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9-1-1998 Production and Propagation of Hermite–Sinusoidal- Gaussian Laser Beams
Lee W. Casperson Portland State University
Anthony A. Tovar Murray State University
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Citation Details Anthony A. Tovar and Lee W. Casperson, "Production and propagation of Hermite-sinusoidal-Gaussian laser beams," J. Opt. Soc. Am. A 15, 2425-2432 (1998).
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Production and propagation of Hermite sinusoidal-Gaussian laser beams
Anthony A. Tovar Department of Physics and Engineering Physics, Murray State University, Murray, Kentucky 42071 -0009
Lee W. Casperson* The Rochester Theory Center and The Institute of Optics for Optical Science and Engineering, University of Rochester, Rochester, New York 14627-0186
Received December 22, 1997; revised manuscript received March 31, 1998; accepted May 1, 1998 Hermite-sinusoidal-Gaussian solutions to the wave equation have recently been obtained. In the limit of large Hermite-Gaussian beam size, the sinusoidal factors are dominant and reduce to the conventional modes of a rectangular waveguide. In the opposite limit the beams reduce to the familiar Hermite-Gaussian form. The propagation of these beams is examined in detail, and resonators are designed that will produce them. As an example, a special resonator is designed to produce hyperbolic-sine-Gaussian beams. This ring resonator contains a hyperbolic-co sine-Gaussian apodized aperture. The beam mode has finite energy and is perturba tion stable. © 1998 Optical Society of America [S0740-3232(98)02809-9] Oe1S codes: 140.3410, 230.5750, 350.5500.
1. INTRODUCTION through free space; and their phase fronts may be much more complicated than those of a spherical wave. It has long been known that the modes of a rectangular Although the sinusoidal beams used previously in the metal waveguide are sinusoidal functions. The ampli tude of the sinusoids is approximately zero at the metal study of rectangular waveguides are well known, these waveguide walls, which impose well-defined boundary beams have been only recently proposed for use in free conditions. When ends are put on these waveguides, space propagation. Since these beams would require in they become resonators, such as those used for the first finite energy, interest has centered on truncated sinu soidal beams for pseudonondiffracting beam applica maser oscillators. With the advent of the laser, it was 6 found that because of the small divergence of coherent la cations. An important alternative to these beams is 7 ser light, the rectangular waveguide walls were no longer given by the newly obtained sinusoidal-Gaussian beams. necessary.! The rectangular-symmetry beam modes of In particular, the much squarer hyperbolic-cosine these open resonators (with slightly spherical end mir Gaussian (cosh-Gaussian) beam shape may be more use rors) were found to be describable in terms of real-valued ful than the Gaussian beam shape at extracting energy Hermite polynomials multiplied by complex-valued from a laser amplifier. Gaussian functions. 2 These Hermite- Gaussian modes In an associated study Hermite-sinusoidal-Gaussian have been highly successful in characterizing the reso beams are obtained as rectangular-symmetried solutions 8 nant fields of low-diffraction-loss resonators. They also to the paraxial wave equation. In the limit of large have the attractive property that they retain their beam Hermite-Gaussian beam size, the sinusoidal factors are shape and spherical phase fronts as they propagate dominant, and the solutions become the modes of the rect through free space. angular waveguide. In the opposite limit of large sinu The real-argument Hermite-Gaussian modes apply to soidal beam size, the beams reduce to the complex laser resonators that do not contain spatial loss varia argument beam modes. The basic theory governing the tions that are due to, for example, Gaussian-profiled propagation of these beams through misaligned optical apodized apertures and laser amplifiers with a radial gain systems representable by complex ABCDGH matrices is profile. For lasers with these elements, alternative discussed in Section 2. The lowest-order beam mode is 7 complex-argument Hermite-Gaussian3,4 beam modes are the sinusoidal Gaussian. However, no resonator has found as solutions to the paraxial wave equation. These been identified previously that produces this mode. In beam modes are called complex argument on the basis Section 3 a variable-reflectivity mirror resonator is de that the arguments of the Hermite polynomials are com signed that has a hyperbolic sinusoidal Gaussian as its plex. The original real-argument modes are a special cavity mode. Variable-reflectivity mirrors and graded ~ase of the complex-argument modes.5 Unlike the real phase mirrors are commonly used to increase the mode 9 11 argument modes, the complex-argument modes do not, in volume of a laser resonator. - In this study the reso general, retain their beam shape as they propagate nator is designed to ensure that the sinusoidal-Gaussian
0740-3232/98/092425-08$15.00 © 1998 Optical Society of America 2426 J. Opt. Soc. Am. A/Vol. 15, No. 9/September 1998 A. A. Tovar and L. W. Casperson A. A. ~ beam profile repeats after a round trip. In addition to where again the curly brackets designate a linear combi wherE satisfying this oscillation condition, however, the cavity nation of function solutions. In deriving Eq. (5), it has tures, mode must also be stable with respect to inevitable also been assumed that the transverse spatial variation of trans' 12 perturbations. In Section 4 it is shown that the de the medium is slow, so that the square of the complex d~a al signed resonator mode is perturbation stable. propagation constant k in Eq. (4) is at most quadratic in x x anc and y. Equation (5) is identical to one in Ref. 8 if a sume, == ~/W, b == -~8IW, a' == D, b' == = ko(z) - k Ix(z)xI2 - k Iy(z)yI2 2 2 - k 2Az)x /2 - k 2y (z)y /2 (4) the Hermite-sinusoidal-Gaussian beam solutions to Eq. (6) (1) are8 2 E'(x, y, z) ~ E~(z)expl - i[Qx~Z) x + Qy~Z) y2 where the 2 subscript indicates the output plane for a given optical element or system. A 1 subscript would in dicate the corresponding input plane. The significance of + Sx(z)x + Sy(z)y + P(z) II the parameters of the Gaussian portion of the beam is In Eo contained in the relations by tl Mx = X [x - 8.(Z)]) Hm(w~) 1 1 Am requ --~-- (7) beall Rx 7TW 2 qx x are I X [y - 8,(Z)]) Hn(w!) tatio 1 1 Am TI cosh[Dx(z)x + (5) 8 y 1 - -- d ya + d;a, (10) f30 qy Vol. 15, No. 9/September 1998/J. Opt. Soc. Am. A 'son A. A. Tovar and L. W. Casperson 2427 Lbi- where R x and R y are the radii of the phase-front curva S x1 + Gx + H xlqx1 las tures, wx and W y are the spot sizes, d xa and d ya are the (19) S x2 = A x + Bxlqx1 Lof transverse displacements of the beam from the z axis, and lex d ~ a and d;a are the slopes of the propagating beam in the S y1 + Gy + HylqY1 x and y directions, respectively. It has also been as S 2 = ~--~-~~~ (20) lX y A y + B ylqY1 a sumed in writing Eqs. (7)-(10) that the input and output ri- planes of the optical system are in a medium with a low These transformations are valid for continuous media, gain per wavelength. thin optical elements, and optical systems. The other re 8 n- With this formalism an optical element is fully charac quired transformations are terized by three matrices: a 3 X 3 generalized beam ma 2 id W x2 = Wx12(Ax + Bx Iq x1)2 + 4iBx(Ax + B x Iq x1)lk 01 , ~d trix for each of the x and y directions and what can be viewed as a 1 X 1 matrix for the plane-wave portion of (21) of 2 D. the beam: Wy2 = WY12(A y + By Iqy1)2 + 4iBy(Ay + B y Iqy1)lk 01 , (22) ~- i- B x 0x2 = ox1(Ax + B xlqx1) + S X1Bxlk01 x) = [ D x (U XU;qx) , (11) ( u:;q ~: ~] + (BxGx - A xH x)lk01 , (23) d~ s I Sxux 2 Gx H x 1 S xux 1 j- Oy2 = oy1(Ay + BylqY1) + S Y1Bxlk01 By .a I + (BxGy - AyHx)/kOl , (24) = [ Dy (12) ( u;fqy) ~: ~] ( u;fql flx1 S yUy 2 G Hy 1 S yUy 1 (25) I~ I y flx2 = A x + B x I qx1 ' (Eb)Z = AJ(Ebh · (13) fly1 I flY2 A B I ' (26) The ABCD portion of the generalized beam matrices are = y + y qy1 identical to ordinary beam matrices. The complex G and I flx1B x (SX1 + Gx + H xlqx1) flx1Hx H terms account for element displacement or 1 Cy + DylqY1 H y , (18) + 2kOl (2Sy1 + Gy + HylqY1) qy2 A y + B ylqY1 J.--- - 2428 J. Opt. Soc. Am. A/Vol. 15, No. 9/September 1998 A. A. Tovar and L. W. Casperson (0) 1 + 2~Ol J:( Gx:,x - Hx :~)dz' 0.8 0.6 (29) 0.4 + 2~J:( Gy :: - Hy ::)dz' 0.2 Transformation equations (21)-(24) govern the propaga -4 -2 o 2 4 tion of the Hermite portion of the beam, and transforma tion equations (25)-(28) govern the propagation of the Input Output (b) Plane Plane sinusoidal portion of the beam. Equation (29) governs the propagation of the axial phase and gain that are due to the Gaussian, Hermite, and sinusoidal portions of the beam. This equation has a + sign. The minus sign is to be used with the hyperbolic sinusoids, and the plus sign is used for the conventional sinusoids. U The general procedure for obtaining the detailed propa gation characteristics for the beam parameter of the Gaussian portion of the beam through an optical system (c) 1 is as follows: 0.8 0.6 1. Determine the system matrix by multiplying the individual optical element matrices (as given in Refs. 14 0.4 and 15, for example) in the reverse of the order in which 0.2 they are encountered by the input beam. 2. Determine the input complex beam parameters q x -4 -2 o 2 4 and qy given an input spot size, radius of curvature, and wavelength from Eqs. (7) and (8). Fig. 1. (a) Field amplitude before the light beam enters the op 3. Determine the output complex beam parameters tical system, (b) example optical system, (c) field amplitude after from the Kogelnik transformation equations (17) and (18). the beam propagates through the optical system. 4. The output spot sizes and radii of curvature may be determined by again using Eqs. (7) and (8). Here we consider an input field whose complex amplitude is se Hermite-sinusoidal-Gaussian beams contain other pa la rameters and associated beam transformations. How (33) ' ever, the basic procedure to analyze these other param eters of the beam is essentially the same as the four steps so that the phase fronts are initially fiat (R 1 = (0); then given above for each parameter. As an example, consider the propagation of a cosh 1 Gaussian beam in a Fourier-transforming lens-free-space (34) optical system. If the lens has a focal length f and the free-space segment of the optical systems has a length of Combining Eqs. (31)-(34), it follows that the complex am wl f, then the system matrix is plitude of the output field is proportional to 2 2 IE~utl = exp[ -x /(", fl7T WI)2]cosh(iDI7T WI ", -If-IX), Th tio (35) tio: where the axial amplitude factor has been ignored, since Note that since the system is aligned, there are no G and only the beam shape is of interest here. If we further E H elements, and thus conventional complex 2 X 2 beam choose the focal length of the lens, f, to be matrices were used. With the beam matrix in Eq. (30), 2 the Kogelnik transformation equation (17) governing the 7TWI wh f= -",-, (36) beam parameter reduces to hYI not then the output field reduces to Gal (31) the bea I: Similarly, Eq. (25) becomes which is identical to the input field except for the imagi terE nary unit in the argument of the cosh factor. These in tern ma1 (32) put and output fields have been normalized and plotted in Fig. 1. roUl son -A. A. Tovar and L. W. Casperson Vol. 15, No. 9/September 1998/J. Opt. Soc. Am. A 2429 3. HYPERBOLIC·SINE·GAUSSIAN·MODE 1 RESONATOR DESIGN E ~ = "2 r mirror taper exp( g oll2)E ~ , o Although it is important to understand the propagation of sinusoidal-Gaussian beams in optical systems, it is also of .(f3o(Cx +Dx/qXl) 2 X exp ( - ~ -2 A + B /q X interest to be able to design laser resonators that can pro x x xl duce these beams. An example of such a design is given here. We begin with the observation that in addition to + [PI - ~ In(Ax + B x /Q xl ) complex sine-Gaussian and cosine-Gaussian beams, there are also hyperbolic-sine-Gaussian (sinh-Gaussian) and x cosh-Gaussian beams, which are governed by similar 2 D;lB 11) + f30A x + B x/q xl transformations. These alternative modes have certain important properties for practical applications, and our 2Dxl x), (41) example will involve a mode of this type. The basic cri X sinh ( A x + B x/q xl terion for determining whether a given field profile is a mode of a laser resonator is the oscillation condition, where r mirror is the amplitude reflectivity of the output which requires that the profile reproduce itself after a coupler, go is the intensity gain coefficient of the ampli round trip through the resonator. Thus we start by as fier, and l is the amplifier length. Equations (17), (25), suming that the initial field configuration corresponds to and (29) have been the basis for the transformations of an x-varying on-axis beam of the form the parameters lIq x ' Dx , and P, respectively, that have r been incorporated in Eq. (41). 2 Examining the argument of the sinh functions in E; ~ E;,o exp[ -i(2:: x + PI) ]sinh(fl x 1x) , (38) I 1 Eqs. (38) and (41), it can be seen that the sinh portion of the field will repeat if, for example, A x = 2 and B x where f30 = 2 7Tn 0 /A represents the dominant real part of = 0. Since the beam matrix must be unimodular the propagation constant k There are several ap I °. (AxDx - B xCx = 1), it follows that the most general form proaches that one could now take in designing a resonator of the beam matrix under these conditions is I that would support the mode given in Eq. (38). The os ) cillation condition mentioned above can be used with Eqs. !r I (17) and (25) to develop constraints on the parameters Mx= [~x 1~21· (42) lIqxl and Dxl ' In this way a resonator can, in principle, be designed to produce the field distribution in Eq. (38) by Our next objective is to find a matrix element Cx that e I using only conventional optical elements that are repre will satisfy the resonator constraints. It will be postu sented by ABCD beam matrices. But, applying the oscil lated, without loss of generality, that this element can be I lation condition to Eq. (25) results in D = 0, which is not written as a beam of the desired type. The novel sinh function in 3 2A I Eq. (6) also suggests the possibility of designing a resona Cx = -- - i --2-' (43) tor that includes optical elements that do not have an 2L 7TW aper I ABeD matrix representation. This possibility is illus If these elements are substituted into Eq. (41), the output trated here. field is given by In our resonator design the field given in Eq. (38) is imagined to immediately strike an apodized aperture [f3o(-3 whose amplitude transmission is E~ = 2-3/2rmirrortaper exp(goll2)E~ , o eXPl-if 4 2L t(X) = t aper cosh( Daperx). (39) This aperture does not have an ABCD matrix representa - i ~ + _1_)x2 + PI]) sinh(Dxlx). (44) 7TW 2q xl tion. Now if we choose Dxl = Daper> the field distribu aper tion after the aperture becomes If we further make the choices 1 1 4A 2 E2 = ~ t.p"E;,o exp[ -i(2:: x + PI) ]sinh(2flx1X) , i --2 ' (45) 1 qxl L 37TW aper (40) r mirror taper exp( g oll2) 2 - 3/2, (46) where we have used a standard product identity for the hyperbolic trigonometric functions. It is important to we find that the right-hand side ofEq. (44) reduces to Eq. note that with this particular aperture choice the sinh (38). Therefore the assumed field distribution does in Gaussian mode remains a sinh-Gaussian mode. Only deed repeat after a round trip through the resonator. the effective width of the sinh factor is changed when the It remains to determine a sequence of optical elements beam is transmitted through the aperture. that can be represented by the matrix in Eq. (42) with the In this ring laser design, the field distribution encoun Cx element given in Eq. (43). The procedure for synthe ters the output coupler, the amplifier, and an optical sys sis of Gaussian beam optical systems has been given tem that is otherwise representable in terms of a beam previously,16 and only the results for one possible realiza matrix before returning to plane 1. Thus the field after a tion will be given here. It can be shown by direct multi round trip is plication that Eq. (42) can be factored as 2430 J. Opt. Soc. Am. A/Vol. 15, No. 9/September 1998 A. A. Tovar and L. W. Casperson taper sinh( Caper n aper x) t(x) = (49) Amplifier caper sinh( n aperx) The denominator represents the input beam, and the nu merator represents the output beam. When caper is cho sen to be 2, Eq. (49) reduces to Eq. (39). The fundamen f= U12 COSh_GaUSSia) tal mode of a resonator whose transmission function is Aperture given by Eq. (49) would be sinh Gaussian. Similarly, if the sinh functions in Eq. (49) were replaced with cosh Distance Between Lenses: Ll3 functions, the corresponding resonator mode would be ~O"I C"ity L".tho L cosh Gaussian. It should also be noted that the cosh-Gaussian beams described here may in some cases have significant advan tages over conventional Gaussian beams. In particular, f= 2UIS the squarer mode profiles that can be obtained resemble Fig. 2. Resonator design that produces a sinh-Gaussian beam. the super-Gaussian field distributions that are known to be more efficient at extracting energy from an amplifying medium.7 As a corollary result, if Eq. (49) were replaced with a I=: .9 0.8 IZl ratio of Hermite polynomials, one could use the synthesis IZl .~ procedure demonstrated in this paper to design a resona I=: tor that would produce a specific Hermite-Gaussian e<:S 0.6 ~ mode. A desired Laguerre-Gaussian beam mode could 4. RESONATOR MODE STABILITY -1 0 2 The basic design procedure has been to assume that the mode at some plane within a resonator has the desired Position, x form and to show that after a round trip the mode repro Fig. 3. Curve (a) Gaussian, curve (b) cosh-Gaussian apodized duces the guessed form. In this way the mode satisfies aperture transmission profiles. the oscillation condition. However, physically realizable beam modes must also possess finite energy and be stable with respect to inevitable perturbations.12 For the sinh Gaussian mode to be stable, it must be stable with respect M x = [ -il\/( :W~p,,) ~ 1[ - ~/L ~ ] [ ~ to both Gaussian and hyperbolic sinusoidal perturba tions. Both of these types of perturbation are considered L/3] here. X [-1~2/L ~ ] [ ~ L:3] [ -15~(2L) ~][~ 1 . (47) A. Mode Stability with Respect to Gaussian Perturbations However, each of these matrices can be represented by ei The Gaussian portion of the beam is unchanged by the ther a length of free space, a lens, or a Gaussian apodized cosh aperture. It is stable ifl2 aperture. This ring resonator configuration is shown in Fig. 2. (50) It should be noted that the two apertures in this design However, from Eq. (42), A x = 2 and E x = 0, and thus are adjacent to each other. Therefore, in practice, they F s = 2, which indicates that the Gaussian portion of the could be combined into a single apodized aperture whose beam is stable. variable transmission is B. Mode Stability with Respect to Sinusoidal 2 ttotal(X) = taper cosh(naperx)exp( -x /w;per)' (48) Perturbations The unperturbed field is given by Eq. (38), and the corre This transmission profile is plotted in Fig. 3, where it is sponding perturbed field is also compared with the more familiar Gaussian transmis sion factor. 2 E; = E;,o exp[ -it 2:: x + PI) ]Sinh[(flx1 + 8)x], Equation (39) represents only one of a class of aperture 1 (51) transmission functions that may be used to design sinusoidal-Gaussian beam modes. In particular, Eq. (39) where 8 is the perturbation of r xl' After striking the is a special case of cosh aperture, this field becomes on A. A. Tovar and L. W. Casperson Vol. 15, No. 9/September 1998/J. Opt. Soc. Am. A 2431 stable. This result is consistent with Fig. 2 of Ref. 7, -,- E' [.( f30 2 9) E 2 - taper 1,0 exp -l 2qx1 X + p)]1 which shows that the cosine-Gaussian beam profile changes as it propagates through free space. In that fig u x sinh[(Dx1 + 8)x]cosh(Dx1x) (52) ure the cosine-Gaussian beam quickly converts (in less o than a Rayleigh length) into a predominantly cosh Gaussian beam. n taper, [.( f30 2 )] is 2 E 1,0 exp -l 2qx1 X + PI if ih x {sinh[ (2Dx1 + 8)x] + sinh( 8x n. (53) 5. SUMMARY )e Thus the aperture converts the perturbed field into two A straightforward procedure to design novel laser resona tors whose beam mode consists of a sinh function multi lS sinusoidal-Gaussian beams. Next we must propagate plied by a Gaussian function has been shown. The sche 1- these beams through the ABeD portion of the optical sys matic for an example ring sinh-Gaussian-mode resonator r, tem. Since Ax = 2 and Ex = 0, it follows from Eq. (25) design is shown in Fig. 2. A prerequisite for physical re Ie that the arguments of each of the sinh terms are reduced alization of the sinh-Gaussian beam mode is that it be ;0 by a factor of 2, so that stable to perturbations, and it has been shown that the .g E' r mirror taper exp( g 01l2) designed resonator produces a perturbation-stable mode. 3 2 E'1,0 However, it has also been shown that the corresponding a is resonator design for sine-Gaussian beams does not pro 2 duce perturbation-stable modes. It is therefore impos ,l- p x ex [ -i( 2:: X + P3) ] sible to design a resonator that produces sine-Gaussian In 3 :d modes by using these methods. X {sinh[(Dx1 + 8/2)x] + sinh(8x/2n. (54) As discussed, the procedure outlined herein may be L- However, the perturbed field after a round trip may be used to design novel laser resonators whose beam mode is written in terms of a new perturbation 8' as cosh Gaussian. Such beams resemble super-Gaussian beams, which are known to be more efficient at extracting energy from a laser amplifier. IE; ~ E;,o exp[ -i( 2:: x2 + PI) ]{Sinh[(fix 1 + 8')x]) , (55) e ' :l I By equating Eqs. (54) and (55), we can compare the initial ACKNOWLEDGMENTS perturbation 8 with its corresponding perturbation after a This work was supported in part by the National Aero ,s round trip, 8'. It follows that the curly bracketed terms nautics and Space Administration through the Kentucky e I in Eq. (54) must equal the curly bracketed term in Eq. Space Grant Consortium under subgrant WKU 521753- e (55): 96-04 and by the National Science Foundation under I grant PHY94-15583. t sinh[(Dx1 + 8')x] = sinh[(Dx1 + 8/2)x] + sinh(Dx1x/2). I (56) *Permanent address, Department of Electrical Engi i neering, Portland State University, P.O. Box 751, Port For small perturbations, 8x ~ 1 and 8'x ~ 1, and Eq. 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