Chapter 2 Angular Sp ectrum Representation of Optical Fields The angular sp ectrum representation is a mathematical technique to describ e optical elds in homogeneous media. Optical elds are describ ed as a sup erp osition of plane waves and evanescentwaves which are physically intuitive solutions of Maxwell's equations. The angu- lar sp ectrum representation is found to b e a very p owerful metho d for the description of laser b eam propagation and light fo cusing. Furthermore, in the paraxial limit, the angular sp ec- trum representation b ecomes identical with the framework of Fourier optics which extents its imp ortance even further. In the subsequent sections we will apply the angular sp ectrum representation to gain an understanding of the limits of far eld optics. 2.1 Basic Formalism Under angular sp ectrum representation we understand the series expansion of an arbitrary eld in terms of plane waves with variable amplitudes and propagation directions. Assume we know the electric eld Eratany p oint r =x; y ; z in space. For example, Er can b e the solution of an optical scattering problem as shown in Fig. 2.1 for which E = E + E .In inc scatt the angular sp ectrum picture, we draw an arbitrary axis z and consider the eld E in a plane z = const transverse to the chosen axis. In this plane we can evaluate the two dimensional Fourier transform of the eld E as 1 Z Z 1 i [k x + k y ] x y ^ E k ;k ; z = Ex; y ; z e dx dy ; 2.1 x y 2 1 1 2 2. ANGULAR SPECTRUM REPRESENTATION OF OPTICAL FIELDS where x; y are the Cartesian transverse co ordinates and k ;k the corresp onding spatial x y frequencies or recipro cal co ordinates. Similarly, the inverse Fourier transform reads as 1 Z Z 1 i [k x + k y ] x y ^ Ek ;k ; z e dk dk : 2.2 Ex; y ; z = x y x y 2 1 Notice, that in the notation of Eqs. 2.1 and 2.2 the eld E =E ;E ;E and its Fourier x y z ^ ^ ^ ^ transform E =E ; E ; E representvectors. Thus, the Fourier integrals hold separately x y z for eachvector comp onent. So far wehave made no requirements ab out the eld E, but we will assume that in the transverse plane the medium is homogeneous, isotropic, linear and source-free. Then, a time-harmonic, optical eld with angular frequency ! has to satisfy the vector Helmholtz equation 2 2 r + k E r=0; 2.3 p where k is determined by k =!=c n and n = " is the index of refraction. In order to get the time-dep endent eld Er;twe use the convention i! t Er;t= RefE re g : 2.4 Inserting the Fourier representation of Er Eq. 2.2 into the Helmholtz equation and de n- ing q 2 2 2 k k k k w ith Imfk g 0 ; 2.5 z z x y ^ we nd that the Fourier sp ectrum E evolves along the z-axis as ik z z ^ ^ Ek ;k ; z =E k ;k ;0e : 2.6 x y x y Einc z Escatt z = const. Figure 2.1: In the angular sp ectrum representation the elds are evaluated in planes z = const p erp endicular to an arbitrarily chosen axis z . 2.1. BASIC FORMALISM 3 The `' sign sp eci es that wehavetwo solutions that need to b e sup erimp osed: the `+' sign refers to a wave propagating into the half-space z>0 whereas the `' sign denotes a wave propagating into z<0. Eq. 2.6 tells us that the Fourier sp ectrum of E in an arbitrary image plane lo cated at z = const can b e calculated bymultiplying the sp ectrum in the object plane at z =0 by the factor expik z . This factor is called pr opag ator in recipro cal space. In z Eq. 2.5 we de ned that the square ro ot leading to k renders a result with p ositive imaginary z part. This ensures that the solutions remain nite for z !1. Inserting the result of Eq. 2.6 into Eq. 2.2 we nally nd for arbitrary z 1 Z Z 1 i [k x + k y k z ] x y z ^ Ek ;k ;0e dk dk Ex; y ; z = 2.7 x y x y 2 1 which is known as the angular spectrum representation. In a similar way,we can also represent the magnetic eld H by an angular sp ectrum as 1 Z Z 1 i [k x + k y k z ] x y z ^ H k ;k ;0e dk dk ; 2.8 Hx; y ; z = x y x y 2 1 1 By using Maxwell's equation H =i! rEwe nd the following relationship o ^ ^ between the Fourier sp ectra E and H 1 ^ ^ ^ H = Z [k =k E k =k E ] ; 2.9 x y z z y 1 ^ ^ ^ [k =k E k =k E ] ; H = Z z x x z y 1 ^ ^ ^ H = Z [k =k E k =k E ] ; z x y y x " p =" " is the wave imp edance of the medium. Although the angular sp ec- where Z = o o " tra of E and H ful ll Helmholtz equation they are not yet rigorous solutions of Maxwell's equations. We still have to require that the elds are divergence free, i.e. rE = 0 and rH = 0. These conditions restrict the k vector to directions p erp endicular to the eld vectors k E = k H = 0. For the case of a purely dielectric medium with no losses the index of refraction n is a real and p ositive quantity. The wavenumber k is then either real or imaginary and turns z the factor expik z into an oscillatory or exp onentially decaying function. For a certain z k ;k pair we then nd two di erentcharacteristic solutions x y i [k x + k y ] i jk j z 2 2 2 x y z P lane waves : e e ; k + k k x y 2.10 i [k x + k y ] jk jjz j 2 2 2 x y z E v anescent w av es : e e ; k + k >k x y Hence, we nd that the angular sp ectrum is indeed a sup erp osition of plane waves and evanescent waves. Plane waves are oscillating functions in z and are restricted by the condi- 2 2 2 2 2 2 tion k + k k . On the other hand, for k + k >k we encounter evanescentwaves with x y x y an exp onential decay along the z axis. Fig. 2.2 shows that the larger the angle b etween the 4 2. ANGULAR SPECTRUM REPRESENTATION OF OPTICAL FIELDS k-vector and the z axis is, the larger the oscillations in the transverse plane will b e. A plane 2 2 wave propagating in direction of z has no oscillations in the transverse plane k + k = 0, x y whereas, in the other limit, a plane wave propagating at a right angle to z shows the highest 2 2 2 spatial oscillations in the transverse plane k + k = k . Even higher spatial frequencies x y are achieved byevanescentwaves. In principle, an in nite bandwidth of spatial frequencies can b e achieved. However, the higher the spatial frequencies of an evanescentwave are, the stronger the eld decay along the z axis will b e. Therefore, practical limitations make the bandwidth nite. So far we know from Eq. 2.6 how the Fourier sp ectrum of E propagates along the z axis. Let us now determine how the elds themselves evolve. For this purp ose we denote 0 0 the transverse co ordinates in the ob ject plane at z =0asx ;y and in the image plane at z = const as x; y . The elds in the image plane are describ ed by the angular sp ectrum ^ Eq. 2.7. We just have to express the Fourier sp ectrum Ek ;k ; 0 in terms of the elds in x y the ob ject plane. Similar to Eq. 2.1 this Fourier sp ectrum can b e represented as 1 Z Z 0 0 1 0 0 i [k x + k y ] 0 0 x y ^ Ex ;y ; 0 e dx dy : 2.11 E k ;k ;0 = x y 2 1 After inserting into Eq. 2.6 we nd the following expression for the eld E in the image plane z = const: 1 1 Z Z Z Z 0 0 1 i [k xx + k y y k z ] 0 0 0 0 x y z Ex; y ; z = e dx dy dk dk : 2.12 Ex ;y ;0 x y 2 4 1 1 This equation describ es an invariant lter with the following impulse resp onse propagator a) z b) z c) ky k k x x plane waves 2 2 2 kx +ky = k ϕ kz kx x evanescent waves E Figure 2.2: a Representation of a plane wave propagating at an angle ' to the z axis. b Illustration of the transverse spatial frequencies of plane waves incident from di erent angles. 2 2 1=2 The transverse wavenumber k + k dep ends on the angle of incidence and is limited to x y the interval [0 :: k ]. c The transverse wavenumb ers k , k of plane waves are restricted to a x y circular area with radius k .Evanescentwaves ll the space outside the circle.