Topic 11: Optical Processing Aim: These two lectures cover basic - PDF document

E I H T Y T Modern Optics O H F G R E U D B I N Topic 11: Optical Processing Aim: These two lectures cover basic optical processing using the 4-f optical system with amplitude filters, phase filters, Fourier holograms and as a joint transform correlator. Finally the practicality of these systems is considered. Contents: 1. Fourier Properties of Lenses 2. Optical Processing System 3. Amplitude Filters 4. Phase contrast filters 5. Fourier Holograms 6. The Vander Lugt Correlator 7. Joint Transform Correlator 8. Practical Optical Processing P T O I C D S E G I R L O P P U A P D S Optical Processing -1- Autumn Term C E P I S A Y R H T P M E o f N T

E I H T Y T Modern Optics O H F G R E U D B I N Fourier Properties of a Lens The Amplitude PSF of a lens is just the scaled Fourier Transform of its Pupil Function, u (x,y) p(x,y) 2 P P’ f z 1 1 P 0 2 Then the amplitude in P 2 (including the quadratic phase factor) be- comes, ı κ � � ( x 2 + y 2 ( x ; y ) = ) u 2 B 0 exp 2 f � ı κ � � Z Z ( s ; t ) exp ( sx + ty ) p d s d t f If we define Z Z � ı 2 π ( ux ( u ; v ) = ( x ; y ) exp ( + vy )) d x d y P p Then � x ı κ � � � ; y ( x 2 + y 2 ( x ; y ) = B 0 exp ) u 2 P λ f λ f 2 f Note: in units, x ; y ! Units of length, mm � 1 ; v ! u Units of Spatial Freq, mm P T O I C D S E G I R L O P P U A P D S Optical Processing -2- Autumn Term C E P I S A Y R H T P M E o f N T

E I H T Y T Modern Optics O H F G R E U D B I N Fourier Transform of Slide ( x ; y ) close to lens Place slide of Amplitude Transmittance f a f (x,y) a u (x,y) 2 p(x,y) P1 P’ f 1 P 2 ( x ; y ) is smaller than the lens, f a ( x ; y ) become effective pupil if f a function, ı κ � � ( x 2 + y 2 ( x ; y ) = ) B 0 exp u 2 2 f � ı κ � � Z Z ( s ; t ) exp ( sx + ty ) f a d s d t f or more simply, � x ı κ � � � ; y ( x 2 + y 2 ( x ; y ) = B 0 exp ) u 2 F λ f λ f 2 f ( x ; y ) plus a quadratic so in P 2 we get the scaled Fourier Transform of f a phase term. The intensity in P 2 is then just � x 2 � � � ; y = B 2 � � ( x ; y ) g � F � λ f λ f � 0 � ( x ; y ) which is the Power Spectrum of f a P T O I C D S E G I R L O P P U A P D S Optical Processing -3- Autumn Term C E P I S A Y R H T P M E o f N T

E I H T Y T Modern Optics O H F G R E U D B I N Fourier Transform Examples P T O I C D S E G I R L O P P U A P D S Optical Processing -4- Autumn Term C E P I S A Y R H T P M E o f N T

E I H T Y T Modern Optics O H F G R E U D B I N Practical System Typical practical system is f (x,y) Film/Detector a f M/S f Laser P2 Collimator FT Lens F(u,v) Liquid Gate Focal length of lenses depends on expected frequency range, eg: � 1 ( x ; y ) : Maximum spatial frequency in f a 100mm � 10mm Size of Fourier plane: Wavelength: 633nm Focal Length FT Lens: 160mm Note: Usually need Liquid Gate to remove phase effect of gelatine to get good results P T O I C D S E G I R L O P P U A P D S Optical Processing -5- Autumn Term C E P I S A Y R H T P M E o f N T

E I H T Y T Modern Optics O H F G R E U D B I N General Case Move the object plane a distance z from the lens, f (x,y) a d u (x,y) f 2 Collimated Beam P P 1 P’ P 0 1 2 ! P 1 1: Propagate from P 0 0 Use Fresnel diffraction to 2: Lens adds a phase factor in P 1 0 ! P 2 3: Propagate from P 1 If we can ignore the finite aperture of the lens, “it-can-be-shown” (see tutorial) that in P 2 we get ı κ � � � � � z ( x 2 + y 2 ( x ; y ) = ) u 2 exp 1 2 f f � ı κ � � Z Z ( s ; t ) exp ( xs + yt ) f a d s d t f = f , then we get so if we take the special case of z � ı κ � � Z Z ( x ; y ) = ( s ; t ) exp ( xs + yt ) u 2 f a d s d t f so we get the exact Fourier transform, without any phase term, (ex- ternal constants ignored) � x � ; y ( x ; y ) = F u 2 λ f λ f P T O I C D S E G I R L O P P U A P D S Optical Processing -6- Autumn Term C E P I S A Y R H T P M E o f N T

E I H T Y T Modern Optics O H F G R E U D B I N Optical Processing System Put two lenses together to get “4-f Optical System”. f (x,y) f f a f f f (-x,-y) a Collimated Beam P P P P P 0 1 2 3 4 Input light at P 0 is collimated, in P 2 we have � x � ; y ( x ; y ) = F u 2 λ f λ f Second lens takes a second Scaled FT, so in P 4 we get ( x ; y ) = f a ( � x ; � y ) u 4 so a mirror image of the input. The intensity measured in P 4 is then given by � y ) j 2 ( x ; y ) = j f a ( � x ; g but we usually rearrange the coordinates in P 4 so that we have ( x ; y ) j 2 ( x ; y ) = j f a g Note this assumes that the PSF is small , so valid for Small Objects & Large Lenses P T O I C D S E G I R L O P P U A P D S Optical Processing -7- Autumn Term C E P I S A Y R H T P M E o f N T

E I H T Y T Modern Optics O H F G R E U D B I N Convolution Filtering In plane P 2 we have for Fourier Transform, so we can add a “filter”. H(u,v) f (x,y) f f f (x,y) o h(x,y) a f f a P P P P P 0 1 2 3 4 ( x ; y ) in plane P 2 we have for and input of f a = x = y ( u ; v ) F u v λ f λ f Apply a filter (slide) in P 2 , modify the distribution to ( u ; v ) H ( u ; v ) F so output plane P 4 is Convolution, giving ( x ; y ) = f a ( x ; y ) � h ( x ; y ) u 4 where Z Z ( � ı 2 π ( ux ( x ; y ) = ( u ; v ) exp + vy )) d u d v h F so the intensity in P 4 is ( x ; y ) j 2 ( x ; y ) = j f a ( x ; y ) � h g ( u ; v ) we can apply different types of filters, which So by changing H ( x ; y ) . are convolved with f a This system is the basis of Optical Image Processing. P T O I C D S E G I R L O P P U A P D S Optical Processing -8- Autumn Term C E P I S A Y R H T P M E o f N T

E I H T Y T Modern Optics O H F G R E U D B I N Practical System To make Fourier plane of sensible size you need long focal length lenses, so typically have to “fold” system. Spatial Filter Polarised Laser Polariser Solid Base f = 30 cm Plate Input Plane CCD 25mm Camera 50 cm 10mm 20 cm f = 50 cm f = 20 cm Fourier Filter Need not have all lenses the same focal length, this system (as set-up = 5 to fit CCD array camera. in Optics Lab), has a magnification of 2 P T O I C D S E G I R L O P P U A P D S Optical Processing -9- Autumn Term C E P I S A Y R H T P M E o f N T

E I H T Y T Modern Optics O H F G R E U D B I N Filtering Examples Low pass Filtered High pass Filtered Low pass Filtered High pass Filtered This is a digital simulation with some enhancement to show details. See Hecht page 268 for examples. P T O I C D S E G I R L O P P U A P D S Optical Processing -10- Autumn Term C E P I S A Y R H T P M E o f N T

E I H T Y T Modern Optics O H F G R E U D B I N Phase Objects Phase object is a transparent object with structure associated with thickness variation. n d Amplitude transmission of the object ( ı φ ( x ; y )) ( x ; y ) = exp f a where φ ( x ; y ) is the Optical Path Difference, so: = 2 π nd ( x ; y ) φ ( x ; y ) λ where n is the refractive index. In imaged in either Coherent or incoherent light, see ( x ; y ) j 2 ( x ; y ) = j f a = 1 g so we don’t see any structure. This problem occurs in: � 98% water 1): Biological cells, 2): Photo-resist on glass, (VLSI, and holograms) Common prob- 3): Finger prints on glass. lem is microscopy. P T O I C D S E G I R L O P P U A P D S Optical Processing -11- Autumn Term C E P I S A Y R H T P M E o f N T

E I H T Y T Modern Optics O H F G R E U D B I N Thin Phase Approximation Phase is periodic of period 2 π , can write = exp ( ı φ 0 ( ı φ ( x ; y )) ( x ; y ) ) exp f a where we have that � π < φ ( x ; y ) � π and φ 0 take no part in the imaging process. ( x ; y ) to get Take the “Weak-Phase” approximation, expand f a � φ 2 ( x ; y ) + ı φ ( x ; y ) ( x ; y ) � 1 f a 2 j Φ j � 1 take the approximately for first order, then so if + ı Φ ( x ; y ) ( x ; y ) � 1 f a This is valid for many practical cases, for example biological cells in water. Take the Fourier transform of this, typically optically, in a 4-f optical system, the Fourier Transform = δ ( u + ı Φ ( u ( u ; v ) ; v ) ; v ) F where Φ ( u f φ ( x ; y ) g ; v ) = F Add Filter in Fourier plane to make phase distribution visible as an Intensity. P T O I C D S E G I R L O P P U A P D S Optical Processing -12- Autumn Term C E P I S A Y R H T P M E o f N T