We theoretically analyze the influence of the Gouy phase shift on the nonlinear interaction between waves of different frequencies. We focus on $\chi^{(2)}$ interaction of optical fields, e.g. through birefringent crystals, and show that focussing, stronger than suggested by the Boyd-Kleinman factor, can further improve nonlinear processes. An increased value of 3.32 for the optimal focussing parameter for a single pass process is found. The new value builds on the compensation of the Gouy phase shift by a spatially varying, instead constant, wave vector phase mismatch. We analyze the single-ended, singly resonant standing wave nonlinear cavity and show that in this case the Gouy phase shift leads to an additional phase during backreflection. Our numerical simulations may explain ill-understood experimental observations in such devices

Lastzka, N

Schnabel, R

Optics Express

arXiv

Lastzka, N.

Schnabel, R.

MPG.PuRe

The Gouy phase shift in nonlinear interactions of waves

Lastzka, Nico

Schnabel, Roman

Max Planck Society eDoc Server

arXiv:physics/0611257 v1   27 Nov 2006The Gouy phase shift in nonlinear interactions of wavesNico Lastzka1 and Roman Schnabel11Institut fu¨r Gravitationsphysik, Leibniz Universita¨t Hannover and Max-Planck-Institut fu¨rGravitationsphysik (Albert-Einstein-Institut), Callinstr. 38, 30167 Hannover, Germany(Dated: November 28, 2006)We theoretically analyze the influence of the Gouy phase shift on the nonlinear interaction betweenwaves of different frequencies. We focus on χ(2) interaction of optical fields, e.g. through birefringentcrystals, and show that focussing, stronger than suggested by the Boyd-Kleinman factor, can furtherimprove nonlinear processes. An increased value of 3.32 for the optimal focussing parameter for asingle pass process is found. The new value builds on the compensation of the Gouy phase shift bya spatially varying, instead constant, wave vector phase mismatch. We analyze the single-ended,singly resonant standing wave nonlinear cavity and show that in this case the Gouy phase shift leadsto an additional phase during backreflection. Our numerical simulations may explain ill-understoodexperimental observations in such devices.PACS numbers: 03.67.-a, 42.50.-p, 03.65.UdNonlinear interactions of waves, in particular those ofoptical fields, have opened new research areas and havefound various applications. In general, waves of differentfrequencies are coupled via nonlinear media, like bire-fringent crystals. Examples are the production of higherharmonics of laser radiation [1], the generation of tunablefrequencies through optical parametric oscillation [2] andthe generation of nonclassical light [3, 4] for high preci-sion metrology [5, 6], fundamental tests of quantum me-chanics and quantum information [7]. The efficiency of anonlinear process depends on parameters of the nonlinearmedium, and generally increases with higher intensitiesof the fields involved and with better phase matching oftheir wave fronts. To achieve strong nonlinear interac-tions, pulsed laser radiation, strong focussing and, espe-cially for continuous wave radiation, intensity build-upin resonators are used. In plane wave theory perfectphase matching is achieved if the wave fronts of inter-acting fields propagate with the same velocity. For fo-cussed laser beams, however, this is not true because ofthe well-known Gouy phase shift. This phase shift oc-curs due to the spatial confinement of a focussed waveand generally depends on the spatial mode as well as thefrequency of the wave [8]. The influence of focussing intoa nonlinear medium has been investigated by Boyd andKleinman in great detail [9]. They discovered that theefficiency of the nonlinear process does not monotonicallyincrease with decreasing focal size. They especially con-sidered the lowest order nonlinearity that enables secondharmonic generation (SHG) and optical parametric am-plification (OPA) and is described by the susceptibilityχ(2), and numerically found an optimum factor betweenthe length of the nonlinear crystal and the Rayleigh rangeof the focussed Gaussian beam for a single pass throughthe crystal.In this Letter we show that focussing stronger thansuggested by the Boyd Kleinman factor can further im-prove nonlinear processes. We show that this effect canbe understood by considering the Gouy phase shift be-tween the interacting waves. We also show that the Gouyphase shift results in a non-trivial phase mismatch prob-lem in standing wave cavities.Boyd and Kleinman have found that the maximumnonlinear coupling between two Gaussian beams of fun-damental (subscript 1) and second harmonic waves (sub-script 2) is achieved for a positive wave vector phase mis-match ∆k=2|k1|−|k2|>0 which increases with decreas-ing waist size of the beam. For a single pass through anonlinear crystal of length L they numerically found theoptimal focussing parameter given by the relationξ :=L2zR= 2.84 , (1)where zR = piw20n/λ is the Rayleigh range of the beamsinside the crystal, and w0, n and λ are the beam’s waistsize, refractive index and wavelength, respectively. Wefirst show that the Boyd-Kleinman factor according toEq. (1) is a consequence of maximizing the intensity ofthe mean pump field inside the nonlinear medium underthe constraint of the Gouy phase shift. In a χ(2) mediumthe nonlinear interaction is described by the following setof differential equations∂zE0,1(z) ∝ E∗0,1(z)E0,2(z) · g∗(z) , (2)∂zE0,2(z) ∝ E20,1(z) · g(z) , (3)g(z) :=ei∆kz1 + i z−z0zR=w0w(z)ei(∆kz+∆φ(z)) , (4)where E0,1, E0,2 are the electrical fields of the fundamen-tal and the harmonic mode in the focal center at positionz0, and ∆k = 4pi/λ∆n is the phase mismatch betweenthe two interacting modes. Here we use the following2abbreviations∆φ(z) = −arctan(z − z0zR), (5)w(z) = w0√1 +(z − z0zR)2, (6)where w(z) corresponds to the beam width at the posi-tion z. In plane wave theory one finds g(z) = exp(i∆kz)and ∆φ = 0, and equal indices of refraction for thetwo interacting modes provide the maximum nonlinear-ity. When focussing the beam into a nonlinear materialhowever, there is a non zero phase difference ∆φ. Sincethe phase difference between a plane wave and a focussedGaussian beam is given by the Gouy phase shift, ∆φshould be the difference of two such phase shifts. Whenconsidering phase shifts between different oscillator fre-quencies, phases have to be frequency normalized. Wetherefore introduce the Gouy phase shift normalized tothe optical frequency of mode iφ˜G(ωi) :=φGωi= −(m+ n+ 1)ωiarctan(z − z0zR), (7)where m and n describe the spatial Hermite-Gaussianmodes (TEMmn). If we now normalize ∆φ to the har-monic frequency we find∆φω2= φ˜G(ω1)− φ˜G(ω2) . (8)We point out, that for an optimized nonlinear interactionof Gaussian beams the Rayleigh ranges are identical forall modes involved. In the case of frequency conversionof a single pump field this is automatically realized bythe nonlinear process. For the χ(2)-processes consideredin Eqs. (2)-(4) (m = n = 0) we find∆φ = φG . (9)From this one can conclude that the Gouy phase shiftleads to a nonperfect matching of the (nonplanar) phasefronts in nonlinear processes. To quantify this effect wedefine the effective nonlinearity κ of the process. Thisquantity is proportional to the conversion efficiency inSHG as well as to the optical gain of OPA. For weakinteraction, i. e. the pump field is not depleted by thenonlinear interaction, κ is given byκ :=∣∣∣∣∫dzg(z)w0∣∣∣∣2, (10)where the integration is taken over the whole interactionlength. For a single pass through a nonlinear medium oflength L the effective nonlinearity is given byκsp =∣∣∣∣∣∫ L0dzei(∆kz+φG(z))w(z)∣∣∣∣∣2. (11)FIG. 1: (Color online) Left: For weak focussing into the non-linear medium (ξ = 0.18) the Gouy phase shift can be com-pensated by choosing ∆k = 1/zR, and perfect phase match-ing can be realized over the full crystal length. Right: Forstronger focussing (ξ = 2.03) a constant ∆k can not provideperfect phase matching. (a) Gouy phase shift ∆φ, (b) com-pensating phase ∆kz, (c) overall phase φ0, where a constantvalue describes perfect phase matching.This quantity is maximized if the averaged fieldstrength inside the crystal is maximized, i. e. if the focusis placed in the crystal center and if the condition∆kz +∆φ(z) = φ0 = const. (12)is satisfied. In this case all partial waves are produced ex-actly in phase to each other, and perfect phase matchingis realized.Curves (a) in Fig. 1 show the differential Gouy phaseshifts ∆φ(z) = φG(z) for weak and strong focussing,respectively. For weak focussing the gouy phase shiftevolves linearly inside the medium, and one can compen-sate this phase mismatch by choosing ∆k = 1/zR > 0, asfound by Boyd and Kleinman [9]. For stronger focussing,however, it is not possible to achieve perfect compensa-tion from ∆k that is constant over the crystal. Curves (b)show the compensating linear phase ∆kz that is due tothe propagation inside the medium and curves (c) showthe total phase φ0. The value of ∆k was chosen to pro-vide the lowest variance of φ0 over the whole interactionrange.We now show that it is possible to realize perfect phasematching for an arbitrary focussing by applying the fol-lowing position dependent index of refraction∆nsp(z) =λ4pi∆k(z) =λ4piφ0 + arctan(z−z0zR)z, (13)where the constant value of the phase φ0 is set by theGouy phase at the entrance surface of the nonlinearmediumφ0 = arctan(z0zR)= φG(0) . (14)Fig. 2 shows ∆nsp(z) for a nonlinear crystal of lengthL for different focussing parameters ξ, with focus placedinto the crystal’s center. Curves (a) to (c) could experi-mentally be realized by applying an appropriate tempera-3FIG. 2: (Color online) Change of refractive index along thecrystal to compensate for the Gouy phase shift, for three dif-ferent strengths of focussing. The temperature scale on theright corresponds to a MgO(7%) : LiNbO3 crystal that hasbeen used in [6]. (a) ξ = 0.55, (b) ξ = 1.14, (c) ξ = 3.32.ture gradient along z-direction. Alternatively, an electri-cal field applied to the crystal could be used. The tem-perature values on the right vertical axis constitute anexample for 7% magnesium-oxide-doped lithium niobate(MgO:LiNbO3). Data was deduced from measurementsof the nonlinear efficiency of crystals used in type I OPAin [6].The effective nonlinearity for the single pass setup withperfect Gouy phase compensation can be written asκsp =1ξln2(√1 + ξ2 + ξ√1 + ξ2 − ξ), (15)with ξ = L/2zR. The numerical optimization of theabove expression leads toξopt = 3.32 . (16)Our result means that with optimum, position dependentphase matching, the optimal waist size is approximatly7.5% smaller than suggested by the Boyd-Kleinman fac-tor, and according to this, the effective nonlinearity isfurther increased by 4.4%.We now analyze if the nonlinear interaction instanding-wave cavities can be similarily improved.Standing wave cavities, in particular in the form of asingly-resonant, single-ended cavity, i. e. with one mir-ror of almost perfect reflectivity, are frequently used inquantum and nonlinear optics [6, 10] . In such cavitieswaves that propagate in two different directions interferewith each other, and the differential phases introducedby the reflections at the cavity mirrors have to be con-sidered. Paschotta et al. have investigated the phasedifference ∆ϕ introduced from back reflection and havesuggested an appropriate design of the high reflectivitydielectric multi-layer coating to annihilate any additionalphase shift. It can be shown that the effective nonlinear-ity for a doublepass of plane waves through a crystaldepends on the ∆ϕ in the following wayκdp,pw =sin2(∆kL2)(∆kL2)2 · cos2(∆kL2+∆ϕ2). (17)For the calculation of the doublepass effective nonlinear-ity in the case of focussed Gaussian beams we model thesystem with a nonlinear medium of length 2L with a thinlense at position L that refocuses the beam. In this waywe obtain two waists at positions z0 and z′0 = 2L− z0 ofsize w0 indicating the way to the endmirror and the wayback, respectively. Now we integrate over 2L and findthe following expression for the effective nonlinearity fora double pass of the fundamental Hermite-Gauss modeκdp =1w20·∣∣∣∣∣∫ L0dz g(z, z0) +∫ 2LLdz g(z, z′0)ei∆ϕ∣∣∣∣∣2=1w20·∣∣∣∣∣∫ L0dz[g(z, z0) + g(z + L, z′0)ei∆ϕ]∣∣∣∣∣2=∣∣∣∣∣∫ L0dzei(∆kz+φG(z))w(z)×(1 +w(z)w′(z)ei(φ′G(z)−φG(z)+∆kL+∆ϕ))∣∣∣∣2,(18)where w′(z) and φ′G(z) belong to the focus at positionz′0. ∆ϕ is again the differential phase that may be in-troduced by the coating of the back reflecting mirror.We first consider the special case of weak focussing, i. e.|z − z0|/zR ≪ 1 ∀ z ∈ [0, L], and simplify the aboveexpression as followsκdp =sin2(∆k′L2)(∆k′L2)2 · cos2(∆k′L2+∆ϕ′2), (19)where ∆k′ := ∆k − 1/zR and∆ϕ′ := ∆ϕ+ 2(L− z0)/zR . (20)This expression has the same form as the one for planewaves as given in Eq. (17). However, an additional phaseshift appears. This phase shift is a result of spatial con-finement and the swapping in sign of the wave front’sradius of curvature during reflection, and corresponds tominus twice the Gouy phase in the limit considered here.From the expression of ∆ϕ′ in Eq. (20) it follows that thisadditional phase jump vanishes if the waist is located ex-actly at the back reflecting surface. In this case we haveplane wave fronts at the end mirror and therefore thesystem is similar to a single pass through a nonlinearmedium of length 2L.4FIG. 3: (Color online) Effective nonlinearity κdp (normal-ized) versus the differential phase introduced by the back re-flecting surface ∆ϕ. For Gaussian beams with waist positionon the reflecting surface, ∆ϕ = 0 generally provides the high-est κdp, (a) (here with strong focussing ξ = 2.84, z0 = L); thesame result is found for plane waves. Contrary, for ξ = 2.84and z0 = L/2 we find ∆ϕ = pi (c). The result for weakerfocussing is shown in (b) (ξ = 0.775, z0 = L/2).We now examine κdp for reflected Gaussian beams ofarbitrary focussing parameters ξ and the waist located inthe center of the medium. Eq. (18) yieldsκdp = 4 cos2(∆kL+∆ϕ2)∣∣∣∣∣∫ L0dzei(∆kz+φG(z))w(z)∣∣∣∣∣2. (21)The first term provides the optimal phase of the end mir-ror of ∆ϕ = −∆kL. In turn ∆k is again found by mini-mizing the variance of the term ∆kz+φG(z). We obtainthe following expression for the optimal differential phase∆ϕ of the end mirror versus focussing parameter∆ϕ = −3ξ2[(1 + ξ2) · arctan(ξ) − ξ]. (22)Fig. 3 shows the effective nonlinearity versus ∆ϕ for threedifferent standing wave cavity arrangements. In all casesthe second harmonic wave is not resonant but simplyback reflected. Curves (a) and (c) use the focussing pa-rameter ξ = 2.84. This value optimizes the effectivenonlinearity of the cavity if the refractive index of themedium does not depend on direction of propagation ofwaves. This is evident from Fig. 2 because the positiondependent refractive indices are not symmetric with re-spect to the focal position and the back reflected wavewould require different values. However, if one transfersthe results from a single pass through the medium andrealizes the required propagation direction dependent re-fractive index the optimum focussing parameter is againξ = 3.32. Curve (a) represents the case for focussing di-rectly onto the back reflecting surface. Curve (c) showsthe effective nonlinearity for a waist position at the crys-tal’s centre. In the latter case the best choice of the dif-ferential phase at the back reflecting surface is ∆ϕ ≈ pi.This is exactly the opposite of what one might expectfrom plane wave theory, where the optimum phase is∆ϕ = 0, similar to curve (a). Trace (b) shows the ef-fective nonlinearity for the focussing parameter that waschosen by Paschotta et. al. [11]. In that paper a fullquantitative comparison between experiment and theoryof the nonlinearity in standing wave cavities was con-ducted. In their experiment the back reflecting mirrorwas designed to prevent a differential phase shift betweenthe two interacting modes and a value of ∆ϕ = 0 waschosen. However, their experimental data revealed theeffective nonlinearity to be ≈ 10% smaller than expectedfrom their calculations. From our calculation it followsthat the optimum phase for their setup was ∆ϕ ≈ 1.55 piand that the chosen value of ∆ϕ = 0 decreased the effec-tive nonlinearity to about 90% of the maximum value.Our considerations are therefore in excellent agreementwith experimental results given in that paper and cansolve the observed discrepancy.In conclusion we have shown how for focussed wavesthe Gouy phase shift produces nonideal phase matchingin case of ∆k = 0. For a single pass through a nonlin-ear medium the optimum focussing parameter is foundto be ξ = 3.32. In this case a position dependent re-fractive index is required to further improve the effectivenonlinearity by 4.4%. For a double pass and for cav-ities an optimum focussing parameter above ξ = 2.84can only be achieved with a refractive index that also de-pends on propagation direction. We have also shown thatthe Gouy phase shift effects the optimum value for thephases introduced by cavity mirrors, with a significanteffect on the achievable effective nonlinearity. Our theo-retical analysis shows exact agreement with experimen-tal data published elsewhere, and may lead to improvedquantitative descriptions of nonlinear cavities.[1] P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinre-ich, Phys. Rev. Lett. 7, 118 (1961).[2] J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14,973 (1965).[3] L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev.Lett. 57, 2520 (1986).[4] R. Ghosh and L. Mandel, Phys. Rev. Lett. 59, 1903(1987).[5] C. M. Caves, Phys. Rev. D 23, 1693 (1981).[6] H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K.Danzmann, and R. Schnabel, Phys. Rev. Lett. 97, 011101(2006).[7] Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001(1986).[8] S. Feng and H. G. Winful, Opt. Lett. 26, 8 (2001)[9] G. D. Boyd and D. A. Kleinman, Journal of Applied5Physics 39, 8 (1968).[10] R. Paschotta, K. Fiedler, P. Ku¨rz, R. Henking, S. Schillerand J. Mlynek, Opt. Lett. 19, 17 (1994).[11] R. Paschotta, K. Fiedler, P. Ku¨rz and J. Mlynek, App.Phys. B 58, 117 (1994).

English

Nico Lastzka

Roman Schnabel

Crossref

The Gouy phase shift in nonlinear interactions of waves

Abstract

Similar works

Full text

Available Versions

MPG.PuRe

Max Planck Society eDoc Server

MPG.PuRe

Crossref