Paraxial Wave Equation and Gaussian Beams

2.4. PARAXIAL WAVE EQUATION AND GAUSSIAN BEAMS 2.4 87 Paraxial Wave Equation and Gaussian Beams So far, we have only treated optical systems operating with plane waves, which is an idealization. In reality plane waves are impossible to generate because of there infinite amount of energy required to do so. The simplest (approximate) solution of Maxwell’s equations describing a beam of finite size is the Gaussian beam. In fact many optical systems are based on Gaussian beams. Most lasers are designed to generate a Gaussian beam as output. Gaussian beams stay Gaussian beams when propagating in free space. However, due to its finite size, diffraction changes the size of the beam and lenses are imployed to reimage and change the cross section of the beam. In this section, we want to study the properties of Gaussian beams and its propagation and modification in optical systems. 2.4.1 Paraxial Wave Equation We start from the Helmholtz Equation (2.18) ¡ ¢e ∆ + k02 E(x, y, z, ω) = 0, (2.214) with the free space wavenumber k0 = ω/c0 . This equation can easily be solved in the Fourier domain, and one set of solutions are of course the plane waves with wave vector |k|2 = k02 . We look for solutions which are polarized in x-direction e e y, z) ex . E(x, y, z, ω) = E(x, (2.215) We want to construct a beam with finite transverse extent into the x-y-plane and which is mainly propagating into the positive z-direction. As such we may try a superposition of plane waves with a dominant z-component of the k-vector, see Figure 2.56. The k-vectors can be written as kz with kx , ky << k0 . q = k02 − kx2 − ky2 , µ ¶ kx2 − ky2 . ≈ k0 1 − 2k02 (2.216) 88 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS y x k z Figure 2.56: Construction of a paraxial beam by superimposing many plane waves with a dominante k-component in z-direction. Then we obtain for the propagating field Z +∞ Z +∞ e e0 (kx , ky ) · E(x, y, z) = E −∞ −∞ ∙ µ ¶ ¸ kx2 + ky2 exp −jk0 1 − z − jkx x − jky y dkx dky , 2k02 Z +∞ Z +∞ e0 (kx , ky ) · = E −∞ −∞ ¶ ¸ ∙ µ 2 kx + ky2 z − jkx x − jky y dkx dky e−jk0 z , (2.217) exp j 2k0 e0 (kx , ky ) is the amplitude for the waves with the corresponding transwhere E verse k-component. This function should only be nonzero within a small range kx , ky ¿ k0 . The function ¶ ∙ µ 2 ¸ Z +∞ Z +∞ kx + ky2 e e z − jkx x − jky y dkx dky E0 (x, y, z) = E0 (kx , ky ) exp j 2k0 −∞ −∞ (2.218) is a slowly varying function in the transverse directions x and y, and it can be easily verified that it fulfills the paraxial wave equation µ 2 ¶ ∂ e ∂ −j ∂2 e0 (x, y, z). E E0 (x, y, z) = + (2.219) ∂z 2k0 ∂x2 ∂x2 Note, that this equation is in its structure identical to the dispersive spreading of an optical pulse. The difference is that this spreading occurs now in the two transverse dimensions and is called diffraction. 2.4. PARAXIAL WAVE EQUATION AND GAUSSIAN BEAMS 2.4.2 89 Gaussian Beams Since the kernel in Eq.(2.218) is quadratic in the transverse k-components using a two-dimensional Gaussian for the amplitude distribution leads to a beam in real space which is also Gaussian in the radial direction because of the resulting Gaussian integral. By choosing for the transverse amplitude distribution ¸ ∙ 2 kx + ky2 e0 (kx , ky ) = exp − , (2.220) E 2kT2 Eq.(2.218) can be rewritten as ¶ ∙ µ 2 ¸ Z +∞ Z +∞ kx + ky2 e E0 (x, y, z) = (z + jzR ) − jkx x − jky y dkx dky , exp j 2k0 −∞ −∞ (2.221) 2 with the parameter zR = k0 /kT , which we will later identify as the Rayleigh range. Thus, Gaussian beam solutions with different finite transverse width in k-space and real space behave as if they propagate along the z-axis with different imaginary z-component zR . Carrying out the Fourier transformation results in the Gaussian Beam in real space µ 2 ¶¸ ∙ 2 + y x j e0 (x, y, z) = . (2.222) exp −jk0 E z + jzR 2(z + jzR ) The Gaussian beam is often formulated in terms of the complex beam parameter or q-parameter. The propagation of the beam in free space and later even through optical imaging systems can be efficiently described by a proper transformation of the q-parameter µ 2 ¶¸ ∙ r 1 e0 (r, z) = exp −jk0 . (2.223) E q(z) 2q(z) Free space propagation is then described by q(z) = z + jzR (2.224) Using the inverse q-parameter, decomposed in real and imagniary parts, 1 1 λ = −j 2 . q(z) R(z) πw (z) (2.225) 90 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS leads to √ ∙ ¸ 2 2 2P r r e0 (r, z) = √ E exp − 2 − jk0 + jζ(z) . πw(z) w (z) 2R(z) (2.226) Thus w(z) is the waist of the beam and R(z) is the radius of the phase fronts. We the beam such that the Gaussian beam intensity ¯ normalized ¯2 ¯e ¯ I(z, r) = ¯E0 (r, z)¯ expressed in terms of the power P carried by the beam is given by ∙ ¸ 2r2 2P exp − 2 , I(r, z) = πw2 (z) w (z) Z ∞ Z 2π I(r, z) rdr dϕ. i.e. P = 0 (2.227) (2.228) 0 The use of the q-parameter simplifies the description of Gaussian beam propagation. In free space propagation from z1 to z2 , the variation of the beam parameter q is simply governed by q2 = q1 + z2 − z1 . (2.229) where q2 and q1 are the beam parameters at z1 and z2 . If the beam waist, at which the beam has a minimum spot size w0 and a planar wavefront (R = ∞), is located at z = 0, the variations of the beam spot size and the radius of curvature of the phase fronts are explicitly expressed as " µ ¶2 #1/2 z w(z) = wo 1 + , (2.230) zR and ∙ R(z) = z 1 + ³ z ´2 ¸ R z , (2.231) where zR is called the Rayleigh range. The Rayleigh range is the distance over which the cross section of the beam doubles. The Rayleigh range is related to the initial beam waist and the wavelength of light according to zR = πwo2 . λ (2.232) 2.4. PARAXIAL WAVE EQUATION AND GAUSSIAN BEAMS 91 Intensity Figure 2.57 shows the intensity of the Gaussian beam according to Eq.(2.227) for different propagation distances. Figure 2.57: The normalized beam intensity I/I0 as a function of the radial distance r at different axial distances: (a) z=0, (b) z=zR , (c) z=2zR . The beam intensity can be rewritten as ∙ ¸ w02 2P 2r2 I(r, z) = I0 2 . exp − 2 , with I0 = w (z) w (z) πw02 (2.233) For z > zR the beam radius growth linearly and therefore the area expands quadratically, which brings down the peak intensity quadratically with propagation distance. On the beam axis (r = 0) the intensity is given by I(r, z) = I0 w02 = w2 (z) 1+ I0 ³ ´2 . (2.234) z zR The normalized beam intensity as a function of propagation distance is shown in Figure 2.58 92 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS Figure 2.58: The normalized Beam intensity I(r = 0)/I0 on the beam axis as a function of propagation distance z [6], p. 84. Power The fraction of the total power contained in the beam up to a certain radius is Z 2π r0 P (r < r0 ) = I(r, z)rdr P P 0 ¸ ∙ Z r0 2r2 4 rdr (2.235) exp − 2 = w2 (z) 0 w (z) ∙ ¸ 2r02 = 1 − exp − 2 . w (z) Thus, there is a certain fraction of power within a certain radius of the beam P (r < w(z)) = 0.86, P P (r < 1.5w(z)) = 0.99. P (2.236) (2.237) Beam radius Due to diffraction, the smaller the spot size at the beam waist, the faster the beam diverges according to 2.230 as illustrated in Figure ??. 2.4. PARAXIAL WAVE EQUATION AND GAUSSIAN BEAMS 93 Planes of const. Phase Beam Waist z/z R Figure 2.59: Gaussian beam and its characteristics. Beam divergence The angular divergence of the beam is inversely proportional to the beam waist. In the far field, the half angle divergence is given by θ= λ , πwo (2.238) see Figure 2.59. Confocal parameter and depth of focus In linear microscopy, only a layer which has the thickness over which the beam is focused, called depth of focus, will contribute to a sharp image. In nonlinear microscopy (see problem set) only a volume on the order of beam cross section times depth of focus contributes to the signal. Therefore, the depth of focus or confocal parameter of the Gaussian beam, is the distance over which the beam stays focused and is defined as twice the Rayleigh range b = 2zR = 2πwo2 . λ (2.239) The confocal parameter depends linear on the spot size (area) of the beam and is inverse to the wavelength of light. At a wavelength of 1μm a beam with a radius of wo = 1cm,.the beam will stay focussed ove distances as long 94 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS 600m. However, if the beam is stronlgy focussed down to wo = 10μm the field of depth is only 600μm. Phase The phase delay of the Gaussian beam is r2 Φ(r, z) = k0 z − ζ(z) + k0 2R(z) µ ¶ z ζ(z) = arctan . zR (2.240) (2.241) On beam axis, there is the additional phase ζ(z) when the beam undergoes focussing as shown in Figure 2.60. This is in addition to the phase shift that a uniform plane wave already aquires. Figure 2.60: Phase delay of a Gaussian beam relative to a uniform plane wave on the beam axis [6], p. 87. This phase shift is known as Guoy-Phase-Shift. This effect is known as Guoy-Phase-Shift. The third term in the phase shift is parabolic in the radius and describes the wavefront (planes of constant phase) bending due to the focusing, i.e. distortion from the uniform plane wave. 2.4. PARAXIAL WAVE EQUATION AND GAUSSIAN BEAMS 95 Figure 2.61: The radius of curvature R(z) of the wavefronts of a Gaussian beam [6], p. 89. 2 r The surfaces of constant phase are detemined by k0 z − ζ(z) + k0 2R(z) = const. Since the radius of curvature R(z) and the additional phase ζ(z) are slowly varying functions of z, i.e. they are constant over the radial variation of the wavefront, the wavefronts are paraboloidal surfaces with radius R(z), see Figures 2.61 and 2.62. Figure 2.62: Wavefronts of a Gaussian beam, [6] p. 88. For comparison, Figure 2.63 shows the wavefront of (a) a uniform plane wave, (b) a spherical wave and (c) a Gaussian beam. At points near the beam center, the Gaussian beam resembles a plane wave. At large z, the beam behaves like a spherical wave except that the phase fronts are delayed by a quarter of the wavlength due to the Guoy-Phase-Shift. 96 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS Figure 2.63: Wavefronts of (a) a uniform plane wave;(b) a spherical wave; (c) a Gaussian beam [5], p. 89.