Fund optics

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Fund optics

  1. 1. Fundamental Opt ics ics 1 Gaussian Beam Op tics Optical Specificatio ns Fundamental of Optics Introduction 1.2 Paraxial Formulas 1.3 Imaging Propertie https://www.cvimellesgriot.com/Products/Documents/TechnicalGuide/Fundamental-Optics.pdf s of Lens Systems 1.6 Lens Combinatio n Formulas Performance Fact -.&$/,.0/1ors&23$/45.6.17#' /)1' !."#.$#$#%&' #()*#+!%*", Lens Shape Lens Combinatio ns 1.17 1.18 Lens Selection 1.20 Spot Size 1.23 Aberration Balanc ing Material Propertie s !"#!$%%&'(#)#*+,%- Diffraction Effect s 1.8 89.$/45.:(5*4&;<;:5=>#3?!8*):@.A<BC#= 1. 2/2554 PHY 340 Applied Optics -.&).#6D)E.4=F 11 1.26 Definition of Term s Paraxial Lens Form ulas Principal-Point Lo cations 1.27 1.29 1.32 1.36 Polarization 1.37 Waveplates 1.41 Etalons 1.46 Ultrafast Theory 1.49 1.52 Optical Coatings Prisms 1.1 http://www.eyedesignbook.com/ch5/eyech5-ab.html
  2. 2. ?3*:+!)G,@!CH)#2'/#3BB!3I# optical system Object (Input) Process Image (Output) &$.87@!%).# (requirement): )J:(%K5.5 7JL9<F%-.> &$.8&8,(" &$.81$F.% #.&.!CH)#2' ?3M+@!!CH)#2' 4(6<6.17#'?
  3. 3. www.cvimellesgriot.com g problem can be ations (first order) agnification, focal ge position. These f this chapter. )#3B$<).#).#!!)LBB#3BB4(6<6.17#' THE OPTICAL ENGINEERING PROCESS araxial values, and ention paid to the lysis for all but the r ray tracing, but the lens selection apes. Determine basic system parameters, such as magnification and object/image distances design constraints, ystem parameters Using paraxial formulas and known parameters, solve for remaining values t be familiar to all erms. pter relates only to aussian beams are Pick lens components based on paraxially derived values Determine if chosen component values conflict with any basic system constraints , experienced wide. The able the user the most hen additional cations engiesitate to n more ts. Estimate performance characteristics of system Determine if performance characteristics meet original design goals )#3B$<).#$/6$)##84(6<6.17#' 1. #3BCH(??(54=F1J&(N *,F< )J:(%K5.5 #353-.> #353$(7OC 2. G,@1P7# paraxial formulas L:3H(??(54=F#P@&F.*>+F!&J<$2&F.!+F<Q 3. *:+!)G,@*:<1'7F.%Q7.8&F.4=F&J<$2I"@ 4. >/?.#2.$F.&F.4=FI"@K("L5@%)(BK@!?J)("K!%#3BB9#+!I8F 5. H#38.2&C2:()E234=F#3BB?34J%.<I"@ 6. >/?.#2.$F.&C2:()E234=F#3BB4J%.<I"@*HR<IH7.87@!%).#9#+!I8F Fundamental Optics 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.4 Typically, the first step in optical problem solving is to select a system focal length based on constraints such as magnification or conjugate distances (object and image distance). The relationship among focal length, object position, and image position is given by Gaussian Beam Optics 1 1 1 = + s s″ f object F1 (1.1) 200 This formula is referenced to figure 1.1 and the sign conventions given in Sign Conventions. Figure 1.2 By definition, magnification is the ratio of image size to object size or m= s″ h″ = . s h (1.2) This relationship can be used to recast the first formula into the following forms: (s + s ″) f =m ( m + 1)2 (1.3) Example 1 (f = 50 mm, s Example 2: Object inside Focal Poin The same object is placed 30 mm left of same lens. Where is the image formed, an (See figure 1.3.) 1 1 1 = − s ″ 50 30 s ″ = −75 mm
  4. 4. f (BFL, denoted fb). ray from object at infinity The ray is broken into three segments by the lens. Two of these are external primary vertex A1 (in the air), and the thirdaxis optical is internal (in the glass). The external segments F can be extended to a common point of intersection (certainly near, and front focal r usually within, the lens). The principal surface is the locus of all such1 The convention in all of the figures (with the exception of a single deliberately reversed ray) is that light travels from left to right. </5.8K!%6(>4' Geometrical Optics H point ff A f tc primary principal point te primary principal surface back focal point ray from object at infinity ray from object at infinity primary vertex A1 optical axis F 1ch_FundamentalOptics_Final_a.qxd front focal point 6/15/2009 secondary principal surface secondary principal point r2 A = front focus to front f = effective focal length; edge distance may be positive (as shown) H H″1.29 secondary vertex F″ A2 Page or negative B = rear edge to rear reversed focus distance focal f = front focal length ray locates front point or primary principal surface f fb = back focal length fb 2:29 PM r1 ff A f B f Figure 1.33 Fundamental Optics Focal length and focal points www.cvimellesgriot.com Definition of Terms.5IHK$.*18! *>+F!G9@).#&J<$2I8F1(B1<L:3*HR<8.7#S.<1.): ;"54(F$IH*#.?3)J9<"G9@L1%*"/<4.%?.)M@ A = front focus to front edge distance f = effective focal length; may be positive (as shown) or negative te = edge thickness FOCAL LENGTH ( f ) B = rear edge to rear focus distance ff = front focal length Two distinct terms describe the focal lengths associated with every lens or lens system. The effective focal length (EFL) or equivalent focal length fb = back focal length (denoted f in figure 1.33) determines magnification and hence the image size. The term f appears frequently in lens formulas and in the tables of standard lenses. Unfortunately, because f is measured with reference to principal Focal length usually inside the lens, Figure 1.33 points which are and focal points the meaning of f is not immediately apparent when a lens is visually inspected. !"#"$%&'( (Focal Length, f ) r1 = radius of curvature of first surface (positive if center of tc = center thickness (F OR F ″curvature is to right) FOCAL POINT ) r2 = radius of curvature point must Rays that pass through or originate at either focal of second be, on the surface optical axis. center is opposite side of the lens, parallel to the(negative if This factof the basis for curvature is to left) locating both focal points. PRIMARY PRINCIPAL SURFACE Let us imagine that rays front )*+$%&'(to(FocalaxisoriginatingF thefrom theF’ point Fside oftherefore Point, at ,!-.focal ) (and the lens) parallel the optical after emergence opposite The second type of &+! ;"54(F$IH8=1!%LBBfocal length relates the focal plane positions directly1J9#(Bare"singly1G"Q ("@.some imaginary<9:(%K!%*:<1'twice refracted (once surface, instead of ) *1@ ?C ;0)( refracted at <9<@.L:3"@.Fundamental Optics < 1.29 to landmarks on the lens surfaces (namely the vertices) which are imme1.effective focalis not simply related to9#+! size but is especially %1=TF.<9#+!lensF8surface) as ";0)(1happens. There is a 6<'G< imaginary length (EFL) image equivalent #( at each *#/ 7@<?.)?C actually ?3K<.<)(BL)<4( unique diately recognizable. It surface, the focal length (#353 f) *HR<7($ about correct lens positioning or .<7#%K@.8)(called;0)(principal surface, at which this can happen. convenient for use when one is concerned)J9<")J:(%K5.5L:3 "@ B?C" 1 *18! To locate this unique surface, consider a single ray traced from the air on K<."-.> #353<=@*HR<#3534=F* this second type of focal length are mechanical clearances. Examples of #.G,@)(<G<1P7#&J<$2 L7F the one side of the lens, through the lens and into the air on the other side. front focal length (FFL, denoted ff I"@ * <+ F !1.33) and the back ?.)length in figure %?.)#353$( " focal *#.I8F 1 .8.#O$( " ?#/ % The ray is broken into three segments by the lens. Two of these are external (BFL, denoted fb). principle points 4=F!5PF-.5G<*:<1' (in the air), and the third is internal (in the glass). The external segments The convention F$("all of the figures (with the exception of a single 2.#353;0)(14= in ?.)>+@<T/$K!%*:<1' (lens surface can be extended to a common point of intersection (certainly near, and deliberately reversed ray) is that light travels from left to right. usually within, the lens). The principal surface is the locus of all such 9#+!*#=5)$F. K(@$K!%T/$=vertices) #353<=@I8F1.8.#O &J<$2I"@;"57#%L7F8=H#3;5,<'7F!?("7JL9<F%K!% $(7OC9#+!-.>?#/%Q ;0)(1<=@I"@L)F front focal length (FFL) L:3 back focal length (BFL) tc primary principal point te primary principal surface ray from object at infinity ray from object at infinity primary vertex A1 optical axis F front focal point r1 ff A f secondary principal surface secondary principal point back focal point r2 H H″ A2 secondary vertex reversed ray locates front focal point or primary principal surface fb B f F″
  5. 5. 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:29 PM Page 1.30 !"/012*3(45'6 (principle plane) Fundamental Optics *8+F!L1%4=F*"/<4.%TF.<#3BB*:<1'4=F8=&$.89<.&F.9<DF% *#. 7@!%).#L)<!@.%!/%4=FG,@)J9<" 4.%5.$;0)(1 #353$(7OC L:3 #353-.> 1.8.#O4JI"@;"5)J9<"#3<.B188C7/4=F*#=5)$F. #3<.B8CK1J&(N (principal planes) L1%4=F*"/<4.%TF.< #3<.B<=@?38=).#*H:=F5<LH:%4/64.%*>=5%&#(@%*"=5$ (I8F7@!% &J<D%OD%*1@<4.%*"/<L1%G<*:<1') 4JG9@%F.57F!).#&J<$2 4=F#353G):@L)<4(6<'>+@<T/$8CK1J&(N (principal surface) ?3 6/15/2009 2:29 PM Page #.BLB<*HR<#3<.B 4JG9@*#.*#=5)8(<I"@$F. #3<.B8CK1J&(N (principal plane) 1ch_FundamentalOptics_Final_a.qxd 1.29 FRONT FOCAL LENG points นแปลงครั้งที!่ " จุดเปลี่ยof intersection of extended external ray segments. The principal surface of a perfectly corrected optical system is a sphere centered on the จุดเปลี่ยนแปลงครั้งที!# ่ focal point. อากาศ This length is the distan vertex (A1). Fundamental Optics อากาศ แก้ว Near the optical axis, the principal surface is nearly flat, and for this reason, it is sometimes referred to as the principal plane. www.cvimellesgriot.com Gaussian Beam Optics Definition of Terms BACK FOCAL LENGT This length is the distanc point (F″). SECONDARY PRINCIPAL SURFACE FOCAL LENGTH ( fThis term is defined analogously to the primary principal surface, OR it ″ ) used ) FOCAL POINT (F but F is Optical Specifications Two distinct terms describe the focal lengths associated with every lens or left and focused through back focal either focal point must be, on the Rays that pass to the or originate at for a collimated beam incident from the EDGE-TO-FOCUS DI lens system. The effective focal length (EFL) or equivalent focal length opposite side of the lens, parallel to the point F″ on the right. Rays in that part of the beam nearest the axis can be optical axis. This fact is the basis for (denoted f in figure 1.33) determines magnification and hence the image locating both focal points. A is the distance from the thought of as once refracted the tables of size. The term f appears frequently in lens formulas and inat the secondary principal surface, instead of B is the distance from t standard lenses. Unfortunately, because f bymeasured with reference to being refracted is both lens surfaces. PRIMARY PRINCIPAL SURFACE principal points which are usually inside the lens, the meaning of f is not point. Both distances are Let us imagine that rays originating at the front focal point F (and therefore immediately apparent when a lens is visually inspected. parallel to the optical axis after emergence from the opposite side of the lens) PRIMARY PRINCIPAL POINT (H) POINT The second type of focal length relates the focal plane positions directlyOR FIRST NODAL some imaginary surface, instead of twice refracted (once are singly refracted at REAL IMAGE to landmarks on the lens surfaces (namely the vertices) which are immeat each lens surface with happens. There is a unique imaginary This point is the intersection of the primary principalsurface) as actuallythe diately recognizable. It is not simply related to image size but is especially surface, called the principal surface, at which this can happen. is one in w A real image optical axis. convenient for use when one is concerned about correct lens positioning or To locate this unique surface, consider a singlewere placed at theon ray traced from the air point mechanical clearances. Examples of this second type of focal length are the one side of the lens, through the lens and into the air on the other side. front focal length (FFL, denoted ff in figure 1.33) and the back focal length (BFL, denoted fb). SECONDARY PRINCIPAL POINT (H ″ ) The ray is broken into three segments by the lens. Two of these are external (in the air), and the third is internal (in the glass). The external segments VIRTUAL IMAGE OR SECONDARY exception POINT The convention in all of the figures (with the NODALof a single can be extended to a common point of intersection (certainly near, and deliberately reversed ray) is that light travels from left to right. usually within, the lens). The principal surface is the locus of all does no A virtual image such This point is the intersection of the secondary principal surface with the optical axis. ″ CONJUGATE DISTANCES (s AND s″ ) tc secondary principal surface primarydistancespointthe object distance, s, and secondary principal point principal are The conjugate image distance, s″. te primary Specifically, s principal surface from the object to H, and s″ is the distance is the distance ray from object the image location. The term infinite conjugate ratio refers to the from H″ to at infinity ray situation in at infinity from object which a lens is either focusing incoming collimated light or is being r primary vertex A optical axis usedFto collimate a source (therefore, either s or s″ is infinity). 2 1 H front focal point rial Properties A virtual image can be v system, such as in the c r1 PRIMARY VERTEX (A 1 ) f f H″ A2 secondary vertex reversed ray locates front focal point or primary principal surface fb B The primary vertex is A intersection of the primary lens surface with the the f f optical axis. F-NUMBER (f/#) The f-number (also know a lens back focal system is defined point clear aperture. Ray f-num by the height at which it F″ f /# = f . CA NUMERICAL APERTU The NA of a lens system
  6. 6. F″ F″ www.cvimellesgriot.com H″ APPLICATION NOTE F″ F″ H″ F″ H″ H″ Figure 1.36 indicates approximately where the principal points fall in relation to the lens surfaces for various standard lens shapes. The exact positions depend on the index of refraction of the lens material, and on the lens radii, and can be found by formula. in extreme meniscus lens shapes (short radii or steep curves), it is possible that both principal Principal-Point Locations F″ H″ H″ H″ F″ H″ Optical Coatings Page 1.36 2:30 PM Gaussian Beam Optics Optical Specifications Material Properties F″ Fundamental Optics For Quick Approximations Much time and effort Figure 1.36saved by ignoring the differences can be Principal points of common lenses among f, fb, and ff in these Fundamental by assuming that f = fb = ff. formulas Optics 1.36 Then s becomes the lens-to-object distance; s″ becomes the lens-to-image distance; and the sum of conjugate distances s=s″ becomes the object-to-image distance. This is known as the thin-lens approximation. 123=)3(: (Nodal Points) ear APPLICATION NOTE Physical Significance of the Nodal Points A ray directed at the primary nodal point N of a lens appears to emerge from the secondary nodal point N″ without change of direction. Conversely, a ray directed at N″ appears to emerge from N without change of direction. At the infinite conjugate ratio, if a lens is rotated about a rotational axis orthogonal to the optical axis at the secondary nodal point (i.e., if N″ is the center of rotation), the image remains stationary during the rotation. This fact is the basis for the nodal slide method for measuring nodal-point location and the EFL of a lens. The nodal points coincide with their corresponding principal points when the image space and object space refractive indices are equal (n = n″). This makes the nodal slide method the most precise method of principal-point location. ./0)123'456&5/%*$7$*%8($.8 9).:)+-;*<123 =)3(:1>?@A5'45$B5<$:*<.:)+- C,5+D8%(;.:)+-8)* 123=)3(:1>?@A5EF?)'(;$(;123&2G+DH(I .&J5?H5* 3(K)48($.8'45?@A5'(F<+?<3F*)G?<.:)+-&4H5*.34@!$() 123=)3(: N1 and N2 1>?@A5EF?)$(;123&2G +DH(I H1 C:> H2 LF*3(K)48($.8'45?@A5'(F<+?< 3F*)&4H5*C,$,5*<$() 123=)3(:1>C@$,(!??$1*$ 123&2G+DH(I C:> .:J5?)6/'*<3F*)'45&4H5*3(K)4 8($.8+A<$!5* 1*$+&;(,"G?<123=)3(: +*&*%L 8*123&2G+DH(I C:>%>)*;&2G+DH(I63F Optical Specifications 46) F″ H″ Fundamental Optics 6/15/2009 F″ Gaussian Beam Optics 1ch_FundamentalOptics_Final_a.qxd H″ H″ F″ Gaussian Beam Optics 50) F″ H″ Fundamental Optics 48) point is at the curved vertex, and the other is approximately one-third of the way to the plane vertex. Optical Specifications ge 1.35 the lens radii, and can be found by formula. in extreme meniscus lens shapes (short radii or steep curves), it is possible that both principal M
  7. 7. Fund points of intersection of extended external ray segments. The principal surface of a perfectly corrected =,+F!*#=5)$F. ;<"( 1I:"'*B<,' (nodal slide bench) 7JL9<F%K!%?C";<"(:1.8.#O9.I"@?.) *&#+F!%8+!4=F8optical system:is a sphere centered on the focal point. FRONT FO This length vertex (A1). Gaussian Beam Optics =)3(:+6:3-.;)K-./0).H%J5?<&J?'45+D8%(;!*<.:)+-=3@'45 Near the optical )6/&*63F?@5*<H:5 surface C$)8&2)EB5<?@A59)C)!3"5<+*&*%L8&2axis, the principal?< is nearly flat, and for this reason, it is &*@(<%>;;.:)+- C:>LA$=M$( :< ,(!.&J5?98FC+<G)*).GF*sometimes referred to as the+principal plane. BACK FOC ;)CN5)C$F!OF* =3@$*%.:J5?).:)+-98FN5*)C$)8&2)C:> This length i 8&2 ) .:)+- $ :( ; 6/&*%?;C$)8&2 )PRINCIPAL SURFACE SECONDARY LF * +( < .$,.80 ) !5 * point (F″). 123%!&C+<6&5&4$*%.:J5?)??$6/'*<3F*)GF*< C+3<!5* C$)8&2)GP>)(F)N5This term is '45 2 N2 'D$*%!(3%>@> primary principal surface, but it is used *)123=)3(: defined analogously to the 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:29 PM from 1.29left and focused to the back focal %>8!5*<C$)8&2)$(for a)collimated $0HJ? %>@>1*$123 N2 the ;CN5 C$F!OF* EB5< beam incident Page EDGE-TOpoint F″ on the right. Rays3&2G+DH(part of the beam nearest the axis can be in that I LB<123 F2 C:>.)J5?<1*$123=)3(:?@A5EF?)$(;12 A is the dista thought LB 12 F2 $0HJ? '*<@*! 3(<)(F) %>@>%>8!5*<123 N2 of<as3once refracted at the secondary principal surface, instead of B is the dist being refracted length) f surfaces. =M$(++&&A: (equivalent focal by both lensG?<%>;; Fundamental Optics point. Both d .:)+www.cvimellesgriot.com PRIMARY PRINCIPAL POINT (H) OR FIRST NODAL POINT Definition of Terms REAL IMA FOCAL LENGTH ( f ) A real image were placed This point is the intersection of the primary principal surface with the optical axis. FOCAL POINT (F OR F ″ ) Optical Specifications Two distinct terms describe the focal lengths associated with every lens or Rays that pass through or originate at either focal point must be, on the SECONDARY or equivalent focal length lens system. The effective focal length (EFL) PRINCIPAL POINT ( H ″ ) side of the lens, parallel to the optical axis. This fact is the basis for opposite (denoted f in figure 1.33) determines magnification and hence the image VIRTUAL I OR SECONDARY NODAL POINT locating both focal points. size. The term f appears frequently in lens formulas and in the tables of standard lenses. Unfortunately, because f is measured with reference to A virtual im This point is the intersection of the secondary principal surface with the PRIMARY PRINCIPAL SURFACE principal points which are usually inside the lens, the meaning of f is not A therefore Let us imagine that rays originating at the front focal point F (and virtual ima optical axis. immediately apparent when a lens is visually inspected. parallel to the optical axis after emergence from the opposite side of the lens)such system, The second type of focal length relates the focal plane positions directly are singly refracted at some imaginary surface, instead of twice refracted (once to landmarks on the lens surfaces (namely the vertices) which are immeat each lens surface) as actually happens. There is a unique imaginary diately recognizable. It is not simply related to image size but is especially ″ CONJUGATE DISTANCES (s ANDsurface, called the principal surface, at which this can happen. s″ ) convenient for use when one is concerned about correct lens positioning or F-NUMBER To locate this and image distance, s″. mechanical clearances. Examples of this second type of focal length are the The conjugate distances are the object distance, s,unique surface, consider a single ray traced from the air on one side of the lens, through the lens and into the air on the other side. front focal length (FFL, denoted ff in figure 1.33) and the back focal length Specifically, s is the distance from the objectisto H, into three segments by the lens. Two of theseThe f-numbe The ray broken and s″ is the distance are external (BFL, denoted fb). a lens system (in the air), and the thirdratio refers to the The external segments is internal (in the glass). from H″ to the image location. The term infinite conjugate The convention in all of the figures (with the exception of a single can be extended to a common point of intersection (certainly near, and clear apertur deliberately reversedsituationlightwhich a lensto right. ray) is that in travels from left is either focusing incoming collimated light or issurface is the locus of all such usually within, the lens). The principal being !"#"('73#*5 (!"#"8'9:*;<"!"#"=0>) by the heigh Material Properties used to collimate a source (therefore, either s or s″ is infinity). f /# = PRIMARY VERTEX ( A 1 ) tc secondary principal surface secondary surface with The primary vertex is the intersection of the primary lensprincipal point the t primary principal surface optical axis. f CA primary principal point e ray from object at infinity ray from object at infinity primary vertex A1 optical axis F SECONDARY VERTEX ( A 2 ) front focal point r1 back focal point r2 H H″ A2 secondary vertex F″ The secondary vertex is the intersection of thereversed ray locates front focal with secondary lens surface point or primary principal surface the optical axis. f f f A f EFFECTIVE FOCAL LENGTH ( E F L , f ) b B f NUMERIC The NA of a marginal ray with the opti The NA can b with the opt NA = nsinv.
  8. 8. Material Properties 2:29 PM (denoted f in figure 1.33) determines magnification and hence the image locating both focal points. a lens system is define from in to formulas and in the tables term infinite conjugate ratio refers to the size. The term f appears frequently H″lensthe image location. Theof clear aperture. Ray f-nu standard lenses. Unfortunately, becausein is measured with either focusing incoming collimated light or is being situation f which a lens is reference to PRIMARY PRINCIPAL SURFACE principal points which are usually inside the lens, the source (therefore, either s or s″ is infinity). by the height at which used to collimate a meaning of f is not Let us imagine that rays originating at the front focal point F (and therefore immediately apparent when a lens is visually inspected. parallel to the optical axis after emergence from the opposite side of the lens) f The second type of focal length relates the focal plane positions directly are singly refracted at some imaginary surface, instead of twice refracted (once f /# = . PRIMARY VERTEX (A 1 ) immeto landmarks on the lens surfaces (namely the vertices) which are CA at each lens surface) as actually happens. There is a unique imaginary diately recognizable. It is not simply related to image size but is especially surface, called the principal surface, the The primary vertex is the intersection of the primary lens surface withat which this can happen. convenient for use when one is concerned about correct lens positioning or To locate this unique surface, consider a single rayNUMERICAL on traced from the air APER optical axis. mechanical clearances. Examples of this second type of focal length are the one side of the lens, through the lens and into the air on the other side. Page 1.30 (FFL, denoted ff in figure 1.33) and the back focal length front focal length The NA are lens system The ray is broken into three segments by the lens. Two of theseof aexternal (BFL, denoted fb). (in the air), and the third is internal (in the glass). The externalray (the ray SECONDARY VERTEX (A 2 ) marginal segments The convention in all of the figures (with the exception of a single can be extended to a common point of intersection (certainly near, and the of all such deliberately reversed ray) is that light travels from left tois the intersection of the secondary lens surface withsurface iswithlocusoptical axis mu The secondary vertex right. usually within, the lens). The principal the The NA can be defined with the optical axis m the optical axis. 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:29 PM Page 1.29 NA = nsinv. EFFECTIVE FOCAL LENGTH ( EFL, f ) Fundamental Optics t secondary principal surface Assuming that the lens is surrounded by air or vacuum (refractive index www.cvimellesgriot.com primary principal point secondary principal point t front focal point (F) to the primary 1.0), this is both surface the distance from the primary principal Fundamental Optics principal point (H) and the distance from the secondary principal point (H″) www.cvimellesgriot.com ray from object at infinity point (F″). Later we use f to designate the paraxial EFL for back focal to the rear focal point ray from object at infinity r the design wavelength (l0).A primary vertex c Optical Coatings e Definition of Terms optical axis F 2 1 H front focal point FRONT principal H″ r1 F″ A2 secondary vertex reversed ray locates front focal FOCAL LENGTH ( f f ) segments. The point or primary F ″ ) FOCAL LENGTH ( f ) FOCAL POINT (F ORprincipal surface f f a sphere centered on the This length is the distance from the front focal point (F) to the primary f b Two distinct terms describe the focal lengths associated with Optics or 1.30 Fundamental every lens A lens system. The effective focal length (EFL) or equivalent focal length vertex (A1). f (denoted f in figure 1.33) determines magnification and hence the image y flat, and for this reason, f appears frequently in lens formulas and in the tables of size. The term standard lenses. Unfortunately, because f is measured with reference to BACK FOCAL LENGTH ( f f ) principal points which are usually inside the lens, the meaning ofb is not immediately apparent when a lens is visually inspected. Rays that pass through or originate at either focal point must be, on the B opposite side of the lens, parallel to the optical axis. This fact is the basis for f locating both focal points. PRIMARY PRINCIPAL SURFACE Let us imagine that rays originating at the front focal point F (and therefore This length is the distance from the secondary vertex (A2axis after emergence from the opposite side of the lens) parallel to the optical ) to the rear focal The second type of focal length relates the focal plane positions directly are edge thickness point (F″). A = front focus surfaces f = effective which are immete =singly refracted at some imaginaryradius of curvature ofrefracted (once r1 = surface, instead of twice first to landmarks on the lens to front (namely the vertices)focal length; at each lens surface) as actually happens. (positive if center of edge distance may be positive (as shown) surface There is a unique imaginary cipal surface, but it isrecognizable. It is not simply related toor negativebut is especially tc = center thickness diately used image size surface, called the principal surface,curvature is to right) at which this can happen. B = rear edge to rear about correct lens positioning or ocused to the convenient for use when one is concerned = front focal length back focal distance focus f r = radius of curvature of second To locate this from the EDGE-TO-FOCUS focal length are the mechanical clearances. Examples of thisf second type ofDISTANCES (A AND B) unique surface,2consider a single ray tracedcenter ofair on surface (negative if m nearest the axis can length (FFL, denoted f in figure=1.33) and the back focal length be fb back focal length one side of the lens, through the lens and intoisthe air on the other side. curvature to left) front focal f A is the distance from the front focal pointThe ray is primary vertex of theby the lens. Two of these are external to the broken into three segments lens. ncipal surface, (BFL, denoted fb). instead of (in the air), andlens to is internal (in the glass). The external segments the third the B is figures (with the exception secondary The convention in all of the the distance from the of a single vertex of the to a commonrear focal can be extended point of intersection (certainly near, and deliberately point. light travels from left Focal length Both distances to right. Figure 1.33 reversed ray) is thatand focal points are presumed always to be positive. usually within, the lens). The principal surface is the locus of all such T NODAL POINT ncipal surface with the incipal surface with Fundamental Optics REAL IMAGE A real image is one in which the light rays actually converge; if a screen t secondary principal surface were placed at the point of focus, an image would secondary principal point be formed on it. primary principal point c te primary principal surface ray from object at infinity VIRTUAL IMAGE ray from object at infinity primary vertex A1 optical F the axis A virtual image does not represent back focal point r2 an actual convergence of light rays. H H″ A secondary vertex A virtual image can be viewed only by looking back through the optical front focal r reversed point system, such as in the case of a magnifying glass. ray locates front focal , and image distance, s″. H, and s″ is the distance 2 1 ff F-NUMBER A (f/#) f point or primary principal surface fb B f The f-number (also known as the focal ratio, relative aperture, or speed) of a lens system is defined to be the effective focal length divided by system F″ 1.29
  9. 9. MAGNIFICATION POWER APPLICATION NOTE Often, positive lenses intended for use as simple magnifiers are rated with a single magnification, such as 4#. To create a virtual image for viewing Technical Reference with the human eye, in principle, any positive lens can be used at an infinite number of possible magnifications. However, there is usually a narrow range For further reading about the definition of magnifications that will be comfortable for the viewer. Typically, when here, refer to the following publication the viewer adjusts the object distance so that the image appears to be essentially at infinity (which is a comfortable viewing distance for most Rudolph Kingslake, Lens Design Fun individuals), magnification is given by the relationship (Academic Press) MAGNIFICATION POWER 254 mm Rudolph Kingslake, System Design magnification = ( in mm . (1.31) APPLICATION NOTE Often, positive lenses intended for usefas simple)magnifiers are rated with f (Academic Press) a single magnification, such as 4#. To create a virtual image for viewing Technical Reference Thus, the human eye, in length positive lens wouldcan be10# magnifier. with a 25.4-mm focal principle, any positive lens be a used at an infinite Warren Smith, Modern Optical Engi number of possible magnifications. However, there is usually a narrow range For furtherHill). (McGraw reading about the defini of magnifications that will be comfortable for the viewer. Typically, when here, refer to the following publicat DIOPTERS Donald C. O’Shea, Elements of Mod the viewer adjusts the object distance so that the image appears to be (John Wiley & Sons) The term diopter is used(which is a comfortableof the focal length, for most essentially at infinity to define the reciprocal viewing distance which Rudolph Kingslake, Lens Design is commonly used for ophthalmic lenses.the relationship length of a lens individuals), magnification is given by The inverse focal Eugene Hecht, Optics (Addison Wes expressed in diopters is (Academic Press) Max Born, Emil Wolf, Principles of O 254 mm Rudolph University System magnificationgreenstone.org ( f in mm ). (1.31) 1000 = (Cambridge Kingslake, Press) Desi (1.32) diopters = ( ff in mm ). (Academic Press) f If you need help with the use of definit http://www.spaceref.com/telescopes/Magnification-and-Using-Eyepieces.html Thus, a 25.4-mm focal length positive lens would be a 10# magnifier. Warren Smith, Modern Optical E presented in this guide, our application Thus, the smaller the focal length is, the larger the power in diopters will be. (McGraw you. pleased to assist Hill). DIOPTERS DEPTH OF FIELD AND DEPTH OF FOCUS The term diopter is used to define the reciprocal of the focal length, which In an imaging system, depth of field refers to the distance in object space is commonly used for ophthalmic lenses. The inverse focal length of a lens over which the system delivers an acceptably sharp image. The criteria for expressed in diopters is what is acceptably sharp is arbitrarily chosen by the user; depth of field increases with increasing f-number. 1000 (1.32) diopters = ( f in mm ). For an imaging system, depth of focus is the range in image space over f which the system delivers an acceptably sharp image. In other words, Thus, the amount that the image surface (such power in diopters will this is the smaller the focal length is, the larger theas a screen or piece ofbe. photographic film) could be moved while maintaining acceptable focus. Again, criteriaFIELD AND DEPTH OF FOCUS DEPTH OF for acceptability are defined arbitrarily. Donald C. O’Shea, Elements of M (John Wiley & Sons) Eugene Hecht, Optics (Addison Max Born, Emil Wolf, Principles (Cambridge University Press) If you need help with the use of def presented in this guide, our applica pleased to assist you. In nonimaging applications, such as laser focusing, depth of focus refers to In an imaging system, depth of field refers to the distance in object space the range in the system delivers an acceptably sharp image. The remainsfor over which image space over which the focused spot diameter criteria below anacceptably sharp is arbitrarily chosen by the user; depth of field what is arbitrary limit. increases with increasing f-number. For an imaging system, depth of focus is the range in image space over which the system delivers an acceptably sharp image. In other words, this is the amount that the image surface (such as a screen or piece of photographic film) could be moved while maintaining acceptable focus. Again, criteria for acceptability are defined arbitrarily. In nonimaging applications, such as laser focusing, depth ofhttp://hyperphysics.phy-astr.gsu.edu/hbase/vision/accom2.html focus refers to the range in image space over which the focused spot diameter remains below an arbitrary limit.
  10. 10. ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.3 Paraxial Ray Approximation Fundamental Optics www.cvimellesgriot.com Paraxial Formulas Sign Conventions The validity of the paraxial lens formulas is dependent on adherence to the following sign conventions: For lenses: (refer to figure 1.1) For mirrors: s is = for object to left of H (the first principal point) f is = for convex (diverging) mirrors s is 4 for object to right of H f is 4 for concave (converging) mirrors s″ is = for image to right of H″ (the second principal point) s is = for object to left of H s″ is 4 for image to left of H″ s is 4 for object to right of H m is = for an inverted image s″ is 4 for image to right of H″ m is 4 for an upright image s″ is = for image to left of H″ m is = for an inverted image m is 4 for an upright image Paraxial Rays and Paraxial Formula When using the thin-lens approximation, simply refer to the left and right of the lens. rear focal point front focal point Figure 1.1 h object f v F H H″ CA F″ image f h″ f s s″ principal points Note location of object and image relative to front and rear focal points. f = lens diameter CA = clear aperture (typically 90% of f) f = effective focal length (EFL) which may be positive (as shown) or negative. f represents both FH and H″F″, assuming lens is surrounded by medium of index 1.0 m = s ″/s = h″ / h = magnification or conjugate ratio, said to be infinite if either s ″ or s is infinite v Figure 1.1 = arcsin (CA/2s) Sign conventions s = object distance, positive for object (whether real or virtual) to the left of principal point H s ″ = image distance (s and s ″ are collectively called conjugate distances, with object and image in conjugate planes), positive for image (whether real or virtual) to the right of principal point H″ h = object height h ″ = image height
  11. 11. 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM (object and image distance). Th 1ch_FundamentalOptics_Final_a.qxd 6/1 position, and image position is g Page 1.3 1 1 1 www.cvimellesgriot.com = + s s″ f 2:28 PM Page 1.4 1.4 Fundamental Optics Fundamental Optics Sign Conventions Gaussian Beam Optics K(@<7!<L#)G<).#L)@H(N9.4.%4(6<6.17#'&+!).# *:+!)&$.85.$;0)(1G9@*98.3)(BK@!?J)("K!%#3BB *,F< )J:(%K5.59#+ #353 conjugate (#353$( Paraxial!Formulas 7OCL:3#353-.>) 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 7.8&$.81(8>(<A'K!%18).# This formula is referenced to f 2:28 PM Page 1.4 in Sign Conventions. 1ch_FundamentalOptics_Final_a.qxd on adherence to the following sign conventions: The validity of the paraxial lens formulas is dependent 6/15/2009 For lenses: (refer to figure 1.1) For mirrors: s is 4 for object to right of H f is = for convex (diverging) mirrors step in optical By definition,Typically, the firstconstraints suchproblm magnification isasth length based on f is 4 for concave (converging) mirrors 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.3 s″ is = for image to right of H″ (the second principal point) s is = for object to left of H s″ is 4 for image to left of H″ s is 4 for object to right of H (object and image distance). The rela position, and image position is given Fundamental Optics s″ h″ Paraxial Formulas = m = 1=1+1 . f s Typically, the firstTypically, the first step solving is to select a solvingfocal select a system focalh s s″ step in optical problem in optical problem system is to m is = for an inverted image m is 4 for an upright image 1ch_FundamentalOptics_Final_a.qxd s″ is 4 for image to right of H″ 6/15/2009 www.cvimellesgriot.com s″ is = for image to left of H″ 2:28 PM Page 1.4 m is = for an inverted image Fundamental Optics Fundamental Optics Fundamental Optics s is = for object to left of H (the first principal point) Gaussian Beam Optics Sign Conventions length based on constraints such as magnification or as magnificationan upright image distances m is 4 for or conjugate length based on constraints such conjugate distancesdependent on adherence objectThis formula is referenced to figure The validity of the paraxial lens formulas is to the following sign conventions: object (object and image distance). The relationship among relationshiptoamong focal length, mirrors: focal length, object For object (object and image distance). The For lenses: (refer figure 1.1) in Sign Conventions. s is for object to of (the first f is = for convex (diverging) mirrors When using the Typically, the first stepsimply refer problem solving is to left select a system focal thin-lens approximation, in optical to the left and=right of the Hlens. principal point) position, and image position is given by s is for position, and image position is given4byobject right of H length based on constraints such as magnificationtoor conjugate distances f is 4 for concave (converging) mirrors s″ is = for image to right of H″ (the second principal point) s is = for object to left of H By definition, magnification is the rat object F1 (object and image distance). The relationship for image to left of H″ length, object s is 4 for object to right of H 1 1 1 s″ is 4 among focal = + position, and 1 (1.1) 1 1 m is = for an inverted image s″ is 4 for image to right of H″ = image (1.1) s s″ focal point + position is given by m is 4 for an upright image point f rear focal front h″ s″ is = for image to left of H″s ″ s s″ f m= = . m is = for an inverted image F s h 1 1 1 h m is 4 for an upright image = + (1.1) 200 This formula is referenced to figure 1.1 and the sign conventions given f When using the thin-lens approximation, simply refer to the left and right of the lens. This s s″ object in Sign Conventions. formula is referenced to figure 1.1 and the sign conventions given This relationship can be used to recas H H″ f CA in Signv Conventions. to figure 1.1 and the F″ conventions given 200 This formula is referenced F forms: Figure 1.2 Example 1 (f = 50 mm, s = 2 ;"5</5.8 magnification is the ratio &+!1( size to h sign (magnification) By definition,)J:(%K5.5Conventions. of image"1F$< object size or 2 in Sign Gaussian Beam Optics This relationship can be used to forms: Fundamental Optics ).#H#38.2K!%*:<1'B.% (Thin-Lens Approximation) 1 Gaussian Beam Optics Gaussian Beam Optics object Optical Specifications (s + s ″) f =m By definition, magnification the ratio of is to select a to object m 1.2 ( Figure 1.2 Exam Typically, the first 7OC &+ optical #39$F.%K<."K!%-.>7F!K<."K!%$(step in ! isproblem solvingimage size system focalsize(or + 1)sExample 1 (f = 50 mm, + s ″) Figure rear focal point front focal point h″ image H H″ F CA F″ sm f = m +1 Optical Specifications Optical Specifications v image ″ ″ principal points Note location of object and image relative to front and rear focal points. f = lens diameter = object distance, positive for object (whether real or virtual) to the left of principal point H CA = clear aperture (typically 90% of f) = effective focal length (EFL) which may be positive (as shown) or negative. represents both FH and H″F″, assuming lens is surrounded by medium of index 1.0 = v f = ″ = ″ / = magnification or conjugate ratio, said to be infinite if either ″ or is infinite = arcsin (CA/2 ) s + s″ 1 ″ = image distance ( and ″ are collectively called conjugate distances, with object and image in conjugate planes), positive for image (whether real or virtual) to the right of principal point H″ = object height 1 m+2+ m ″ = image height Optical Coatings Gaussian Beam Optics f Material Properties s( m + 1) = s + s ″ l Properties where (s=s″) is the approxima ecifications Optical Specifications Optical Specifications s ″ hBy definition, magnification is the ratio of image size to object size or Example 2:fObject inside Focal Point ″ length based on constraints such as magnification or conjugate distances =m (1.2) m= = . object ( m + 1)2 h f f Example 2: Objec (objects ″ image distance). The relationship among focal length, object and h ″ s h m =″ and image position is given by = . The same (1.2) is2: Object inside Focal Po object Example placed 30 mm left of the s h″ f f position, (1.2) m= = .h sm s s s″ s s f = is the image formed, same lens. Where object is placed 30 mm and p s h The same objectleft o is w This relationship can be used to recast the first formula into the following The same m + 1 F 1 1 1 principal points (See same lens. Where is ?.)&$.81(8>(locationrelationshipimage be to recast5theand rear focal points. following figure 1.3.) Where is the image formed, a <This of object and be *#.1.8.#O*K= < first formula into the A'K!% = + used used front (1.1) s same lens. forms: Note This relationship can can relative to to recast the first formula into the following s s″ f s + #353 1J9#(B*:<1'?#/%4=F8=&$.89<.&F.9<DF% figure 1.3.)(See figure 1.3.) #353-.> s ″ 1 s (See f = s 1 1 18).#!+F<fQ I"@"forms: (%<=@ = forms: lens diameter s = object distance, positive for object s(whether real f = − 1 f ( s + s ″ ) This formula is referenced to figure 1.1 and the sign conventions the left″ principal 301 H 2 + 1 200 1 or virtual) to given ?3!@ 50 1 point + 1 $(7OC L:3&$.85.$;0)(1s of .%!/%?.) m principle 1 = − f = m CA = clear aperture (typically 90% of f) m = − (1.3) s + s ″) s 50 75″ 1.1.) MDF% m s /s h h ( m + 1)2fin=Sign(Conventions. points = (?C" Hdistance (H”s ″ G<″ are collectively calledI8Fs,F 50 30 L:3 s andhs= −Fig.mm 30 G ″ s″ image m (s + s ″) 2 (EFL) which may be positive (1.3) with object = −+ 1) = in + s ″ f = effective= m length f focal + 1) s s conjugate distances, h s s( and75 Example ″ m image s (m Figure 1.2 mm By definition, magnification is both FH of image <5' toconjugate planes), positive″ image .%#39$F. ″ 1 (f = 50 mm, object size or for sm (as shown) or negative. )2representsthe ratio and ?C"6Psize):.%K!%*:<1' m .*#.I8F&/"−75 (whether s% = −75 mm (m + 1f O@ = s (1.3)#3539F= −755 real s = s ″ H″2. H″F″, assuming lens is surrounded by medium (1.4) f = or virtual) to the right of principal point − sm s m= = m + 1 of index 1.0 sm h ″ (1.4) f = Sign conventions Figure 1.1 principle points (*#=5)$F. hiatus)Example 2: Objectapproximate obj #353 30s 30 = < = = −75 = ?3*HR inside where (s=s″) is the−2.5s ″ Focal Po m + s″ m (1.2) m=1 = . (1.4) f ″ = = magnification or or virtual image is 2.5 mm high and upright. h = object height s s m +=s ″ ″/s = h / h s 1 h 30 imageFundamental Optics 1.3 is 2.5 #353#39$F.%$(7OCOD%-.> ).#I8ForThe sameJG9@#3BB*HRplacedthickness, the &/"virtualrealobjectof finite highmm left o #353<=@4 lens is <mm s30 and uprig m +said s+ f = With a (1.5) conjugate ratio,s ″ to be infinite if (1.5) 1 f =″ or s is infinite same lens is h″ This relationship into following LBB!5F=.%%F.5 theL:3*#.*#=5)).#H#38.2<=Whereallisreferenced to thea m + 2 +either s m + 2s++1 ″can be used to recast the first formulaimage height In this case, thelens. theor ).#used as a magni focal case, @$F. lens the image formed, are virtual image is 2.5 s In this lengthis being being used as a m m forms: (See figure 1.3.) the f = (1.5) m viewed only back By neglecting the center 1 v = arcsin (CA/2s) H#38.2K!%*:<1'B.% viewed only back the lens.through the lens. lens (thin-lens 1 of throughthis lens. the dist approximation) case, 1 In 1 s( m + 1) = s + s (″m + 1m + 2 s ″ (1.6) s )=s++ (1.6) − known = 6the hiatus, s=s″ becomes t as <6.17#'I"@ m 4JG9@*#.1.8.#O&J<$2&F.7F.%Q G<#3BB4( 50 viewed only back thr (s + s ″) s″ 30 f =m 2 (1.3) plification, called the thin-lens appro Figure 1.1 Sign conventions is = +approximate object-to-image distance. s ″ = −75 mm s (s=s″) O s + (1.6) where (s=s″) iswhere ( m + 1)OCthe -.> ″ the approximate object-to-image distance.!5F.%%F.5 *8+F! *HR<#353?.)$(7 ( mD% 1) s (;"5H#38.2) when dealing −75 simple optical syst s ″ with sm m= = =− With a real = of finite image distance, object distance, and thickness, the image distance, object distance, and (1.4) f lens Fundamental Optics 2.5 1.3 With a real lens of finite thickness, the approximate object-to-image distance. s 30 m + is the where (s=s″)1 referenced to the principal points, not to the physical Example 1: Object outside Foc focal length are all focal length are all referenced sto the principal points, not to the physical or virtual image is 2.5 mm high and uprig + neglecting the distance between the lens’ principal points, s″ center of the lens. By = the of finite thickness, lens’ principal points, (1.5) With a freal lensdistance between the the image distance, object distance, and center of the lens. By neglecting A 1-mm-high objectFis placed on the 1 With a real lens of finite thickne
  12. 12. Gaus Optical Sp Example 2: Object inside Focal Point In this case, the lens is being used (1.2) m Typically, the first step in optical problem solving is to select a system focalThe same object is placed 30 mm left of the left principal point of back through the lens. viewed only the m= m+2+ s″ h″ = . s h same lens. Where is the image formed, and what is the magnification? This s( m as magnification or conjugate distances(See figure 1.3.) Page 1.4 length based on constraints relationship1) = used to recast the first formula into the following such + can be s + s ″ (1.6) forms: 7($ s1″ 50 − 30object (object and $!5F.%distance).4=F!5PFI+):!!)IH?.)?C";0)(1 length, object!5F.=%1 2:1 $(7OC4=F-.5G<#353;0)(1 image 1: $(7OC Thes relationship among focal 7( ( s ″) = wheregiven) by (1.3) s ″ = −75 (m + 1 position, and image positionf is m(s=s″) is the approximate object-to-image distance.mm 2:28 PM 2 s″ 75 $(7OC*"=5$)(=< distance, and $(7OC4=F8=&$.81P%With=asm !5PFB<L)<4(6<'thickness, axis) 1 f mm lens of finite (optical the image distance,mobject−!5P=F4−=F7.JL9<F% 30 mm 4.%M@.5K!%?C" = 25 select a system focal real s 30 Fundamental Optics F m +1 1 1F#353 1 conjugate distances 200 mm ?.)4.%M@.5K!% referenced to the principal principle image isthemm high5$)(< )J:(%K5.5K!%#3BB?38= s + s″ = 4= + focal length are all principle points points, not to 2.5 physical (1.1)or virtual K!%*:<1'*"= and upright. www.cvimellesgriot.com f = object s f In this case, focal length, object s″ 4=F8=&$.85.$;0)(+11lens. By neglecting%-.>L:3 between*HR<*4F.G"the lens is being used as a magnifier, and the image can be centerm + them 50 mm 7JL9<F the distance of 2 principal points, &F. the lens’back through the lens. K!%*:<1' viewed only ( + )=s known1as the F image )J:(% referenced sOCmfigure+ s ″hiatus, s=s″ becomes the object-to-image distance. This sim200 This formula is K5.5K!%$(7to!5PF4=FG" 1.1 and the sign conventions given Optical Specifications (1.4) 1 (1.5) (1.6) in Sign Conventions. Material Properties g is to select a system focal (1.1) ion or conjugate distances mong focal length, object 2 plification,the approximate object-to-image distance. F1 where (s=s″) is called the thin-lens approximation, can speed up calculation when dealing with simple optical systems. and With a real lens of finite thickness, the image distance, object distance, focal length are all referenced to the principal points, not to the physical center of the lens. By neglecting the distance between the lens’ principal points, image known as the hiatus, s=s″ becomes the object-to-image distance. This sim200 F2 plification, called F1 thin-lens approximation, can speed up calculation the when dealing with simple optical systems. object By definition, magnification is the ratioObject outside Focal Point Example 1: of image size to object size or n conventions given Figure 1.2 66.7 F1 F1 object image Example 1 (f = 50 mm, s = 200 mm F2 Material Properties Optical Coatings Optical Coatings Example 2 (f = 50 (1.1) image of the 2: A 1-mm-high object is placed on the optical axis, 200 mmExampleleft Object inside Focal Point left s″ h″ (1.2) m= = . principal point of a 1 (f66.750 mm, s = 200 mm, s″ 66.7 is the Example 1: Example Focal = Point Figure 1.2 200Object outside LDX-25.0-51.0-C (f = 50 mm).=Wheremm) image s he sign conventionsor o object size given h Example 3: Example 2 (f object is placed 475 mm) left of at left prin Figure 1.3 The same = 50 mm, s = 30 mm, s″= 30 mmObject the Focal P A 1-mm-high object is placed on the optical axis, 200 mm left of the left formed, and what is the magnification? (See figure 1.2.) principal (f of amm, s 200 mm,(f 66.7 Where mm). 50 LDX-25.0-51.0-C same lens. Where is A 1-mm-high object and whaton th the image formed, is placed is th e size This relationshipFigure 1.2 used to1recast the first formula mm) the following to object size or Example 3: Object at Focal Point can be Example point=Object=insides″==50figureinto is the image Example 2: is the magnification? (See Focal Point formed, and what 1.2.) (1.2) A 1-mm-high(See placed on the optical object 50 mm left the first $/A=4J ?.)1P7# isfigure 1.3.) axis,principalofpoint of an LDK-50.0-52.2 $/A ?.)1P 2: Object inside1Focal 1 forms: (1.2) =4J Example7# 1 1 Point 1 1= principal point of the The same= object − of the left principal point of the the left principal point ofan LDK-50.0-52.2-C (f =450 mm). Where is the image is − mm left placed 30 mm left of The same object is placed 30 f 1 s s s″ ″ f s formed, and what is 1 magnification? 1 figure 1.4.) the (See formed, and what is the magnificat Where formula into thefollowing same lens. Wherelens.image formed, and what is the magnification? a into the following s + (See ) same is1the 1 1 is the image formed, and what is the magnification? = 50 − 30 1 1 1 s″ ( s ″ figure 1.3.) 1 − 1 = − 1 1 1 1 (See 1 s″ = 50 200 figure 1.3.) f =m 1 1 (1.3) s″ −50 50s ″ = −75 mm = − ( m + 1)s2 = 50 − 30 s″s=″1 = mm − 200 s ″ = −25 mm 66.7 50 ″ s ″ −50 50 1 1 (1.3) s ″ −25 s ″ = −75 mm = s ″ − .7 66 m= = = −0.5 s ″ −75 = mm m =50 =66.7 0.33 s.″ = −25 mm sm 30 s 50 m = s ″ s75 s ″ s= 200 −″ = = −2 5 m= = = −2.5 (1.4) f = (1.4) (1.3) s s30 = −75 mm s 30 ″ real image is 0.33 mm high and inverted. s ″ −25 m + 1 or virtual imageor 2.5 mm high″ upright.7 or virtual image is 0.5 mm high and upright. s and 66. is m= = = −0.5 = 0.33 m″= −75 = (1.5) s upright. I"@-.>*18+!or virtual image ismm mm high and50 <9($7(@%K<." 2.5 2.5 s s + sIn this case, the lens is being used as a2002.5and the image can be ″ magnifier, m= =s f = (1.4) viewed only back through the lens. = − (1.5) s 30 1 G<)#2=<=@*:<1'?3OP)G,@LBBL$Flens is being used!%OP)magnifier, and Fundamental Optics 1.4 (1.6) In this case, the <K5.5L:3-.>?37@ as a0.5 mm high and m +!)J:(%K5.58=Kreal image 2.5 mm mm -.>9($):(B or virtual image is or <." 0.33 is 0.33 high and and inverted. 9#+ 2 + m or virtual image is mm L:3*HR< high upright. ge distance. viewed only back through the lens. 8!%?.)"@.<9:(%K!%*:<1'*4F.<(@< (1.5) In tance, object distance, and = s + s ″ this case, the lens is being used as a magnifier, and the image can be s( m + 1) (1.6) Figure 1.3 object points, not to the physical viewed only back through the lens. een the lens’ principal points, 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM F1 F2 (1.6) o-image distance. This simobject can speed up calculation Page 1.5 where (s=s″) is the approximate Fundamental Optics object-to-image distance. 1.4 7($!5F.% 3: $(7OC4=F?C";0)(1 image tance. a real lens of finite thickness, the image distance, object distance, and With Fundamental Optics $( 1P% all $.81P% 1 the 7(@%!5P points, 6<' !% 4(@%<= (rays) 5( focal length7are4=F8=&referenced to mmprincipalFB<L)<4(notKto the physical @$/A=).#*K=5<#(%1= Fundamental%1.8.#OG,@9.7JL9<F% Optics *:<1'*$@.&$.85.$;0)(1 50 mm 4=F#353 50 mm principal points, %K5.5K!%-.>G<#3BB paraxial system I"@ L:3)J:( center of the lens. By neglecting the distance between the lens’?.) www.cvimellesgriot.com F known as the hiatus,point 4=F 1 K!%*:<1' 7JL9<F%-.>?3*)/"KD@< This sim- $/A=).#*K=5<<=@KD@<!5PF)(B18B(71/1!%H#3).#K!% s=s″ becomes the object-to-image distance. principle ).#G,@ object F plification, called the thin-lens approximation, can speed up calculation 4=FG" L:3)J:(%1 1 <*4F.G" K5.5*HR #3BBI"@L)F object image peed up calculation when dealing with 1simple optical systems. = − Figure 1.3 Example 2 (f = 50 mm, s = 30 mm, s″= 475 mm) xis, 200 mm left of the left object distance, and mm). Where is the image ,gure 1.2.) the physical Example 3: Object at Focal Point not to lens’ principal points, A 1-mm-high object is placed on the optical axis, 50 mm left of the first principal point of an LDK-50.0-52.2-C (f =450 mm). Where is the image F1 ge distance. This sim- formed, and what is the magnification? (See figure 1.4.) 2 1.#(%1=7@!%*"/<4.%?.)K<.<)(BL)<4(6<' TF.<*:<1' Example 1: Object outside Focal Point L:3:PF*K@. (9#+!:PF!!)1J9#(B*:<1'*$@.) ?C";0)(1 Figure 1.3 Example 2 (f = 50 mm, s = 30 mm Figure 1.3 Example 2 (f 50 mm s = of mm, s″= %1=7@!%*"/< A 1-mm-high object is placed on the upright. axis,=200 mm, left 30the left 475 mm) 4.%*K@.1PF 1st principle point K!% optical 2.#( 0 mm left of the left or virtual image is 0.5 mm high and principal point L:37@!%*"/< 3: Object Focal Point Where is the image of a LDX-25.0-51.0-C (f = 50 mm). Where is the image #3BB Example 4.%!!)?.)at2nd principle Example 3: Object at Focal Point .2.)formed, and what is the magnification? (See figure 1.2.) point A K!%#3BB G<4/64.%*"=5$)(<)(B4=F*K@.optical axis, 50 m 8. 1-mm-high object is placed on the A 1-mm-high=object iss placed on 425 mm) the optical axis, 50 mm left ofand numerical aperture Figure 1.5 F-number the first Figure 1.4 Example 3 (f 450 mm, = 50 mm, s″= ():F.$&+image 8 point J)(BL)<4(6<'?37@!%*HR< (f ! 8C 1 1 1 principal point of an LDK-50.0-52.2-C (f =450 mm). Where is theprincipal4=F#(%1=4of an LDK-50.0-52.2-C8C8=450 mm). W = $/A− J A simple graphical method can also be used to determine paraxial image ?.)1P7# and what is Ray f-numbers can *"=5$)(<4( for any arbitrary ray formed, 1.4.) also be defined s ″ f =4 slocation and magnification. This graphicalthe magnification? (See figure and the diameter at which@%it*K@.L:3!!)) if its is the magnification? (See figure 1.4 formed, and what conjugate distance intersects the principal surface of approach relies on two simple the optical system are known. properties of an 1 optical system. First, a the system parallel 1 $/A=).#<=@OP)G,@G<4(@%1.87($!5F.%4=FTF.<8.G<#PH 1.2-1.4 1 1 1 optical axis crosses the optical1 rayatthat enterspoint. Second, a ray 1 1 1 = − axis the = − to the the first principal point of50 system focal the system from the = − that enters exits s ″ −50 the s ″ 50 200 principal point parallel to its original direction (i.e., its exit angle &$#>D%#3$(%I$@$sF.″ −50 50 approximation ?3 ).#G,@ thin-lens second s ″ = −25 mm NOTE ″ = −25 mm with the optical axis is the same as its entrance angle). This method has s ″ = 66.7 mm applied to the three previous examples illustrated in figures 1.2 4JG9@18B(7/K@!4=s !%%F.5KD@<*<+F!%?.) 1st principle F1 been s ″ −25 Because the sign convention given previously is not used universally m= = = −0.5 through 1.4. Note that by using the thin-lens approximation, this second in all optics texts, the reader may notice differences in the paraxial s ″ 66.7 point L:3 2nd m = s ″ =point *HR<?C"*"=5$)(< principle −25 = − 50 formulas. However, results will be correct as long as a consistent set 0.5 = 0.33 s m = = property reduces to the statement that a ray passing through the center s ″ −50 50 s ″ = −25 mm object s ″ −25 m= = = −0.5 s 50 F2 image image f CA 2 v F1 Gaussian Beam Optics principal surface of the 200 lens is undeviated. 9#+!?3I"@-.>*18+!<9(image is 0.50.5 mm and upright. or virtual $7(@%K<." mm high F-NUMBER AND NUMERICAL APERTURE or real image isThe paraxial calculationsand inverted.the necessary element 0.33 mm high used to determine diameter are based on the concepts of focal ratio (f-number or f/#) and numerical aperture (NA). The f-number is the ratio of the focal length of the lens to its “effective” diameter, the clear aperture (CA). f-number = f . (1.7) of formulas and sign conventions is used. s 50 Optical Specifications s or virtual image is 0.5 mm high and upright.
  13. 13. second principal point parallel to its original direction (i.e., its exit angle with the optical axis is the same as its entrance angle). This method has been applied to the three previous examples illustrated in figures 1.2 through 1.4. Note that by using the thin-lens approximation, this second property reduces to the statement that a ray passing through the center of the lens is undeviated. f-number = f . CA NOTE (1.7 Because the sign convention given previously is not used uni in all optics texts, consider a lens notice positive focal the p To visualize the f-number, the reader may with a differences in lengt formulas. However, results light. The f-number defines consis illuminated uniformly with collimatedwill be correct as long as athe ang of the coneformulas and sign lens which ultimately forms the image. Th of of light leaving the conventions is used. F-NUMBER AND NUMERICAL APERTURE is an important concept when the throughput or light-gathering powe of an optical system /5.8 cone focusing G<4J<!%*"=5$)(<*#.1.8.#OG9@is<critical, such as whenangle light into a mono ).#&J<$<LBB paraxial 4=FG,@NUMERICAL.APERTURE F-NUMBER AND &J<$2&F.7F %?3KD@<!5PF chromator or projecting a high-power image. "@$5&F. Numerical Aperture (NA) MDF%*HR<&F. sine )(BL<$&/"K!%The paraxialratio (9#+!*#=5)F.to f-numberthe necessary element focal calculations used determine The other term used commonly in defining this cone angle is numeric are based on focal K!%8C 9#+! f/#) diameter "1F$<#39$F.the concepts of the ratio of (f-number or f/#)#39$F.%aperture. The NA ray sine of the angle made byHthe marginal ray wit MDF%*HR<1( aperture (NA).%&$.85.$;0)(17Fratio the focal length8ofand marginal is the )(BL)<4(6<' ?.)#P ! numerical The f-number is the the 5<&F NA referring to figure “effective” diameter “effective”aperturethe clear aperture (CA). 1.5 *#.1.8.#O*K=optical.axis. ByI"@7.818).# 1.5 and using simple trigonometr lens to its = clear diameter, (CA) it can be seen that 2:28 PM f-number = Page 1.5 f . CA (1.7) NA = sinv = CA 2f (1.8 To visualize the f-number, consider a lens with &F. f-number G,@</5.88C8*H/" (angle of cone 9#+! a positive focal length and illuminated uniformly with collimated light. The f-number defines the angle 1 cone angle) of the cone ofF*"/<4.%!!)?.)#3BB*:<1'L:3 forms the image. This NA = 2(f-number ) . K!%L1%4= light leaving the lens which ultimately Fundamental Optics is an "<=@1J&( concept F!).##$8L1%K!% www.cvimellesgriot.com 1#@.%-.> L<$&/importantN8.)*8+when the throughput or light-gathering power of an optical system is critical, such as when focusing light&F. a monointo f-number 1.8.#O</5.8I"@*18!1J9#(B#(%1=G"Q #3BB4(6<6.17#'*K@.G):@?C"$/)U7 *,Fhigh-power image. F chromator or projecting a < ).#;0)(1L1%1P O@. monochromatorother!term used%-.>4=FOP)in defining this cone angle is*#.#P@#353 conjugate L:3*1@<TF.<6P<5'):.%4=F>+@<T/$ The 9#+ ).#1#@. commonly K5.5KD@< numerical 9<@.K!%#3BB4(6<6.17#' aperture. The NA is the sine of the angle made by the marginal ray with (1.9 Fundamental Optics the optical axis. By referring to figure 1.5 and using simple trigonometry, f CA it can be seen that 2 1ch_FundamentalOptics_Final_a.qxd v 1ch_FundamentalOptics_Final_a.qxd F 7/6/2009 1:42 PM CA NA = sinv = 1 7/6/2009 Page 1.6 2f Gaussian Beam Optics principal surface and F-number and numerical aperture Figure 1.5 mine paraxial image relies on two simple s the system parallel l point. Second, a ray s the system from the on (i.e., its exit angle le). This method has trated in figures 1.2 imation, this second g through the center Ray f-numbers can also be defined for any arbitrary ray if its conjugate distance and the diameter at which it intersects the principal surface of the optical system are known. NA = 1 . 2( f-number ) (1.9) Fundamental Optics www.cvimellesgriot.com Imaging Properties of Lens Systems Fundamental Optics Fundamental Optics mm, s″= 425 mm) 98.5*97C Page %1.6 optics 7F.%Q ).#*:+!)G,@ G<9<( 1+! *&#+F!%98.5 + 9#+! - !.?8=&$.8L7)7F.%)(<IH (7.8 (1.8) 8C88!%K!%TP@*K=5<) L7FT:4=FI"@?37@!%*98+!<)(< 1:42 PM Imaging Properties of Lens Systems Fundamental Opti NOTE )J:(%K5.5K!%#3BB8=&F.*4F.)(B1("1F6/15/2009 NA K!%$(7Page 1.7 $<#39$F.% 2:28 PM OC in all optics texts, the reader may notice differences in the paraxial To understand the importance of the NA, consider its relation to magnification. Two very! NAapplications of simple optics involve coupling light into 7F common K!%-.> T:<=@I8FKD@<!5PF)(B#3BB L:31.8.#OG,@ formulas. However, Referring to figure 1.6, results will be correct as long as a consistent set an optical fiber or into the entrance slit of a monochromator. Although these THE OPTICAL of formulas and sign conventions is used. INVARIANT Example: System with &J<$29.&F.*1@<TF.<6P<5'):.%K!%*:<1'4=F<@!54=F1C"G<)#2=4=F Fixed Input problems appear to be quite different, they both have the same limitation CA NA (object side) = sin v = 8= relation *)= magnification. I"@ (1.10) To understand the importance of Page consider its K="?J)("to F5$)(BK<." aperture Two usually expressed 2s 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM the NA,1.7 — they have a fixed NA. For monochromators, this limit is very common applications of simple op side) = sin v = CA = 2s sin v (1.12) and NA″ (image side) = sin v ″ = Gaussian Beam Optics CA = 2s ″ sin v ″ leading to CA 2s CA 2s ″ (1.13) which can be rearranged to show s ″ sin v NA = = . s sin v ″ NA ″ CA = 2s sin v (1.14) s″ Since is simply thand e magnification of the system, s we arrive at m = NA . NA ″ CA = 2s ″ sin v ″ leading to (1.15) s ″ sin v NA = = . s s sin v equal to the ratio of the NAs NA ″ The magnification of the system is therefore″ *8+F!#3BB*:<1'OP)<J8.G,@G<).#1#@problems *:<1'4=F8to be quite different, they .%-.> appear =*1@<TF.< 6P<5'):.%G9NF?35!8G9@#(%1=TF.<I"@8.)KD@< (CA 8.)KD@<) 4JG9@ *#.I"@-.>4=F1$F.%8.)KD@< L7F*>#.3&$.81(8>(<A'#39$F.%)J:(% K5.5)(B CA !.??38=K="?J)("4.%4UEV=4=F4JG9@).#*>/F8K<." K!%#3BBI8F8=T:7F!&$.81$F.%K!%-.> it is necessary, using a singlet le Suppose Suppose it is necessary, using a singlet lens, to couple the output of an (1.10) — they optical fixed incandescent bulb with a filament 1 mm in diameter into anhave afiber NA. For monochromato as shown in figure 1.7. Assume that the fiber has terms diameter of in a core of the f-number. In addition to the f 100 mm and an NA of 0.25, and that the design requires that the total Fundamental Optics entrance pupil (image) size. (1.11) distance from the source to the fiber be 110 mm. Which lenses are www.cvimellesgriot.com appropriate? By definition, the magnification must be 0.1. Letting s=s″ total 110bulb with a filament 1 mm in incandescent mm (using the thin-lens approximation), we can use equation 1.3, *<+F!%?.) NA = CA/2s O@.&$.85.$;0)(1L:3)J:(%K5.5OP)Assume that the (1.12) as shown in figure 1.7. ) ( s + s ″$ &F. NA )R?3OP)G,@)J9<" CA "@$5*,F5)(< "(%<(@< 100 mm(see eq. 1.3) of 0.25, and that the and an NA )J9<"L:@ , f =m (m + 1) sdistance from the source to the fiber be 9.)#3BBI"@)J9<"&F. NA L:@$ ).#*>/F8K<."K!%#3BB*:<1')R CA (1.13) 2 to determine that the focal length is 9.1 mm. To .%G" *97C).#2'<=@*#.*#=5)$F. appropriate? ?3I8F,F$5*>/F818##O<3L7F*>=5%!5Fdetermine the conjugate v distances, s and s″, we utilize equation 1.6, CA optical invariant By definition, the magnification must be 0. 2 we arrive at ecifications When a lens or optical system is used to create an image of a source, it is NA object side Fundamental the lens, 1.5 natural to assume that, by increasing the diameter (f) ofOptics thereby m = . increasing its CA, we will be able to collect more light and thereby produce NA ″ a brighter image. However, because of the relationship between magnification and NA, there can be a theoretical limit beyond which increasing the Gaussian Beam Optics (using the thin-lens approximation), we ca (see eq. 1.6) s ( m + 1) = s + s ″, s″ (1.14) CA object side on the object and image sides of the system. This powerful and useful result 2 s″ (s + s ″) mm. vSince is completely independent of the specifics of the optical system, and it can is simply the magnification and thethat s = 100 mm and s″ = 10 v″ of find system, f =m , CA s often be used to determine the optimum lens diameter in situations (m + 1)2 We can now use the relationship NA = CA/2s image = CA/2s″ to derive or NA″ side (1.9) involving aperture constraints. CA, the optimum clear aperture (effective diameter) of the lens. (1.8) Optical Coatings Optical Specifications e angle is numerical he marginal ray with simple trigonometry, (1.11) 2s ″ which can be rearranged NA (object to show in terms of the f-number. In addition to the fixed NA, they both have a fixed an optical fiber or into the entrance slit of a m entrance pupil (image) size. Fundamental Optics positive focal length mber defines the angle orms the image. This ght-gathering power ng light into a mono- ReferringCA figure 1.6, to NA″ (image side) = sin v ″ = Material Properties Gaussian Beam Optics (1.7) 1ch_FundamentalOptics_Final_a.qxd Example: System with Fixed Input NA Optical Specifications necessary element f-number or f/#) and he focal length of the A). THE OPTICAL INVARIANT given previously is not used universally Because the sign convention Figure 1.6 Numerical aperture determine With an image NA of 0.25 and an image distancetoand10 mm, that the focal length is 9.1 mm (s″) of magnification 0.25 = CA 20 distances, s and s″, we utilize equation 1.6 (1.15)
  14. 14. PM Figure 1.6 Page 1.6 Numerical aperture and magnification NA (object side) = sin v = CA 2s CA 2s ″ Gaussian Beam Optics 7($!5F.%).#*)/" optical invariant (1.11) Suppose it is necessary, using a singlet lens, to couple the output of an incandescent bulb with a filament 1 mm in diameter into an optical fiber 1ch_FundamentalOptics_Final_a.qxd 0.1! magnification = 2h″sin v = s = 0.1 = 6/15/2009 2:28 PM Page 1.9 Fundamental Optics CA cation. 1.0 Two very commonhapplications of h″ 0.1 optics involve(1.12) optical systemfigure 1.7. Assume that the fiber has a core diameter of simple couplingas shown in light into magnification = = = 0.1! and www.cvimellesgriot.com 100 mm and f = 9.1 mm an NA of 0.25, and that the design requires that the total h 1.0 an optical fiber or into the entrance slit of a monochromator. Although optical system these distance from fiber 110 CA = 2 ″ of Lens Systemss ″sin vquite=different, they both have the same limitation the source to theNA″ =beCA =mm. Which lenses are problems appear to be NA CA = 0.025 appropriate? leading to 0.25 (1.13) f = 9.1 mm Optical Specifications Material Properties Fundamental Optics Optical Coatings andmuch simplerwill not result in anyeffectivesystem throughput because of It is diameter lens to calculate the greater (combined) focal length and the limited input NA of the optical fiber. The singlet lenses in this catalog lens principal-point locations1and then use these results in any subsequent that meet these criteria are LPX-5.0-5.2-C, which is plano-convex, and paraxial calculations (see figure 1.8).which are biconvex. used in the optical LDX-6.0-7.7-C and . CA = 5 mmLDX-5.0-9.9-C, They can even be invariant calculations described in the preceding section. Symbols fc = lens combination focal length (EFL), positive if combination finalcombination to the right of the combination secondary focal point falls principal point, negative otherwise (see figure 1.8c). Making some simple calculations has reduced to just Accomplishing thischapter, Gaussian Beamour a single lens therefore requiresfocal length of the first element″(see figure 1.8a). imaging task with choice of lenseshow to f1 = an zH three. The following Optics1 discusses f, optic with FOCAL LENGTH length and a 5-mm diameter. Using a larger a choice of lenses based on various performance criteria. 9.1-mm focal EFFECTIVE make a final (a) (c) f2 = focal length of the second element. fc diameter lens will not resultto calculate the effective focal length The following formulas show how in any greater system throughput because of s1 = d4f d = distance and limited input NA of the optical fiber. The singlet com- 1 the principal-point locations for a combination of any two arbitrarylenses in this catalog from the secondary principal point of the first Hc H than two lenses is very 2” H2 H simple: Calculate H1″ element to the primary principal point of the second element, ponents. The approach for more 1 that meet these criteria are LPX-5.0-5.2-C, which is plano-convex, and positive if the primary principal point is to the right of the the values for the first two elements, then perform the same calculation for LDX-6.0-7.7-C and LDX-5.0-9.9-C, whichuntil all lenses in the secondary principal point, negative otherwise (see figure 1.8b). this combination with the next lens. This is continued are biconvex. lens 1 system are accounted for. lens Making some simple calculations has reduced our choice of lensessto = distance from the primary principal point of the first just and 1″ combination lens 2 The expression for the combination focal length is the same whether lens element to the final combination focal point (location of the three. The following chapter, Gaussian Beam Optics, discusses how to separation distances are large or small and whether f1 and f are positive final image for an object at infinity to the right of both lenses), make a final choice of lenses based on various2performance criteria. or negative: f f f = (b) 1 2 . f1 + f2 − d d positive if the focal point is to left of the first element’s zH primary principal point (see figure 1.8d). s2″(1.16) This may be Lens combination Figure 1.8 more familiar in the form focal length and principal planes d 1 1 1 = + − . f f1 f2 f1 f2 (1.17) (d) fc s2″ = distance from the secondary principal point of the second element to the final combination focal point (location of the final image for an object at infinity to the left of both lenses), positive if the focal point is to the right of the second element’s secondary principal point (see figure 1.8b). Fundamental Optics 1.9 Optical Coatings Optical Specifications nd NA, H c″ Material Properties Gaussian Beam Optics Many optical tasks require several lenses in order to achieve an acceptable CA level0.25 = 5 mm. One possible approach to lens combinations is to conof performance. CA = H1 H1″ 20 sider each image formed by each lens as the object for the next lens and so Accomplishing this imaging task with a single lens therefore requires an on. This iswith a 9.1-mm focal length timea 5-mm diameter.unnecessary. optic a valid approach, but it is and consuming and Using a larger Optical Specifications magni. Thus, either nification and NA, s value ce (i.e., rred to .25 = Figure 1.9 “Extreme” meniscus-form lenses with ).#G,@*0:<1'#208)(<9:.57($1.8.#O,F$5*1#/818##O<3K!%#3BB4(6<6.17#'I"@ planes"(drawing not.>K!%*:<1'L#)*HR< F$ external principal L<$&/ &+!).#G,@- to scale) With an image NA of 0.25 and an image distance (s″) of 10 mm, $(7OCK!%*:<1'O("IH #3BB4=OpticsNL1"%G<#PH4=F 1.8 F1J&( Fundamental 1.6 and Gaussian BeamOptical Coatings Optics roduce ength and magnivalue of CA. Thus, agnifistrained on either ing the value beyond this performance htness. (i.e., etimes referred to Lens Combination Formulas We With an image NA of 0.25 and an image distance (s″) of 10 mm, NA″ = CA/2s″ to derive can now use the relationship NA = CA/2s or CA, the optimum clear aperture (effective diameter) of the lens. CA Fundamental Optics Properties Material ations ge of a source, it is f the lens, thereby d thereby produce between ce, it ismagnifihich increasing the hereby image brightness. Optical Coatings #353#39$F.%$(7OC)(B-.> fixed --> optical invariant Material Properties filament CA CA 2s 2s NA = (1.10) — they havesa fixedvNA.NAfilament For=monochromators, = 0.025 is usually expressedthe magnification mustNA″0.1.2s″″ = 0.25 total 110 mm By definition, be = Letting s=s″ 2s this limit h = 1 mm sin ″ h . 1 mm = = (using s sin v ″ NA ″ (1.14) Fundamental Optics in terms of the f-number. In addition to the fixed NA, they both have a the thin-lens approximation), we can use equation 1.3, fixed CA 5 mm CA = 5 mm=( s + s ″ ) s ″ with Fixed Input NA Example: System simply the magnification of the system, Since is entrance pupil (image) size. (see eq. 1.3) f =m , fiber core www.cvimellesgriot.com (1.11) s fiber core (m + 1)2 n to magnification. h″ = 0.1 mm Two very common applications of simple optics involve coupling light into h″ = 0.1 mm we arrive at Suppose itfibernecessary, using a monochromator. Although these the output of an the focal length is 9.1 mm. To determine the conjugate an optical is or into the entrance slit of a singlet lens, to couple to determine that s = 100 mm s″ = 10 mm NA problems appear to be quite a filament mm the same limitation have m = . distances, s and s theysbothmm in s″ = 10 incandescent bulb withdifferent,=s100 1 110 mm diameter into an optical fiber s″, we utilize equation 1.6, mm + ″= (1.10) — they have a fixed NA ″ monochromators, this limit is usually expressed(1.15) NA. For (1.12) s + s″ 110 Assume that they fiber has a as shownofin figure=1.7. mm to the fixed NA, theboth have a fixed core diameter of in terms the f-number. In addition (see eq. 1.6) s ( m + 1) = s + s ″, Figure pupil (image) of1.7 Optical that equal to the ratio of the the 100entrance The magnification of the system is thereforegeometry for focusingNAs that thean incandescent bulb into an optical fiber mm and an NA size.0.25, andsystemthe design requires output of total (1.11) COMBINATION SECONDARY PRINCIPAL-POINT LOCATION on the object and image sides of the system. This powerful and useful result and find are it is necessary, using a singlet lens, to of an distance Optical approximation the specifics of the optical system, most can lenses that s = 100 mm into = 10 mm. Figure Supposefrom theindependent ofis obviouslycouple the outputoutput of an incandescent bulb and s″ an optical fiber 1.7 Because the is completely source to the for focusing the and Which thin-lens system geometry fiber be invalid mm. it highly 110 for (1.13) incandescent bulbused tofilament 1 mm in diameter into an optical fiber often ability toa determine the optimum lens secondary princibe with determine the location of the diameter in situations We can now use the relationship NA 2 CA/2s or NA″4 CA/2s″ to derive combinations, the appropriate? 1 = 3 = (1.12) as shown in figure 1.7. Assume that the fiber has a core diameter of involving aperture constraints. CA, the optimum clear aperture (effective diameter) of the lens. pal pointmmvital an NA of 0.25, and that the designwhen another element is and for accurate determination of d requires that the total 100 When or optical system calculates the distance from the By definition, the source to for this is used110create0.1. Lettingare it is total 110 mm NA of 0.25 and an image distance (s″) of 10 mm, is added. Thefrom a lensformula the fiber be to mm. an image of a source, With an image distance simplest magnification must be Which lenses s=s″ the natural point of that, by increasing the diameter (f) of secondary (1.13) secondary principalto assume the final (second) element to the the lens, thereby appropriate? (using theincreasing its CA, we will be able to collect more light use equation 1.3, thin-lens approximation), we can and thereby produce CA (1.14) 0.25 = principal point of the combination (see figure 1.8b): 20 a brighter image. However, because of the relationship between magnifiBy definition, the magnification must be 0.1. Letting s=s″ total 110 mm (s + ″) cation ands approximation), we can use equation which (using the thin-lensNA, there can be a theoretical limit beyond1.3, increasing the (1.14) and brightness. (see eq. 1.3) z f s2″ mf . = = −diameter has no,effect on light-collection efficiency or image (1.19) (m +″1)2 (s + s ) em, d>0 f = m the NA of a ray is given by CA/2s, once a focal (see eq. 1.3) , Since length and magniCA = 5 mm. (m + 1)2 COMBINATION EXAMPLES fication have been selected, the value of NA sets the value of CA. Thus, Accomplishing 4 1 2 to determine thatdealing with system in which9.1 NA is PMTo Pageon1.8 the conjugate this imaging task with a single lens therefore requires an that the focal length is 2:28 constrained mm. determine 3 1ch_FundamentalOptics_Final_a.qxd a6/15/2009 todetermine the conjugateeither to determine a lens combination or system exhibit principal planes It is possible if one is the focal length is 9.1 mm. To the for Fundamental Optics 1.7 optic with a 9.1-mm focal length and a 5-mm diameter. Using a larger theand s″,or image side, increasing the lens 1.6, beyond this value objects″, we utilize equation diameter distances, s andfrom the system. When such systems are themselves that distances, s are far removed we utilize equation 1.6, diameter lens will not result in any greater system throughput because of will increase system but will not improve performance (1.15) (1.15) combined, negative values of dsize and cost Probably the simplest example(i.e., the limited input NA of the optical fiber. The singlet lenses in this catalog throughput or imagemay occur. This concept is sometimes referred to brightness). that meet these criteria are LPX-5.0-5.2-C, which is plano-convex, and (see eq. with s ( m as ) = s + s ″ of a negative+ 1the optical, invariant. d-value situation is shown in figure 1.9. Meniscus lenses1.6) e ratio of the NAs (see eq. 1.6) and LDX-5.0-9.9-C, which are biconvex. s ( m + have s + s ″, LDX-6.0-7.7-C steep surfaces1) = external principal planes. When two of these lenses are ul and useful he NAs result and find that s = a negative s″ = of d can brought into contact,100 mm andvalue10 mm. occur. Other combined-lens Fundamental Optics Making some simple calculations has reduced our choice of lenses to just 1.7 system, and it can SAMPLE CALCULATION l result Fundamental three. The following chapter, Gaussian Beam Optics, discusses how to Optics in relationship NA = CA/2s or eter in situations examples are shown thefigures 1.10 through 1.13. NA″ = CA/2s″ to derive To understand how to use this relationship mm. andWe canthatuse= 100 mm and s″ = 10between magnification and NA, find now s make a final choice of lenses based on various performance criteria. d<0 d it can CA, the optimum the following example. diameter) of the lens. www.cvimellesgriot.com consider clear aperture (effective Optical Specifications Example:hich can be rearranged to showInput NA System with Fixed w problems appear to be quite different, they both have the same limitation — they have a fixed NA. For monochromators, this limit is usually expressed in terms of the f-number. In addition to the fixed NA, they both have a fixed entrance pupil (image) size. Optical Specifications NA″ (image side) = sin v ″ = (1.10)
  15. 15. H1 Symbols ptable o conngthso and ry. omlate th and n for quent the optical www.cvimellesgriot.com fc = combination focal length (EFL), positive if combination final focal point falls to the right of the combination secondary f1 principal point, negative otherwise (see figure 1.8c). (a) COMBINATION SECONDARY PRINCIPAL-POINT LOCATION (b) .16) of a negative d-value situation is shown in figure 1.9. Meniscus lenses with element to the primary principal point of the second element, culate s2″ have external principal secondary principal point of the from the planes. steep surfaces= distance primary principal When two of these lensesthe positive if the point is to the right of are ion for second element to the final d can occur. Other point (location brought into contact, aprincipal point, ofcombination focalcombined-lens negative value negative otherwise (see figure 1.8b). secondary in the examplesof the final in figures 1.10 through 1.13. are shown image for an object at infinity toFundamental Optics the left of both mulas the next (1.16) ptable o conand so ned (1.17) ry. ond th and of the equent e next optical .18) mbined length econd y comculate ion for in the (1.18) er g lens ositive (1.16) (1.17) (c) fc H1c H 1 2 H1″ 3 4 lens com lens combination lens 1 zH (d) d>0 fc (a) f1 1 3 4 H1 H1″ Optical Coatings f1 thin-lens approximation is obviously highly invalid for Because the= focal length of the first element (see figure 1.8a). most H1 H 2 H 2” H1″ combinations, the ability to determine the location of the secondary princif2 = focal length of the second element. pal point is vital for accurate determination of d when another element Symbols is added. d = distance formula for this calculates the distance the first The simplest from lens 1 the secondary principal point of from the and final (second) element to the secondary secondary principal point of the element to the primary2principal point of the second element, lens principal point of the combination (see figure 1.8b): fc = combination focal length (EFL), positive if right of the positive if the primary principal point is to thecombination final focal point falls to the right of the combination secondary secondary principal point, negative otherwise (see figure 1.8b). z = s2″principal point, negative otherwise (see figure 1.8c). (1.19) −f . d Figure 1.9 external pri zH″ s1 = d4f1 s1″ = distance from the primary principal point of thes2″ first f1 = focal length of the lens COMBINATIONto the final combination focal point (location of the element EXAMPLES first element (see figure 1.8a). e 1.8 tive It is possible Figurelens combination at infinity to exhibit principal planes final image for an object or system the right of principal planes for a 1.8 Lens combination focal length and both lenses), f2 = focal length of the second element. positive if the focal point is to left of the first element’s length that are far removed from the system. When such systems are themselves combined, negative values of dthe secondary principal point of the first primary principal point (see figure 1.8d). simplest example d = distance from may occur. Probably the y com- .17) er lens ositive www.cvimellesgriot.com lens combination Material Properties tical H1″ Fundamental Optics lens 1 and mulas uent cifications able cond so combined, negative values of d may occur. Probably the simplest example of a negative d-value situation is shown in figure 1.9. Meniscus lenses with steep surfaces have external principal planes. When two of these lenses are H c″ brought into contact, a negative value of d can occur. Fundamental Optics Other combined-lens examples are shown in figures 1.10 through 1.13. 2 s1 = d4f1 H2 H2” Fundamental Optics 1.9 lens 1 and lens 2 d<0 lenses), distance if the the primary is to the right of the second s1″ = positive from focal point principal point www.cvimellesgriot.com of the first d element’s secondary principal point (see figure 1.8b). of the element to the final combination focal point (location (b) s2″ final image for an object at infinity to the right of both lenses), zH = distance to the combination primary principal point positive if the focal point is to left of the first element’s Figure 1.9 “Extreme” meniscus-form lenses with measuredprincipal point (see figure 1.8d). of the first element, Figure 1.8 Lens combination focal length and principal planes primary from the primary principal point external principal planes (drawing not to scale) positive if the combination secondary principal point is to s ″ = distance from principal point of second element the2right of secondarythe secondary principal point of the second element to the final combination focal point (location (see figure 1.8d). H infinity to the left of both of the final image H1 an object at 1″ for H c″ zHSymbols toif the focal point issecondary principal point ″ = distance lenses), positive the combination to the right of the second element’s secondary principal point (see figure the second measured from the secondary principal point of1.8b). element, positive if the combination secondary principal point lens lens 1 zc the right of to focal length (EFL), positiveof the second f H = distance the secondary principal point if combination combination is to = combinationthe combination primary principal point measured from the primary principal the combinationelement, final focal point falls to the right of point of the first secondary element (see figure 1.8c). positive ifpoint, negative otherwise (see figure 1.8c). is to principal the combination secondary principal point the right of secondary principal point coaxial combinations Note: These paraxial formulas apply to of second element zH″ f1 f1 thick and thin of(see=figure length of the first element (see figure 1.8a). both focal 1.8d). lenses immersed in air or any other (a) (c) fc fluid with refractive index independent of position. They zfH″= focal lengththethe second element. principal point = distance to of combination secondary 2 assume that light propagates from left to right through = d4f1 s1 an measured from the secondary principal point of the second opticaldistance from the secondary principal point of the first system. Hc d= H2 H2” element, positive if H1 combination secondary principal point the H1″ element to the the secondary principal of the second element, is to the right of primary principal point point of the second positive if the primary principal point is to the right of the element (see figure 1.8c). lens 1 secondary principal point, negative otherwise (see figure 1.8b). lens and Note: These paraxial formulas apply to coaxial combinations combination lens and of 1″ = distance from the primary principal point of other s both thick2 thin lenses immersed in air or any the first fluid withto the finalindex independent of position. They the element refractive combination focal point (location of assume that for an object at infinityleft theright through an zH final image light propagates from to to right of both lenses), optical system. focal point is to left of the first element’s positive if the (d) d (b) s2″ fc primary principal point (see figure 1.8d). s ″ = distance from the secondary principal point of the Figure 1.8 2 Lens combination focal length and principal planes second element to the final combination focal point (location of the final image for an object at infinity to the left of both lenses), positive if the focal point is to the right of the second element’s secondary principal point (see figure 1.8b). (c) Fundamental Optics 1.9 (d)
  16. 16. Optical Specifications The following formulas show how to calculate the effective focal length and principal-point locations for a combination of any two arbitrary components. The approach for more than two lenses is very simple: Calculate d>0 the values for the first two elements, then perform the same calculation for this combination with the next lens. This is continued until all lenses in the 1 2 3 4 system are accounted for. The expression for the combination focal length is the same whether lens separation distances are large or small and whether f1 and f2 are positive or negative: f = f1 f2 . f1 + f2 − d (1.16) d<0 This may be more familiar in the form d 1 1 1 Figure 1.9 “Extreme” meniscus-form lenses with = + − . f f1 f2 f1 2 external principalfplanes (drawing not to scale) (1.17) Notice that the formula is symmetric with respect to the interchange of the lenses (end-for-end rotation of the combination) at constant d. The next two formulas are not. Hc″ COMBINATION FOCAL-POINT LOCATION The expression for the combination focal length is the same whether lens f1 = focal length of the first element (see figure 1.8a). separation distances are large or small and whether f1 and f2 are positive or negative: focal length of the second element. f = 2 f1 f2 H the H1″ f = d = distance. from 1 secondary principal point of the first (1.16) f1 + f2 − d element to the primary principal point of the second element, positive familiar in the form This may be moreif the primary principal point is to the right of the secondary principal point, negative otherwise (see figure 1.8b). d 1 1 1 = + lens 1 − . (1.17) f sf1″ = distanceffrom the primary principal point of the first f2 f1 2 1 element to the final combination focal point (location of the Notice that the formula is symmetric with respect to the interchange of the final image for an object at infinity to the right of both lenses), lenses (end-for-end the focalof the combination) at constant d. The next positive if rotation point is to left of the first element’s two formulas are principal point (see figure 1.8d).f1 primary not. (a) s2″ = distance from the secondary principal point of the COMBINATION FOCAL-POINT LOCATION s1 = second element to the final combination focal point (locationd4f1 For all values offinal2image for an objectof the focal point of theof both f1, f , and d, the location at infinity to the left combined of the H H2 H2” H1″ system (s2″), measured from1the secondaryis to the right ofof the second lenses), positive if the focal point principal point the second lens (H2″), is given secondary principal point (see figure 1.8b). element’s by lens 1 z f=(distance to the combination primary principal point f1 − d ) s2″ = H 2 and . (1.18) measured− d the primary principal point of the first element, f1 + f2 from lens 2 positive if the combination secondary principal point is to This cantheshownof secondary principal point of second solving be right by setting s1=d4f1 (see figure 1.8a), and element (see figure 1.8d). 1 1 1 d = (b) =zH″ + distance to the combination secondary principal point s2″ f2 s1 s2″ Optical Coatings Optical Coatings combination COMBINATION SECONDARY PRINCIPAL-POINT LOCATION (1.19) Fundamental Optics 2 1.9 d>0 COMBINATION EXAMPLES It is possible for a lens combination or system to exhibit principal planes that are far removed from the system. When such systems are themselves combined, negative values of d may occur. Probably the simplest example of a negative d-value situation is shown in figure 1.9. Meniscus lenses with steep surfaces have external principal planes. When two of these lenses are brought into contact, a negative value of d can occur. Other combined-lens examples are shown in figures 1.10 through 1.13. s2″ = secon of the lenses eleme zH = d measu positiv the rig (see fi zH″ = measu eleme is to th eleme Note of bot fluid w assum optica 3 4 Optical Coatings z = s2 ″ − f . 1 final im positi prima 1 3 4 2 Gaussian Beam Optics Because the thin-lens approximation is obviously highly invalid for most combinations, the ability to determine the location of the secondary princifor s2″. zH pal point is vital for accurate determination of d when another element (d) is added. The simplest formula for this calculates the distance from the s2″ fc secondary principal point of the final (second) element to the secondary Fundamental Optics (see figure 1.8b): 1.8 principal point of the combination rincipal planes s1″ = Figure 1. eleme externa Fundamental Optics measured from the secondary principal point of the second element, positive if the combination secondary principal point to the right of the secondary principal point of the second Figure 1.8 isLens combination focal length and principal planes zH″ element (see figure 1.8c). for s2″. f ( f − d) Fundamental Optics s2″ = 2(c)1 . (1.18) fc f1 + f2 − d Note: These paraxial formulas apply to coaxial combinations www.cvimellesgriot.com of both thick and thin lenses immersed in air or any other s1 = d4f1 Fundamental Optics 1.8 fluid with refractive index independent of position. They Hc This can be shown by setting s1=d4f1 (see figure 1.8a), and solving assume that light propagates from left to right through an optical system. 1 1 1 = + f2 s1 s2″ lens system (s2″), measured from the secondary principal point of the second lens (H2″), is given by Material Properties Material Properties 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.9 lens For all values of f1, f2, and d, the location of the focal point of the combined combination system are accounted for. principal point, negative otherwise (see figure 1.8c). Optical Specifications Optical Specifications cipal planes themselves est example s lenses with se lenses are mbined-lens EFFECTIVE FOCAL LENGTH Material Properties (1.19) paraxial calculations (see figure 1.8). 2 can even4be used in the optical 1 They 3 invariant calculations described in the preceding section. Gaussian Beam Optics Gaussian Beam Optics ndary princiher element ce from the e secondary d<0 H1 H1″ Hc″ lens Optical Specifications Figure 1.9 “Extreme” meniscus-form lenses with external principal planes (drawing not to scale)
  17. 17. Fundamental Optics Fundamental Opt z s2″ d f1 f2 f<0 f1 f2 f<0 combination secondary principal plane focal plane Gaussian Beam Optics z<0 s2″ d Figure 1.10 Positive lenses separated by distance greater combination focal plane than f1 = f2: f is negative and both s2″ and z are positive. Lens secondary symmetry is not required. principal plane Figure 1.10 Positive lenses separated by distance greater than f1 = f2: f is negative and both s2″ and z are positive. Lens symmetry is not required. f1 Optical Specifications Optical Specifications Gaussian Beam Optics z H2 H2″ H2 H1″ H2″ d f2 tc n d f Figure 1.11 Achromatic combinations: Air-spaced lens combinations can be made nearly achromatic, even though both elements are made from the same material. Achieving Figure 1.11 Achromatic combinations: Air-spaced lens achromatism requires that, in the thin-lens approximation, combinations can be made nearly achromatic, even though 2 Material Properties Material Properties Figure 1.12 Telep combination characteristic of the secondary the image size, can principal plane the first lens surface positive lens followe Figure 1.12 Telephoto lens shapes shown i characteristic of the teleph f2 = 4f1/2. Then f i the image size, can be mad d = f1/2 (Galilean te the first lens surface to the d larger than f1 by T positive lens followed/2.a lens shapes shown inLDX-5 catalog lenses the fi f2 = 4f1/2. Then f iswill yie d = 78.2 mm, negat d = f1/2 (Galilean telescope d larger than f1/2. To make catalog lenses LDX-50.8-13 d = 78.2 mm, will yield s2″ f1 H1″ combination secondary principal plan both elements are made from the same material. Achieving ( f 1 + f2 requires that, in the thin-lens approximation, achromatism ) d= d= 2 . ( f 1 + f2 ) . 2 This is the basis for Huygens and Ramsden eyepieces. This is the basis for Huygens and Ramsden eyepieces. This approximation is adequate for most thick-lens situations. The signs of f1, f2, and adequate for most thick-lens situations. a This approximation is d are unrestricted, but d must have value that guarantees theare unrestricted, but d must have a The signs of f1, f2, and d existence of an air space. Element shapes are unrestricted the existence chosen to compensate for value that guarantees and can be of an air space. Element Figure 1.13 Cond vertices of a pair of i Figure 1.13 Condenser on contact. identica vertices of a pair of(The len d = 0, = f1 lenses cou on contact. f (The /2 = f2/2 pal f1/2 of the 1/2 d = 0, f =point= f2/2, fseco point the second ele pal point ofof the combin

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