. Jenkins &amp; H.E. White, “Fundamentals of Optics”,4th Ed., McGraw-Hill, 1976, pp.162-166
  • #98 This can be found in V.N. Mahajan, ”Aberration Theory Made Simple”, SPIE, 1991, pg.23 M.V. Klein &amp; T. Furtak, “Optics”, 2nd Ed., John Wiley &amp; Sons, 1986, pg.245 F.A. Jenkins &amp; H.E. White, “Fundamentals of Optics”,4th Ed., McGraw-Hill, 1976, pp.162-166