924 20. Diffraction – Fourier Optics This amounts to replacing the propagator impulse response h(r⊥,z)of Eq. (19.2.9) 20 by the approximation of Eq. (20.1.2): jk −jkz −jk|r⊥|2/2z jk −jkz −jk(x2+y2)/2z h(r⊥,z)= e e = e e (Fresnel) (20.1.4) Diffraction – Fourier Optics 2πz 2πz Noting that k = 2π/λ, the constant factor in front is often written as: jk j = 2πz λz The accuracy of the Fresnel approximation can be quantified by considering the higher-order terms in the expansion of the square root in, | − |2 | − |4 k r⊥ r⊥ k r⊥ r⊥ kR = k z2 +|r⊥ − r⊥|2 ≈ kz + − +··· 2z 8z3 This chapter continues the discussion of diffraction and emphasizes Fourier optics top- ics, Fresnel and Fraunhofer approximations, Talbot effect, Fourier properties of lens The approximation will be accurate if the third term is small. For an aperture of | − | systems, 1-D and 2-D apodizer design, aperture synthesis concentrating on narrow typical size D, the quantity r⊥ r⊥ remains of the order of D, thus, we obtain the beamwidths and low sidelobe designs, and concluding with a discussion of superres- following condition on the distance z at which the Fresnel approximation is accurate, olution, superdirectivity, and superoscillations. kD4 1 1 1 ⇒ z3 kD4 ⇒ kz (kD)4/3 (20.1.5) 8z3 8 2 20.1 Fresnel Approximation The Fresnel approximation can also be understood from the plane-wave spectrum point of view. The Fourier transform of (20.1.4) is obtained from the following Fourier The Fresnel approximation for planar apertures is obtained from the Rayleigh-Sommerfeld integral, which is a special case of Eq. (3.5.18): formula (19.1.6). Using (19.1.7), we have: ∞ jk − 2 2 −jkR ejkxx e jkx /2z dx = ejkx z/ 2k (20.1.6) 2z 1 e 2 E(r⊥,z)= E(r⊥, 0) jk + d r⊥ (20.1.1) 2πz −∞ S R R 4πR where j should be understood as ejπ/4. More generally, where R = (x − x )2+(y − y )2+z2 = |r⊥ − r⊥|2 + z2. The Fresnel approximation assumes that z is large enough such that |r⊥ − r⊥|z, which can be realized if the ∞ jk jk x −jk(x−x )2/2z jk x jk2 z/ 2k | | e x e 0 dx = e x 0 e x (20.1.7) aperture has dimension d so that r⊥ <d, and one assumes that the observation point 2πz −∞ r⊥ remains close to the z-axis (the paraxial approximation) such that |r⊥| <d, and z is chosen such that z d. Then, we can approximate R as follows: Applying (20.1.6) with respect to the x and y integrations, we obtain the two-dimensional spatial Fourier transform of h(r⊥,z): 2 2 2 |r⊥ − r⊥| 1 |r⊥ − r⊥| |r⊥ − r⊥| ∞ ∞ 2 2 R = |r⊥ − r⊥| + z = z 1 + z 1 + = z + jk⊥·r⊥ 2 jk −jkz −jk|r⊥|2/2z jk⊥·r⊥ 2 z2 2 z2 2z h(ˆ k⊥,z)= h(r⊥,z)e d r⊥ = e e e d r⊥ , or, −∞ 2πz −∞ √ where we used the Taylor series expansion 1 + x 1 + x/2. Assuming also that R or −jkz j|k⊥|2z/2k −jkz j(k2+k2 )z/2k z is much greater than the wavelength of the wave, z λ, so that k 1/R we obtain, h(ˆ k⊥,z)= e e = e e x y (20.1.8) −jkR −jk(z+|r⊥−r⊥ |2/2z) 2z 1 e e jk 2 Then, Eq. (20.1.3) can be written in the wavenumber domain as, + = −jkz −jk|r⊥−r⊥ | /2z jk jk e e (20.1.2) R R 4πR 2πz 2πz ∞ 2 ∞ 2 −jk⊥·r⊥ d k⊥ −jk⊥·r⊥ d k⊥ E(r⊥,z)= E(ˆ k⊥,z)e = E(ˆ k⊥, 0) h(ˆ k⊥.z) e where we set R z in the amplitude factors, but kept the quadratic approximation in −∞ (2π)2 −∞ (2π)2 the phase e−jkR. The Fresnel approximation is finally: with, jk 2 ∞ 2 ∞ = −jkz −jk|r⊥−r⊥ | /2z 2 d k⊥ E(r⊥,z) e E(r⊥, 0)e d r⊥ (Fresnel) (20.1.3) ˆ −jk⊥·r⊥ ˆ jk⊥·r⊥ 2 2πz E(r⊥, 0)= E(k⊥, 0)e E(k⊥, 0)= E(r⊥, 0)e d r⊥ S −∞ (2π)2 −∞ 923 20.1. Fresnel Approximation 925 926 20. Diffraction – Fourier Optics Thus, we may write the Fresnel diffraction formula convolutionally and in its plane- Introducing cylindrical coordinates in the space and wavenumber domains, similar 2 2 2 wave spectrum form, where we expanded, |r⊥ − r⊥| =|r⊥| +|r⊥| − 2r⊥ · r⊥, in the to Eqs. (19.11.1) and (19.12.8) of Chap. 19, third expression, r⊥ =|r⊥|= x2 + y2 x = r⊥ cos φ ∞ 2 2 d k⊥ −jkz ˆ j|k⊥| z/2k −jk⊥·r⊥ φ = atan2(y, x) y = r⊥ sin φ E(r⊥,z)= e E(k⊥, 0)e e −∞ (2π)2 (20.1.10) 2 2 k = k⊥ cos ψ k⊥ =|k⊥|= kx + ky x jk − − | − |2 = jkz jk r⊥ r⊥ /2z 2 k = k⊥ sin ψ e E(r⊥, 0)e d r⊥ (Fresnel) ψ = atan2(ky,kx) y 2πz S =| | jk −jkz −jk|r⊥|2/2z −jk|r⊥ |2/2z jkr⊥·r⊥ /z 2 and assuming that E(r⊥, 0) depends only on, r⊥ r⊥ , all angular integrations can be = e e E(r⊥, 0)e e d r⊥ 2πz S done with the help of the following integral of Eq. (19.12.22), (20.1.9) 2π jk⊥r⊥ cos(ψ−φ) dψ The last integral is recognized as the two-dimensional Fourier transform of the prod- = ⊥ ⊥ e J0(k r ) (20.1.11) −jk|r⊥ |2/2z uct, E(r⊥, 0)e , evaluated at k⊥ = kr⊥/z. 0 2π −jk z Eq. (20.1.8) can also be obtained from the exact form h(ˆ k⊥,z)= e z by assum- then, E(ˆ k⊥, 0) will depend only on k⊥ =|k⊥|, and will satisfy the following Hankel ing that for large z the evanescent modes will have already decayed and assuming the transform relationships, 2 + 2 2 approximation kx ky k for the propagating modes. Then, we can write: ∞ ˆ = 2 2 2 E(k⊥, 0) E(r⊥, 0)J0(k⊥r⊥) 2πr⊥dr⊥ |k⊥| 1 |k⊥| |k⊥| 0 k = k2 −|k⊥|2 = k 1 − k 1 − = k − z 2 2 ∞ (20.1.12) k 2 k 2k k⊥dk⊥ E(r⊥, 0) = E(kˆ ⊥, 0)J0(k⊥r⊥) 2π and, hence 0 −jk z −jkz j|k⊥|2z/2k e z e e Then, the cylindrical symmetry is preserved by the propagation process, and (20.1.9) can be written as follows, where we replaced, r⊥ · r⊥ = r⊥r cos(φ − φ ), and the φ Because of the assumption |k⊥| <k, the maximum transverse wavenumber will ⊥ integration was done with (20.1.11), be |k⊥|=k = 2π/λ, and correspondingly the smallest achievable transverse spatial ∼ | |∼ resolution will be Δr⊥ 1/ k⊥ λ, that is, about one wavelength. This is the basic ∞ 2 k⊥dk⊥ diffraction limit of optical instruments, such as lenses and microscopes. Implicit in −jkz jk⊥ z/ 2k E(r⊥,z)= e E(kˆ ⊥, 0)e J0(k⊥r⊥) Eq. (20.1.9) then, is that the range of integration in k⊥-space must be restricted to the 0 2π ∞ propagating range, |k⊥|≤k. jk 2 2 kr⊥r −jkz −jkr⊥/2z −jkr⊥ /2z ⊥ = e e E(r⊥, 0)e J0 2πr⊥dr⊥ Near-field optics methods [534–553], where the evanescent modes are not ignored, 2πz 0 z overcome this limitation and can achieve much higher, subwavelength, resolutions. Al- (20.1.13) though ordinary lenses are diffraction-limited, it has been shown recently [398] that Another special case is when the aperture field E(x,y, 0) depends only on one “superlenses” made from metamaterials having negative refractive index can achieve transverse coordinate, say, E(x, 0). Then, the dependence of (20.1.3) on the y direction perfect resolution. can be integrated out using the integral ∞ jk − − 2 Circularly Symmetric and One-Dimensional Cases e jk(y y ) /2z dy = 1 (20.1.14) 2πz −∞ In the special case when the fields have circular symmetry, as it happens often when working with circular apertures and scalar fields, the above expressions can be written and we obtain the following 1-D version of the propagation impulse response, as 0th-order Hankel transforms, i.e., with respect to the J0 Bessel function. ∞ ∞ jk −jkz −jkx2/2z −jky2/2z h(x, z)= h(r⊥,z)dz= e e e dy , or, −∞ 2πz −∞ jk − − 2 − 2 jkz jkx / 2z ˆ jkz jkxz/ 2k h(x, z)= e e h(kx,z)= e e (20.1.15) 2πz 20.1. Fresnel Approximation 927 928 20. Diffraction – Fourier Optics −jkz and the corresponding one-dimensional Fresnel formula, written convolutionally and in The incident field is a uniform plane wave, Einc(x, z)= E0e , whose values on the slit its plane-wave spectrum form, are E(x , 0)= E0. The diffracted field at distance z is given by Eq. (20.1.16): ∞ ∞ a − 2 − dkx jk − − − 2 jk − − − 2 jkz ˆ jkxz/ 2k jkxx jkz jk(x x ) /2z jkz jk(x x ) /2z E(x, z) = e E(kx, 0)e e E(x, z)= e E(x , 0)e dx = E0 e e dx −∞ 2π 2πz −∞ 2πz −a ∞ jk −jkz −jk(x−x)2/2z The integral can be reduced to the Fresnel integral F(x) of Appendix F by making the = e E(x , 0)e dx (20.1.16) 2πz −∞ change of variables: ∞ jk − − 2 − 2 k π k = e jkz e jkx /2z E(x , 0)e jkx /2z ejkxx /z dx (x − x)= u, v± = (±a − x) 2πz −∞ 2z 2 πz so that where again, the kx-integration may be restricted to |kx|≤k for distances z that are a v+ large enough so that the evanescent modes have decayed.