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Topic 3: Operation of Simple Lens V N I E R U S E I T H Y Modern Optics T O H F G E R D I N B U Topic 3: Operation of Simple Lens Aim: Covers imaging of simple lens using Fresnel Diffraction, resolu- tion limits and basics of aberrations theory. Contents: 1. Phase and Pupil Functions of a lens 2. Image of Axial Point 3. Example of Round Lens 4. Diffraction limit of lens 5. Defocus 6. The Strehl Limit 7. Other Aberrations PTIC D O S G IE R L O P U P P A D E S C P I A S Properties of a Lens -1- Autumn Term R Y TM H ENT of P V N I E R U S E I T H Y Modern Optics T O H F G E R D I N B U Ray Model Simple Ray Optics gives f Image Object u v Imaging properties of 1 1 1 + = u v f The focal length is given by 1 1 1 = (n − 1) + f R1 R2 For Infinite object Phase Shift Ray Optics gives Delta Fn f Lens introduces a path length difference, or PHASE SHIFT. PTIC D O S G IE R L O P U P P A D E S C P I A S Properties of a Lens -2- Autumn Term R Y TM H ENT of P V N I E R U S E I T H Y Modern Optics T O H F G E R D I N B U Phase Function of a Lens δ1 δ2 h R2 R1 n P0 P ∆ 1 With NO lens, Phase Shift between , P0 ! P1 is 2p F = kD where k = l with lens in place, at distance h from optical, F = k0d1 + d2 +n(D − d1 − d2)1 Air Glass @ A which can be arranged to|giv{ze } | {z } F = knD − k(n − 1)(d1 + d2) where d1 and d2 depend on h, the ray height. PTIC D O S G IE R L O P U P P A D E S C P I A S Properties of a Lens -3- Autumn Term R Y TM H ENT of P V N I E R U S E I T H Y Modern Optics T O H F G E R D I N B U Parabolic Approximation Lens surfaces are Spherical, but: If R1 & R2 h, take parabolic approximation h2 h2 d1 = and d2 = 2R1 2R2 So that kh2 1 1 F(h) = kDn + (n − 1) + 2 R1 R2 substituting for focal length, we get kh2 F(h) = kDn − 2 f So in 2-dimensions, h2 = x2 + y2, so that (x2 + y2) F(x;y; f ) = kDn − k 2 f the phase function of the lens. Note: In many cases kDn can be ignored since absolute phase not usually important. The difference between Parabolic and Spherical surfaces will be con- sidered in the next lecture. PTIC D O S G IE R L O P U P P A D E S C P I A S Properties of a Lens -4- Autumn Term R Y TM H ENT of P V N I E R U S E I T H Y Modern Optics T O H F G E R D I N B U Pupil Function The pupil function is used to define the physical size and shape of the lens. p(x;y) ) Shape of lens so the total effect of the lens of focal length is p(x;y)exp(ıF(x;y; f )) For a circular lens of radius a, p(x;y) = 1 if x2 + y2 ≤ a2 = 0 else The circular lens is the most common, but all the following results apply equally well for other shapes. Pupil Function of a simple lens is real and positive, but it will be used later to include aberrations, and will become complex. PTIC D O S G IE R L O P U P P A D E S C P I A S Properties of a Lens -5- Autumn Term R Y TM H ENT of P V N I E R U S E I T H Y Modern Optics T O H F G E R D I N B U Fresnel Image of Axial Point We will now consider the imaging of an axial point using the Fresnel propagation equations from the last lecture. Consider the system. u (x,y) u (x,y) f 2 0 z z P P' 1 P P0 0 1 1 2 In plane P0 we have that u0(x;y) = A0d(x;y) so in plane P1 a distance z0, we have that, u(x;y;z0) = h(x;y;z0) A0d(x;y) = A0h(x;y;z0) where h(x;y;z) is the free space impulse response function. If we now assume that the lens is thin, then there is no diffraction 0 0 between planes, P1 and P1, so in P1 we have 0 u (x;y;z0) = A0h(x;y;z0)p(x;y)exp(ıF(x;y; f )) so finally in P2 a further distance z1 we have that 0 u2(x;y) = u (x;y;z0) h(x;y;z1) = A0h(x;y;z0)p(x;y)exp(ıF(x;y; f )) h(x;y;z1) PTIC D O S G IE R L O P U P P A D E S C P I A S Properties of a Lens -6- Autumn Term R Y TM H ENT of P V N I E R U S E I T H Y Modern Optics T O H F G E R D I N B U Take the Fresnel approximation, where exp(ıkz) k h(x;y;z) = exp ı (x2 + y2) ılz 2z and k F(x;y; f ) = knD − (x2 + y2) 2 f so we get that A0 exp(ıkz0) k 2 2 u(x;y;z0) = exp ı (x + y ) ılz0 2z0 then by more substitution, 0 A0 exp(ık(z0 + nD)) u (x;y;z0) = p(x;y) ılz0 k 1 1 exp ı − (x2 + y2) 2 z0 f Finally!, we can substitute and expand to get 1 A0 u2(x;y) = 2 exp(ık(z0 + z1 + nD)) zl z0z1 }| { 2 k exp ı (x2 + y2) z 2z1}| { 3 k 1 1 1 p(s;t)exp ı (s2 +t2) + − Z Z z 2 }|z0 z1 f { k exp −ı (sx +ty) dsdt z1 4 PTIC | {z } D O S G IE R L O P U P P A D E S C P I A S Properties of a Lens -7- Autumn Term R Y TM H ENT of P V N I E R U S E I T H Y Modern Optics T O H F G E R D I N B U Look at the term: 1. Constant, amplitude gives absolute brightness, phase not mea- surable. 2. Quadratic phase term in plane P2, no effect on intensity, and usually ignored. 3. Quadratic phase term across the Pupil, depends on both z0 and z1. 4. Scaled Fourier Transform of Pupil Function. If we select the location of plane P2 such that 1 1 1 + − = 0 z0 z1 f Note same expression as Ray Optics, then we can write k 2 2 u2(x;y) = B0 exp ı (x + y ) 2z1 k p(s;t)exp −ı (sx +ty) dsdt Z Z z1 so in plane P2 we have the scaled Fourier Transform of the Pupil Function, (plus phase term). Intensity in plane P2 is thus 2 g(x;y) = ju2(x;y)j k 2 = B2 p(s;t)exp −ı (sx +ty) dsdt 0 Z Z z 1 Being the scaled power spectrum of the Pupil Function. PTIC D O S G IE R L O P U P P A D E S C P I A S Properties of a Lens -8- Autumn Term R Y TM H ENT of P V N I E R U S E I T H Y Modern Optics T O H F G E R D I N B U Image of a Distant Object For a distant object z0 ! ¥ and z1 ! f u (x,y) p(x,y) 2 z P1 P' f 0 1 P2 Then the amplitude in P2 becomes, k u (x;y) = B exp ı (x2 + y2) 2 0 2 f k p(s;t)exp −ı (sx +ty) dsdt Z Z f which being the scaled Fourier Transform of the Pupil function. The intensity is therefore: k 2 g(x;y) = B2 p(s;t)exp −ı (sx +ty) dsdt 0 Z Z f which is known as the Point Spread Function of the lens Key Result Note on Units: x;y;s;t all have units of length m. Scaler Fourier kernal is: 2p exp −ı (sx +ty) l f PTIC D O S G IE R L O P U P P A D E S C P I A S Properties of a Lens -9- Autumn Term R Y TM H ENT of P V N I E R U S E I T H Y Modern Optics T O H F G E R D I N B U Simple Round Lens Consider the case of round lens of radius a, amplitude in P2 then becomes, k u (x;y) = Bˆ exp −ı (sx +ty) dsdt 2 0 Z Z f s2+t2≤a2 the external phase term has been absorbed into Bˆ0 This can be integrated, using standard results (see Physics 3 Optics notes or tutorial solution), to give: ka J1 x ˆ 2 f u2(x;0) = 2pB0a ka f x where J1 is the first order Bessel Function.
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