The concept of the Iwasawa decomposition [1] of ABCD ray-matrices is used for a systematic calculation of field distributions appearing in spherical optical systems. Examples of optimization for applications are calculate

Glur, H.

Kireev, A.

Weber, H.

Wyss, E.

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Appl. Phys. B 81, 59–63 (2005) Applied Physics BDOI: 10.1007/s00340-005-1843-0 Lasers and OpticsH. GLUR✉A.N. KIREEVE. WYSSH.P. WEBERSystematic engineering of paraxial coherentfields with spherical optical systemsInstitute of Applied Physics, University of Bern, Sidlerstrasse 5, 3012 Bern, SwitzerlandReceived: 1 February 2005 / Revised version: 4 April 2005Published online: 3 June 2005 • © Springer-Verlag 2005ABSTRACT The concept of the Iwasawa decomposition [1]of ABCD ray-matrices is used for a systematic calculationof field distributions appearing in spherical optical systems.Examples of optimization for applications are calculated.PACS 41.85.Ew; 42.15.Eq1 IntroductionIf the amplitude and phase of a light beam areknown over its cross-section at one location, it allows oneto calculate the field at any position. This article shows anefficient way to get systematically the possible field distri-butions in a spherical optical system for stigmatic, coherentand monochromatic beams. It allows the derivation of crite-ria for the optical system needed to come close to a desiredfield distribution. The method is applied for optimizing beamshaping (like e.g., in [2]) and the application of custom modesfor material processing.To calculate the second intensity moment σ and the phase-front curvature radius R the q-parameter formalism is wellsuited [3, 4]. The beam width σ and the curvature radiusof the phase-front R can be described by the the complexq-parameter which transforms in an optical system as [4]q2 = Aq1 + BCq1 + D . (1)A, B, C and D denote the ray-matrix elements and q is definedby the relation [3]1q= 1R(z) − iM22kσ 2(z) . (2)M2 corresponds to the beam propagation factor and z to thedistance in the direction of propagation. The field distributionhas to be calculated, e.g., by evaluating the Collins integral(3, 4) according to [5]. It is valid for beams propagatingthrough optical elements that can be described by 4 × 4 ray-matrices [5]. In our case the 4 × 4 matrix can be reduced toa 2 × 2 matrix due to rotational symmetry. To get the cal-culations more systematic and intuitive, the Iwasawa decom-position [1] of the 2 × 2 ray-matrices is used. It has to be✉ Fax: +41 (0)31-631-37-65, E-mail: glur@iap.unibe.chmentioned that the validity of the Collins integral (3, 4) islimited to cases with no loss or gain in the optical system andwhere the Fresnel approximation of the Kirchhoff integral isvalid.2 TheoryThe Collins integral can be written in the form [1]u(x, y) = −iλBexp(iπ DλBr2)×∫ ∫exp(− 2π iλB(xx ′ + yy′))× u′(x ′, y′) exp(iπ AλBr ′2)dx ′dy′ if B = 0 (3)u(x, y) = 1Aexp(iπCλAr2)u′(xA,yA)if B = 0 (4)where A, B, C and D are the beam propagation matrix el-ements and r2 = x2 + y2. The calculation of the field dis-tributions appearing in an optical system by means of theCollins integral (3, 4) can be done more systematically usingthe Iwasawa decomposition [1] of the ABCD matrices, whichis schematically described in Fig. 1.For stigmatic beams without loss of generality the ele-ments B and C can be scaled with the Rayleigh rangez0 = 2kσ20M2(5)to get a dimensionless AB ′C ′ D matrix.1 σ0 corresponds to thesecond intensity moment in the waist. How the correspondingparameters σ , σ0, M2 and R can be calculated from a givenfield distribution is shown in [3]. From now on we use thedimensionless parametersB ′ = Bz0, C ′ = Cz0, f ′ = fz0. (6)In the Iwasawa decomposition the only part of the matrixwhich can change the shape of the field is formally equivalentto a GRIN lens [6]. Mathematically this corresponds to aphase-space rotation or a fractional Fourier transformation.1Of course the vector (x, α) has to be transformed to (x/z0, α) and theq-parameter has also to be scaled with z0.60 Applied Physics B – Lasers and OpticsFIGURE 1 Schematic description of the matrix decomposition. A phase-space rotation (PR) followed by a pseudo magnification (Mag.) and a lensare equivalent to the transformation via the ABC D matrixThe complete decomposition of the dimensionless matrix hasthe form( A B ′C ′ D)=1 0−1f ′ 1︸ ︷︷ ︸(lens)(Mag.)︷ ︸︸ ︷b 001b(cos φ sin φ− sin φ cos φ)︸ ︷︷ ︸(PR), (7)where (PR) corresponds to a phase-space rotation and (Mag.)represents a magnification which scales up positions andscales down angles by the same factor. The operation (PR)is the only one for which Eqs. (3) and (4) have to be eval-uated. The other two parts can be calculated by Eq. (4). Byevaluating the expression (2) and comparing the left and theright sides, one obtainsφ = arctan(B ′A), b =√A2 + B ′2,f ′ = B′AA2+B ′2 − D= −AB ′A2+B ′2 + C ′. (8)The phase-space angle φ, which is a parameter of the shapeof the propagating field, only depends on the matrix elementsA and B ′ with the last parameter describing essentially thedeviation from the perfect imaging plane. This fact can alsobe concluded directly by looking at the part inside the integralof expression (3). The decomposition can be comprehended ina more intuitive way. Every beam propagation matrix can bedivided into three operations changing at least one of the prop-erties like the relative intensity distribution, the beam widthand the spherical curvature of the phase front. The first oper-ation (PR) is the only one which is able to change the relativefield distribution,2 the second one (Mag.) only changes σ andthe third one (lens) only changes R. If the field is starting inits waist, the operation (PR) does not change q (σ and R).3The systematic field propagation through the phase-space al-lows the investigation of the possible field distributions thatcan be found in the whole optical systems. This feature canbe even further generalized. The corresponding optical ele-ments of the system can be found after optimization for thebest shape of the field distribution for the given application.2Since u is complex, it is valid for the intensity profile as well as for thenon-spherical part of the phase-front.3Without any loss of generality, we can assume the field to be in its waist.If the field is not in the waist, we can calculate the location of the waist andadapt the ray-matrix.The phase-space rotations do not modify the shape of individ-ual spherical modes. However it modifies the superpositionof fields of individual modes. We consider in the followingtwo types of superpositions with well defined amplitude andphase relation.3 ExamplesWe concentrate on the influence of this phase angleonly since the other operations are mathematically trivial.3.1 The in-phase super-mode of a three-core fiberAs a first example the electromagnetic field distri-bution of the in-phase super-mode of a strongly-coupled fiberlaser array (ASM) is investigated. The generation of thesesuper-modes is aimed to solve the power and the brightnessup-scaling problem in fibers [7]. The calculated phase-spacerotations of the ASM are shown in Fig. 2.3.1.1 Detecting intensity peaks. A possible application is thesystematic search of maximum intensity peaks in optical sys-tems since it is important to know where and how large theyare to avoid material damages. Figure 3 shows the relativemagnitude of the maximum intensity peak compared to itsnear field value at different phase-space angles for the ASM.Using relations (8) we obtain for the intensity peak relativeto the original intensity peak in the optical systemII0= Irel(φ)b2= Irel(arctan( B ′A))A2 + B ′2 , (9)where I0 stands for the maximum initial intensity, I for themaximum local intensity and Irel for the function plotted inFig. 3. The maximum intensity peak with a fixed σ appearsat φ = 90◦. In a similar way e.g., the width of the 90% en-ergy mark in the whole optical system can be calculated in asystematic way.3.1.2 Material processing. Because at φ = 90◦ the field ofthe ASM is closest to radial symmetry, it is convenient to usethis field distribution for material processing. Inserting intoEq. (8) leads to the condition A = 0 for the ray-matrix. Ifadditionally the field has to be in its waist, D has to be 0 aswell. The simplest optical system that fulfills both conditionsis the f − f image 0 f′i1f ′i0 , (10)where f ′i stands for the focal length of the imaging lens andcorresponds to the parameter b in Eq. (7). Inserting the prop-agation matrix (10) into Eq. (2) leads tof ′i =σ ′2σ ′1(11)for the focal length of the imaging lens with an initial beamwidth σ ′1 and the desired beam width σ ′2.GLUR et al. Systematic engineering of paraxial coherent fields with spherical optical systems 61FIGURE 2 Intensity profiles of the ASM for different phase-space angles φFIGURE 3 Maximum intensity peak of the ASM in the phase-space3.2 The super-Gaussian beamAs second example the super-Gaussian field distri-bution is investigated as a practical approximation of the flattop profile which is desirable e.g., in second harmonic gen-eration. We investigated the 12th order super-Gaussian mode(SG M) which is defined asu(r ) = u0 exp{−(rw0)n}, (12)where n = 12 is the super-Gaussian index. Figure 4 showsthe intensity profiles for the SGM for different phase-spaceangles φ = 0◦or 90◦.4 ApplicationThe theory described above can be applied to dif-ferent problems. If we want to avoid non-spherical elementsin a laser resonator with the three-core fiber the phase-spaceangle has to be φ = 0◦ or 90◦ since only there the field has nonon-spherical aberrations.4 We try to find the optimum res-onator configuration for beam shaping. The deviation fromthe spherical phase-front only depends on φ as shown inFig. 5 for the SGM.This deviation is most pronounced between φ = 10◦ −30◦. Intuitively beam-shaping (like e.g., in [8]) is expected tobe done the best in this region, since the phase-front distortionis strongest. A physically more meaningful crucial parameteris the energy theoretically needed to superpose to the field toreach a phase conjugation∫dr · r · I (r )|e2iθ(r ) − e2iθ0 |2, (13)where I (r ) stands for the intensity distribution, θ (r ) for thephase of the field and θ0 for an arbitrary constant phase whichunderlies the condition∂∂θ0∫dr · r · I (r )|e2iθ(r ) − e2iθ0 |2 = 0. (14)4This does not automatically mean an f − f or a telescope image. Sphericalmirrors are allowed.62 Applied Physics B – Lasers and OpticsFIGURE 4 Intensity profiles of the SG M for different phase-space angles φFIGURE 5 Phase front of the SG M as a function of the phase-space rota-tion angle.Evaluating condition (14) results intan(2θ0) =∫dr · r · sin(2θ (r ))I (r )∫dr · r · cos(2θ (r ))I (r ) . (15)Evaluating Eq. (13) with θ0 from Eq. (15) leads to Fig. 6.As expected this relative overlap intensity gets zero for φ = 0◦and φ = 90◦ if the field has a plane phase-front at φ = 0◦.From Fig. 6, we can conclude that the optimum point togenerate the SG M with a adaptive mirror would be at φ ≈ 25◦which quantitatively underlines the intuitive guess. It is mostconvenient to have the desired field distribution in its waistsince in this case the theoretical maximum deformation theFIGURE 6 Overlap between the phase front distortion and the intensityprofile of the SG Mmirror has to manage is λ/2. The optimum scaling factor is atrade-off between exploiting the full resolution of the adaptivemirror and avoiding losses. It depends on the dimensions ofthe laser rod and the adaptive mirror and the transversal reso-lution of the adaptive mirror. The ABC D matrix is completelydefined by these three conditions, and we have to think abouta corresponding experimental setup. The ray-matrix can beobtained by inserting into Eq. (7) and getsb cos φ b sin φ− sin φbcos φb (16)GLUR et al. Systematic engineering of paraxial coherent fields with spherical optical systems 63FIGURE 7 Setup for the generation of the SG MThe simplest way to get a well-defined phase-space angleφ in the waist is to under-compensate the thermally inducedlens of the Nd:YAG laser as shown in Fig. 7.Then the resulting waist can be imaged with a telescope tothe adaptive mirror. The ray-matrix before the telescope hasapproximately5 the form1 − L′f ′ L′− 1f ′ 1 (17)Inserting Eq. (17) into Eq. (16) leads to b = cos φ,L ′ = cos φ sin φ and f ′ = cot φ. Inserting φ = 25◦ andassuming σ0 ≈ 0.5 mm at the output coupler, we getz0 ≈ 1.65 m and therefore f ≈ 1.1 m and L ′ ≈ 0.63 m. Notethat f stands for the resulting net lens effect, not for the focallength of the compensating lens itself. The magnificationfactor of the intra-cavity telescope is chosen to be M = 12.These values lead to Fig. 7.5The distance from the output coupler to the compensating lens has to besignificantly smaller than the focal length of the thermal lens and the distanceto the telescope.5 ConclusionThe Iwasawa decomposition of the ABC D ma-trix was shown to be an useful way to investigate the prop-agation of arbitrary coherent, monochromatic and stigmaticbeams through spherical optical systems. It is based on thedecomposition of the ray-matrix into a phase-space rotation,a magnification and a spherical lens transformation. Only thephase-space rotation provides information about the intensitydistribution and the non-spherical part of the phase front of thepropagating beam. The q-parameter formalism [3] has beenapplied for the calculation of the second intensity moment andthe spherical part of the phase-front. As practical examplesthe method has been applied for the characterization of thein-phase super-mode of a fiber array and the optimization ofa setup for the generation of a super-Gaussian mode with anadaptive mirror.REFERENCES1 R. Simon, N. Mukunda, J. Opt. Soc. Am. A 15, 8 (1998)2 C. Pare´, P.A. Be´langer, IEEE J. Quantum Electron. 30, 4(1994)3 P.A. Be´langer, Opt. Lett. 16, 4 (1991)4 N. Hodgson, H. Weber, Optical Resonators (Springer-Verlag, London,1997)5 S.A. Collins, J. Opt. Soc. Am. 60, 9 (1970)6 L.M. Bernardo, Opt. Eng. 35, 3 (1996)7 A.N. Kireev, T. Graf, H.P. Weber (submitted for publication in J.Lightwave Technol)8 M. Gerber, T. Graf, A. Kudryashov (submitted for publication in Appl.Phys. B)

Systematic engineering of paraxial coherent fields with spherical optical systems

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Systematic engineering of paraxial coherent fields with spherical optical systems

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