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A review of linear systems theory through an electrical engineers perspective of Fourier optics

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- 1. Application of Linear SystemsAnalysis to 2-D Optical Images (Fourier Optics) An Intuitive Approach Andrew Josephson ajosephson@comcast.net Pg. 1
- 2. Fourier Transforms of Electrical Signals• As electrical engineers, we conceptualize the Fourier transformation and Fourier synthesis of voltages and currents frequently in circuit analysis• Why do we use the Fourier Transform?• There are other transforms – Hilbert – Hankel – Abel – Radon• What makes the Fourier Transform special? Pg. 2
- 3. Linear Time Invariant Systems Linear Time x(t ) Invariant System y(t ) h(t ) x(t ) h(t ) Linear TimeLinearity x1(t ) x2 (t ) Invariant System y(t ) x1 (t ) h(t ) x2 (t ) h(t ) h(t ) Linear TimeTime Invariance x(t d ) Invariant System y(t d ) h(t ) x(t d ) h(t ) Pg. 3
- 4. Fourier Analysis of LTI Systems x(t ) X ( ) When a system can be Linear Time classified as LTI, we can Invariant System h(t ) H ( ) analyze it easily with Fourier Analysis…why? y(t ) Y ( ) X ( ) H ( ) Pg. 4
- 5. Eigen Function of LTI Systems• The Eigen function of an LTI system is a mathematical function of time, that when applied as a system input, results in a system output of identical mathematical from – The output equals the input scaled by a constant „A‟ – Delayed in time by „d‟ Linear Time (t ) Invariant System y(t ) h(t ) (t ) A (t d ) h(t ) Pg. 5
- 6. Eigen Function of LTI Systems• Complex exponentials are the Eigen functions of LTI systems (t ) e jt• Complex exponentials are also the Kernel of the Fourier Integral 1 X ( ) jt x(t )e dt 2 Pg. 6
- 7. Fourier Analysis of LTI Systems Revisited• Conceptually, when we analyze an LTI system, we represent the input signal x(t) as a summation x(t ) c0 c1e j1t c2e j2t ... of linearly scaled Eigen functions (complex exponentials) – Fourier Decomposition Linear Time – The signal‟s “spectrum” Invariant System• We can then run each complex exponential through the system h(t ) easily because they give rise to a linearly scaled output that is delayed in time y(t ) K0c0 K1c1e j1 (t d ) K2c2e j2 (t d ) ...• The output y(t) is the summation of these scaled and delayed complex exponentials Pg. 7
- 8. Answer to the Million Dollar Question• We use the Fourier Transform to analyze LTI systems because the Eigen function of an LTI system IS the Kernel of the Fourier Integral• When we do not have an LTI system, we usually assume it is closely approximated by one, or force it to operate in a well-behaved region – Linearization• The complex exponentials we deal with in circuits are single variable functions with independent variable „t‟• Where else in electrical engineering do we use complex exponentials? Pg. 8
- 9. Plane Waves• Complex exponentials are also used to describe plane waves – The plane defines a surface of constant phase• These are multivariate functions with independent variables R = (x,y,z) z k jkR R E(R) E0e x y Pg. 9
- 10. Plane Waves as Eigen Functions• Plane waves are Eigen functions of certain system types as well – Since the independent variable is no longer time, we aren‟t interested in Linear Time Invariant Systems• We are now interested in the more general Linear Shift Invariant System – Shift refers to spatial movement since the independent variables now describe position instead of time Pg. 10
- 11. Fourier Optics• Fourier Optics is the application of linear shift invariant system theory to optical systems• The plane wave in an optical system is represented by the multivariate complex exponential just like the sine wave in an LTI system is represented by the jt single variable e• Just like an electrical signal can be represented as summation of sine waves, an optical image can be represented as a summation of plane waves – Angular Plane Wave Spectrum Pg. 11
- 12. Concept of Spatial Frequency• Assume a plane wave propagates in z-direction (down the optical axis) y jkz E( x, y, z) E0e k x z • The image plane (x-y) is normal to the optical axis• The projection of lines of constant phase onto the x- y plane is zero (spatial frequency equivalent to DC) Pg. 12
- 13. Concept of Spatial Frequency• Deflecting the wave-vector at an angle other than zero gives a projection of the plane wave intensity into the image plane (max and min) y k x z • This example deflects k into the y-direction creating a nonzero spatial frequency in the y-direction Pg. 13
- 14. Concept of Spatial Frequency• Increasing the angle of deflection increases the spatial frequency of intensity maximum/minimum y k x z y k x z Pg. 14
- 15. Concept of Spatial Frequency Pg. 15
- 16. Concept of Spatial Frequency• Spatial frequency in the y-direction can be denoted as f y and has units 1/cm 1 fy Pg. 16
- 17. Angular Plane Wave Spectrum• An arbitrary 2-D field distribution (image) can be decomposed into a spectrum of plane waves – Assuming monochromatic light• Just like the Fourier Transform of an electrical signal represents the magnitude and phase of each sinusoid in the signal spectrum, the 2-D Fourier Transform of an image represents the magnitude and phase of each plane wave in the image spectrum• High spatial frequency -> plane wave at large angles• Low spatial frequency -> plane wave at small angles Pg. 17
- 18. 2-D Fourier Transform• Consider an arbitrary 2-D black and white image in the XY-Plane• The image can be described mathematically by some function U(x,y,z=0) – „U‟ is optical intensity versus position 2 – Optical intensity is just proportional to E• The angular plane wave spectrum of the image is related to the 2-D Fourier Transform A( f X , fY ) U ( x, y, z 0) exp j 2 f X x fY y df X dfY Pg. 18
- 19. LSI Optical Systems• In Linear Shift Invariant optical systems, we can use Fourier analysis to decompose an image into its spectrum, multiply the spectrum by the optical transfer function(s), and inverse transform to get the resulting output – Note: Free space propagation of optical images can be modeled as an LSI system – This technique correctly models diffraction – This technique produces identical results to the full Rayleigh-Sommerfeld solutions Pg. 19
- 20. Thin Lens• A thin lens can be modeled as a phase shifting device – Assumes that no optical power is absorbed – Using the refractive index, n, and the radius of curvature, a mathematical transfer function can be calculated R1 R2 y x y x D1 D2 n Pg. 20
- 21. Thin Lens• To determine the optical transfer function of the simple lens system – Free space propagate D1 – Multiple by lens transfer function – Free space propagate D2 R1 R2 y x y x D1 D2 n Pg. 21
- 22. Fourier Transforming Lenses• A special value of D1 exists where many terms in the optical transfer function simplify 1 D1 1 1 n 1 R R 1 2 – This special value is called the focal length – When the input image is placed one focal length away, the optical transfer function at one focal length after the lens becomes a 2-D Fourier Transformation of the input image – Most of us already kinda knew that… Pg. 22
- 23. Fourier Transforming Lenses• A delta function and sine wave (complex exponential) form a Fourier Transform pair• What is the image equivalent of a delta function? – A point of light• We know that a plane wave is a complex exponential – A point of light and a plane wave form a Fourier Transform Pair• This is exactly what happens when we place a point of light one focal length away from a lens Pg. 23
- 24. Fourier Transforming Lenses• An optical delta function placed one focal length away is transformed into a plane wave one focal length away (and always) y x y x F F• This point source has been „collimated‟ Pg. 24
- 25. Spatial Filtering – A simple 2 Lens System• With two lenses, we can construct a system that produces the Fourier transform of the input image and then transform this again to create the original image y x y x y x F F F F Fourier Transform of Input Image Pg. 25
- 26. Spatial Filtering – A simple 2 Lens System• We now have direct access to the image spectra and can filter it physically with apertures• The low frequency components (small y x angular deflection from optical axis) are contained within the center of the image spectrum• Using a circular aperture and blocking out a portion of spectrum re-creates the image with the higher frequency components blocked – low pass filter Pg. 26
- 27. Spatial Filtering – A simple 2 Lens System• http://micro.magnet.fsu.edu/primer/java/digitalimaging/processing/fouriertr ansform/index.html• The link above gives many interactive images and spatial filtering examples – High pass • Block out the image spectra around origin – See high resolution portion of image remain unchanged – Low pass • Allow low frequency planes waves (small angles) to pass through the aperture – Blurs image by removing high frequency plane waves – Can be used to balance versus higher frequency image noise – Band Reject • Find an input image with a periodic grating (Black Knot Fungus) • Image spectra is periodic • Band reject the aliases • Recreate image without grating presents Pg. 27

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