Gaussian Beams and Resonators

2.6 2.6.1 Gaussian Beams and Resonators Gaussian Beam Propagation The propagation of Gaussian beams through paraxial optical systems can be efficiently evaluated using the ABCD-law [4], which states that the q- parameter of a Gaussian beam passing a optical system described by an ABCD-marix is given by Aq1 + B q2 = , (2.259) Cq1 + D where q1 and q2 are the beam parameters at the input and the output planes of the optical system or component, see Figure 2.72 Figure 2.72: Gaussian beam transformation by ABCD law, [6], p. 99. To proove this law, we realize that it is true for the case of free space propagation, i.e. pure diffraction, comparing (2.259) with (2.229) and (2.246). If we can proove that it is additionally true for a thin lens, then we are finished, because every ABCD matrix (2x2 matrix) can be written as a product of a lower and upper triangular matrix (LR-decomposition) like the one for free space propagation and the thin lens. Note, the action of the lens is identical to the action of free space propagation, but in the Fourier-domain. In the Fourier domain the Gaussian beam parameter is replaced by its inverse (2.222) µ 2 ∙ ¶¸ x + y2 j exp −jk0 . q(z) 2q(z) ¶¸ ∙ µ 2 kz + ky2 e0 (kz , ky , z) = 2πj exp −jq(z) E 2k0 e0 (x, y, z) = E (2.260) (2.261) But the inverse q-parameter transforms according to (2.259) D q11 + C 1 , = 1 q2 B q1 + A (2.262) which leads for a thin lens to 1 1 1 = − . q2 q1 f (2.263) This is exactly what a thin lens does, see Eq.(2.225), it changes the radius of curvature of the phase front but not the waist of the beam according to 1 1 1 = − . R2 R1 f (2.264) With that finding, we have proven the ABCD law for Gaussian beam propagation through paraxial optical systems. The ABCD-matrices of the optical elements discussed so far including nonnomal incidence are summarized in Table 2.6. As an application of the Optical Element Propagation in Medium with index n and length L Thin Lens with focal length f Mirror under Angle θ to Axis and Radius R Sagittal Plane Mirror under Angle θ to Axis and Radius R Tangential Plane Brewster Plate under Angle θ to Axis and Thickness d, Sagittal Plane Brewster Plate under Angle θ to Axis and Thickness d, Tangential Plane ABCD-Matrix ¶ µ 1 L/n ¶ µ 0 1 1 0 −1/f 1 µ ¶ 1 0 −2 cos θ 1 R µ µ µ 1 −2 R cos θ 1 nd 0 1 1 0 d n3 1 0 1 ¶ ¶ ¶ Table 2.6: ABCD matrices for commonly used optical elements. Gaussian beam propagation, lets consider the imaging of a Gaussian beam with a waist w01 by a thin lens at a distance d1 away from the waist to a beam with a different size w02 , see Figure 2.73. d1 d2 zR1 zR2 Figure 2.73: Focusing of a Gaussian beam by a lens. There will be a new focus at a distance d2 .The corresponding ABCD matrix is of course the one from Eq.(2.257), which is repeated here ³ ´ ! ¶ à µ 1 − df2 1 − df2 d1 + d2 A B = . (2.265) C D −1 1 − d1 f f The q-parameter of the Gaussian beam at the position of minimum waist is πw2 πw2 purely imaginary q1 = jzR1 = j λ01 and q2 = jzR2 = j λ02 , where q2 = A q1 + B jzR1 A + B jzR1 A + B = = = jzR2 . C q1 + D jzR1 C + D jzR1 C + D (2.266) In the limit of ray optics, where the beam waists can be considered to by zero, i.e. zR1 = zR2 = 0 we obtain B = 0, i.e. the imaging rule of classical ray optics Eq.(2.256). It should not come at a surprise that for the Gaussian beam propagation this law does not determine the exact distance d2 of the position of the new waist. Because, in the ray analysis we neglected diffraction. Therefore, the Gaussian beam analysis, although it uses the same description of the optical components, gives a slightly different and improved answer for the position of the focal point. To find the position d2 , we request that the real part of the right hand side of (2.266) is zero, 2 AC = 0 BD − zR1 (2.267) which can be rewritten as 1 1 1 = − . z2 d2 f d1 + d1R1 −f (2.268) Again for zR1 → 0, we obtain the ray optics result. And the imaginary part of Eq.(2.266) leads to or ¢ 1 1 ¡ 2 2 = C2 , D + zR1 zR2 zR1 µ µ ¶2 " ¶2 # zR1 d1 1 1 1+ . = 2 1− 2 w02 w01 f d1 − f (2.269) (2.270) With the magnification M for the spot size, with is closely related to the Magnification Mr of ray optics, we can rewrite the results as ¯ ¯ p ¯ f ¯ 2 zR1 Magnification M = Mr / 1 + ξ , with ξ = d1 −f and Mr = ¯ d1 −f ¯ Beam waist Confocal parameter Distance to focus Divergence w02 = M · w01 2zR2 = M 2 2zR2 d2 − f = M 2 (d1 − f ) θ02 = θ01 /M (2.271) 2.6.2 Resonators With the Gaussian beam solutions, we can finally construct optical resonators with finite transverse extent, i.e. real Fabry-Perots, by inserting into the Gaussian beam, see Figure 2.74, curved mirrors with the proper radius of curvature, such that the beam is imaged upon itself. L R2 R1 z1 z2 Figure 2.74: Fabry-Perot resonator with finite beam cross section by inserting curved mirrors into the beam to back reflect the beam onto itself. Any resonator can be unfolded into a sequence of lenses and free space propagation. Here, we replace the curved mirrors by equivalent lenses with f1 = R1 /2, and f2 = R2 /2, see Figure 2.75. Figure 2.75: Two-mirror resonator unfolded. Note, only one half of the focusing strength of mirror 1 belongs to a fundamental period describing one resonator roundtrip. The product of ABCD matrices describing one roundtrip of the beam in the resonator according to Figure 2.75 is M= µ 1 −1 2f1 0 1 ¶µ 1 L 0 1 ¶µ 1 −1 f2 0 1 ¶µ 1 L 0 1 ¶µ 1 −1 2f1 0 1 ¶ . (2.272) To carry out this product and to formulate the cavity stability criteria, it is convenient to use the cavity parameters gi = 1−L/Ri , i = 1, 2. The resulting cavity roundtrip ABCD-matrix can be written in the form ¶ µ ¶ µ (2g1 g2 − 1) A B 2g2 L . (2.273) M= = C D 2g1 (g1 g2 − 1) /L (2g1 g2 − 1) Resonator Stability The ABCD matrices describe the dynamics of rays propagating inside the resonator. The resonator is stable if no ray escapes after many round-trips, which is the case when the magnitude of the eigenvalues of the matrix M are less than one. Since we have a lossless resonator, i.e. det|M| = 1, the product of the eigenvalues has to be 1 and, therefore, the stable resonator corresponds to the case of a complex conjugate pair of eigenvalues with a magnitude of 1. The eigenvalue equation to M is given by ¯µ ¶¯ ¯ (2g1 g2 − 1) − λ ¯ 2g2 L ¯ ¯ = 0, (2.274) det |M − λ · 1| = det ¯ 2g1 (g1 g2 − 1) /L (2g1 g2 − 1) − λ ¯ The eigenvalues are λ1/2 λ2 − 2 (2g1 g2 − 1) λ + 1 = 0. (2.275) q = (2g1 g2 − 1) ± (2g1 g2 − 1)2 − 1, (2.276) ½ exp (±θ) , cosh θ = 2g1 g2 − 1, for |2g1 g2 − 1| > 1 = (2.277) . exp (±jψ) , cos ψ = 2g1 g2 − 1, for |2g1 g2 − 1| ≤ 1 The case of a complex conjugate pair corresponds to a stable resontor. Therefore, the stability criterion for a stable two mirror resonator is |2g1 g2 − 1| ≤ 1. (2.278) The stable and unstable parameter ranges are given by stable : 0 ≤ g1 · g2 = S ≤ 1 unstable : g1 g2 ≤ 0; or g1 g2 ≥ 1. (2.279) (2.280) where S = g1 · g2 , is the stability parameter of the cavity. The stability criterion can be easily interpreted geometrically. Of importance are the distances between the mirror mid-points Mi and the cavity end points, i.e. gi = (Ri − L)/Ri = −Si /Ri , as shown in Figure 2.76. Figure 2.76: The stability criterion involves distances between the mirror mid-points Mi and the cavity end points. i.e. gi = (Ri − L)/Ri = −Si /Ri . The following rules for a stable resonator can be derived from Figure 2.76 using the stability criterion expressed in terms of the distances Si . Note, that the distances and radii can be positive and negative stable : 0 ≤ S1 S2 ≤ 1. R1 R2 (2.281) The rules are: • A resonator is stable if the mirror radii, laid out along the optical axis, overlap. • A resonator is unstable if the radii do not overlap or one lies within the other. Figure 2.77 shows stable and unstable resonator configurations. Figure 2.77: Illustration of stable and unstable resonator configurations. For a two-mirror resonator with concave mirrors and R1 ≤ R2 , we obtain the general stability diagram as shown in Figure 2.78. Figure 2.78: Stabile regions (black) for the two-mirror resonator. There are two ranges for the mirror distance L, within which the cavity is stable, 0 ≤ L ≤ R1 and R2 ≤ L ≤ R1 + R2 . It is interesting to investigate the spot size at the mirrors and the minimum spot size in the cavity as a function of the mirror distance L. Resonator Mode Characteristics The stable modes of the resonator reproduce themselves after one round-trip, i.e. Aq1 + B q1 = (2.282) Cq1 + D The inverse q-parameter, which is directly related to the phase front curvature and the spot size of the beam, is determined by µ ¶2 µ ¶ 1 A−D 1 1 − AD + = 0. (2.283) + q B q B2 The solution is µ ¶ q 1 A−D j =− (A + D)2 − 1 ± q 1/2 2B 2 |B| (2.284) If we apply this formula to (2.273), we find the spot size on mirror 1 µ ¶ q j λ 1 =− (A + D)2 − 1 = −j 2 . (2.285) q 1/2 2 |B| πw1 or w14 µ ¶2 g2 1 g1 1 − g1 g2 µ ¶2 µ ¶ λR1 R2 − L L = . π R1 − L R1 + R2 − L = 2λL π (2.286) (2.287) By symmetry, we find the spot size on mirror 3 by switching index 1 and 2: w24 µ 2λL π ¶2 g1 1 g2 1 − g1 g 2 µ ¶2 µ ¶ R1 − L λR2 L = . π R2 − L R1 + R2 − L = (2.288) (2.289) The intracavity focus can be found by transforming the focused Gaussian beam with the propagation matrix µ ¶µ ¶ 1 0 1 z1 M = −1 1 0 1 2f1 ¶ µ z1 z1 1 − 2f 1 , (2.290) = −1 1 2f1 to its new focus by properly choosing z1 , see Figure 2.74. A short calculation results in g2 (g1 − 1) 2g1 g2 − g1 − g2 L(L − R2 ) , = 2L − R1 − R2 z1 = L (2.291) (2.292) and, again, by symmetry g1 (g2 − 1) 2g1 g 2 − g1 − g2 L(L − R1 ) = L − z1 . = 2L − R1 − R2 z2 = L (2.293) (2.294) The spot size in the intracavity focus is wo4 µ ¶2 g1 g2 (1 − g1 g2 ) λL = π (2g1 g2 − g1 − g2 )2 µ ¶2 λ L(R1 − L)(R2 − L)(R1 + R2 − L) = . π (R1 + R2 − 2L)2 (2.295) (2.296) All these quantities for the two-mirror resonator are shown in Figure 2.79. 2 g1, g 1/2 20 0 5 10 Cavity Length, L / cm 15 20 0 5 10 Cavity Length, L / cm 15 20 0 5 10 Cavity Length, L / cm 15 20 0 5 10 Cavity Length, L / cm 15 20 w 0 / ( λ (R 1 *R 2 ) 1/2 /π ) 1.0 0.8 0.6 0.4 0.2 0.0 z2 / L 15 2 10 Cavity Length, L / cm xg 5 1 0 0.6 1.0 0.8 0.6 0.4 0.2 0.0 S=g 1.0 0.5 0.0 -0.5 -1.0 0.4 0.2 w 2 / ( λ R 2/π ) 1/2 w 1 / ( λ R 1 /π ) 1/2 0.0 6 4 2 0 6 4 2 0 z1 / L 1.0 0.8 0.6 0.4 0.2 0.0 Figure 2.79: From top to bottom: Cavity parameters, g1 , g2 , S, w0 , w1 , w2 , z1 and z2 for the two-mirror resonator with R1 = 10 cm and R2 = 11 cm. Hermite-Gaussian-Beams (TEMpq -Beams) It turns out that the Gaussian Beams are not the only solution to the paraxial wave equation (2.219). The stable modes of the resonator reproduce themselves after one round-trip, ∙ ∙√ ¸ ¸ "√ # 2x 2y w 0 e El,m (x, y, z) = Al,m Gl Gm · (2.297) w(z) w(z) w(z) µ 2 ¶ ¸ ∙ x + y2 + j(l + m + 1)ζ(z) exp −jk0 2R(z) where ∙ 2¸ u Gl [u] = Hl [u] exp − , for l = 0, 1, 2, ... 2 (2.298) are the Hermite-Gaussians with the Hermite-Polynomials H0 [u] H1 [u] H2 [u] H3 [u] = = = = 1, 2u, 4u2 − 1, 8u3 − 12u, (2.299) and ζ(z) is the Guoy-Phase-Shift according to Eq.(2.241). The lower order Hermite Gaussians are depicted in Figure 2.80 Figure 2.80: Hermite-Gauissians Gl (u) for l = 0, 1, 2 and 3. and the intensity profile of the first higher order resonator modes are shown in Figure 2.81. Figure 2.81: Intensity profile of TEMlm -beams, [6], p. 103. Besides the different mode profiles, the higher order modes experience greater phase advances during propogation, because they are made up of k-vectors with larger transverse components. Axial Mode Structure As we have seen for the Fabry-Perot resonator, the longitudinal modes are characterized by a roundtrip phase that is a multiple of 2π. Back then, we did not consider transverse modes. Thus in a resonator with finite transverse beam size, we obtain an extended family of resonances, with distinguishable field patterns. The resonance frequencies ω pmn are determined by the roundtrip phase condition φpmn = 2pπ, for p = 0, ±1, ±2, ... (2.300) For the linear resonator according to Figure 2.74, the roundtrip phase of a Hermite-Gaussian Tpmn -beam is φpmn = 2kL − 2(m + n + 1) (ζ(z2 ) − ζ(z1 )) , (2.301) where ζ(z2 ) − ζ(z1 ) is the additional Guoy-Phase-Shift, when the beam goes through the focus once on its way from mirror 1 to mirror 2. Then the resonance frequences are c (2.302) ωpmn = [πp + (m + n + 1) (ζ(z2 ) − ζ(z1 ))] . L If the Guoy-Phase-Shift is not a rational number times π, then all resonance frequencies are non degenerate. However, for the special case where the two mirrors have identical radius of curvature R and are spaced a distance L = R apart, which is called a confocal resonator, the Guoy-Phase-shift is ζ(z2 ) − ζ(z1 ) = π/2, with resonance frequencies πi ch πp + (m + n + 1) . (2.303) ωpmn = L 2 In that case all even, i.e. m + n, transverse modes are degenerate to the longitudinal or fundamental modes, see Figure 2.82. Figure 2.82: Resonance frequencies of the confocal Fabry-Perot resonator, [6], p. 128. The odd modes are half way inbetween the longitudinal modes. Note, in contrast to the plan parallel Fabry Perot all mode frequencies are shifted by π/2 due to the Guoy-Phase-Shift.