Topic 4: Imaging of Extended Objects Aim: Covers the imaging of - PDF document

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Topic 4: Imaging of Extended Objects Aim: Covers the imaging of extended objects in coherent and inco- herent light. The effect of simple defocus is also considered. Contents: 1. Imaging of two points 2. Coherent and Incoherent points. 3. Extended Objects 4. Coherent imaging of extended objects 5. Incoherent imaging of extended objects. 6. Optical Transfer Function 7. OTF of Simple Lens 8. OTF under defocus O P T I C D S E G I R L O P P U A P D S Images of Extended Objects -1- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Image of Two Points If we assume the system is Space Invariant, c ,d z 0 0 1 a ,b 0 0 a ,b 2 2 c ,d 2 2 P z 0 1 P P 2 0 ( a 0 ; b 0 ( c 0 ; d 0 ) in P 0 we get in P 2 Two ) , and Two points sources at ( a 2 ; b 2 ( c 2 ; d 2 ) and ) where PSF located at � z 1 � z 1 ; a 2 a 0 b 2 b 0 = = z 0 z 0 � z 1 � z 1 ; c 2 c 0 d 2 d 0 = = z 0 z 0 So in P 2 we get amplitude Au 2 ( x � a 2 ; y � b 2 Bu 2 ( x � c 2 ; y � d 2 ) PLUS ) | {z } ? where A and B are the brightness of the points. What does PLUS mean Depends on the physical properties of the two sources. O P T I C D S E G I R L O P P U A P D S Images of Extended Objects -2- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Coherent Sources If the two sources originate from the same source, (eg. Y oung’s Slits), then their amplitudes will sum. Two Amplitude PSFs Two Holes Point P P Source P 1 0 2 In P 2 Intensity will be, ) j 2 g ( x ; y ) j Au 2 ( x � a 2 ; y � b 2 + Bu 2 ( x � c 2 ; y � d 2 = ) These points are said to be Coherent Incoherent Sources Two point sources completely independent, (2 stars, 2 light bulbs, 2 LED), then their INTENSITIES sum. In P 2 Intensity will be, ) j 2 ) j 2 g ( x ; y ) j Au 2 ( x � a 2 ; y � b 2 j Bu 2 ( x � c 2 ; y � d 2 = + These points are said to be Incoherent Coherent and Incoherent are two extremes, the mixture is covered by Partial Coherence. (Not part of this course). O P T I C D S E G I R L O P P U A P D S Images of Extended Objects -3- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Extended Objects Consider an extended object to be an array of δ -functions. Picture contains 128 2 points. Each point of the object is imaged through the optical system and forms PSF. Output image is “Combination” of these PSFs, either Coherently, or Incoherently. Remember Convolution Relation: (for Fourier Transform Booklet) = f(x) s(x) f(x) s(x) O P T I C D S E G I R L O P P U A P D S Images of Extended Objects -4- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Coherent Imaging All points illuminated from a single point source. f (x,y) a v(x,y) z 1 z 0 P P P 1 2 0 = u 2 ( x ; y ) Amplitude PSF = 2 f Take special case of Unit Magnification, z 0 = z 1 Also reverse the direction of the coordinates in Plane P 2 . So at point ( x 0 ; y 0 ) we get )+ ∑ Parts of other PSFs v ( x 0 ; y 0 = f a ( x 0 ; y 0 ) so we get that Z Z d s d t v ( x ; y ) f a ( x � s ; y � t ) u 2 ( s ; t = ) | {z } PSF so we have that v ( x ; y ) = f a ( x ; y ) � u 2 ( x ; y ) so the intensity distribution in P 2 is given by ( x ; y ) j 2 g ( x ; y ) j f a ( x ; y ) � u 2 = O P T I C D S E G I R L O P P U A P D S Images of Extended Objects -5- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Cont: Apply the Convolution Theorem, we get that V ( u ; v ) = F ( u ; v ) U ( u ; v ) the effect of the lens is to Multiply by the Filter Function U ( u ; v ) Define: U ( u ; v ) = Coherent Transfer Function, (CTF) CTF is the Fourier Transform of the amplitude PSF, so CTF is a scaled version of the Pupil Function. ( u λ z 1 ; v λ z 1 U ( u ; v ) = p ) � 1 (Spatial Frequency), while the Note on Units, u & v have units m Pupil function has units of m, (physical size). Ideal Lens: x 2 + y 2 � a 2 1 p ( x ; y ) = 0 = else so that the CTF will be u 2 + v 2 � w 2 1 U ( u ; v ) = 0 0 = else where we have the Spatial Frequency limit = a w 0 λ z 1 to the lens acts like a “Low Pass Filter” with W 0 < Spatial Frequency passed W 0 > Spatial Frequency blocked O P T I C D S E G I R L O P P U A P D S Images of Extended Objects -6- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N cont: For a distant object, we have z 1 ! f , so maximum spatial frequency passed by a lens, = a w max λ f which can be written as 1 w max = 2F No λ Y on the F No of the lens. so the CTF depends ONL Example: 100 mm focal length, F No = 4 lens (25 mm diameter). for λ = 550 nm = 227 cycles = mm w max ie 227 cycles = mm imaged < Grating of Frequency 227 cycles = mm not imaged > Grating of Frequency Coherent imaging is investigated in detail in Optical Processing sec- tion. (Little more complicated when we include phase). O P T I C D S E G I R L O P P U A P D S Images of Extended Objects -7- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Incoherent Imaging (Camera) Assume NO interference between points, reasonable model for pho- tographic images of a natural scene. Input intensity image f ( x ; y ) , f(x,y) g(x,y) z 0 z P P 1 P 1 0 2 The PSF of the system, in incoherent light, is ( x ; y ) j 2 h ( x ; y ) j u 2 = Imaging as for the coherent case, is that g ( x ; y ) = f ( x ; y ) � h ( x ; y ) | {z } | {z } | {z } Image Object PSF So in Fourier space we have that G ( u ; v ) = F ( u ; v ) H ( u ; v ) where H ( u ; v ) is known as the “Optical Transfer Function” (OTF). The OTF acts like a Fourier Space filter and determines the imaging characteristics of the lens. O P T I C D S E G I R L O P P U A P D S Images of Extended Objects -8- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Cont: Note that ( x ; y ) j 2 � � H ( u ; v ) f h ( x ; y ) g j u 2 = F = F so from the Correlation Theorem, we have that H ( u ; v ) = U ( u ; v ) � U ( u ; v ) OTF is Auto-correlation of CTF So for a lens of pupil function p ( x ; y ) the OTF is given by ( u λ z 1 ; v λ z 1 ( u λ z 1 ; v λ z 1 H ( u ; v ) = p � p ) ) Again for a distant object, z 1 ! f then the OTF becomes ( u λ f ; v λ f ( u λ f ; v λ f H ( u ; v ) = p � p ) ) so we can determine the OTF from the Pupil Function of the lens. Note: This is true for all pupil functions, even if they include aberra- tions. Since the OTF is the auto-correlation of the CTF it will be “wider” then the CTF. So optical system will pass higher frequency grating in incoherent light. Better Resolution in Incoherent Light O P T I C D S E G I R L O P P U A P D S Images of Extended Objects -9- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Summary of Optical Measures Pupil Fn p ( x ; y ) Real space 6 Scaling ? � Coherent CTF PSF U ( u ; v ) u ( x ; y ) f g F Fourier Real space - space jj 2 � ? ? � Intensity OTF PSF H ( u ; v ) h ( x ; y ) f g F Fourier Real space - space Note if you know p ( x ; y ) or u ( x ; y ) you can calculate H ( u ; v ) and h ( x ; y ) but NOT VICE-VERSA. We are not able to determine the properties of a lens (or optical sys- tem), in coherent light from measures taken in incoherent light . O P T I C D S E G I R L O P P U A P D S Images of Extended Objects -10- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N OTF of Round Lens We have that the OTF is given by H ( u ; v ) = U ( u ; v ) � U ( u ; v ) and for a simple circular lens, u 2 + v 2 � w 2 1 U ( u ; v ) = 0 0 = else Pictorial Example: Area of overlap of two shifted circles. w 0 w 0 w 0 w 0 w w=2w 0 where w 2 = u 2 + v 2 So OTF will be circularly symmetric. Also: = 0 > 2 w 0 H ( u ; v ) w so the frequency limit for incoherent light is, = 2 a = d 2 w 0 λ z 1 λ z 1 This is TWICE the limit for coherent light O P T I C D S E G I R L O P P U A P D S Images of Extended Objects -11- Autumn Term C E P I S A Y R H T P M f E o N T