In any focussed nonlinear interaction the focus induced phase shift, known as the Gouy phase shift, provides an imperfection in phase matching for any linearly invariant material. However, using an appropriately designed quasi-phase matched structure it is theoretically possible to compensate for the deleterious effects of the Gouy phase shift, allowing a symmetric frequency response and tighter optimal focussing than in a uniform material

Belmonte

Boyd

Corin B.E. Gawith

Feng

Huw E. Major

Myers

Paschotta

Peter G.R. Smith

Ruffin

English

Major, Huw E.

Gawith, Corin B.E.

Smith, Peter G.R.

Southampton (e-Prints Soton)

Gouy phase compensation in quasi phase matchingHuw E. Major,∗ Corin B.E. Gawith, and Peter G.R. SmithOptoelectronics Research Centre, University of Southampton, UK(Dated: August 6, 2007)In any focussed nonlinear interaction the focus induced phase shift, known as the Gouy phaseshift, provides an imperfection in phase matching for any linearly invariant material. However, usingan appropriately designed quasi-phase matched structure it is theoretically possible to completelycompensate for the deleterious effects of the Gouy phase shift, allowing a symmetric frequencyresponse and tighter focussing for higher conversion efficiencies.Nonlinear optics provide an essential source of laserlight at wavelengths that are otherwise difficult to obtain.These wavelengths can be created with nonlinear effectssuch as second harmonic generation (SHG) [1, 2], sumand difference frequency generation and optical paramet-ric oscillation (OPO) [3]. The conversion efficiency ofsuch processes are highly dependent on the fundamentallaser power, with harmonic output generally increasingquadratically with the fundamental intensity. The largeelectric fields required for efficient operation normally re-quire focusing to a tight waist.The effects of focussing laser beams, in particularGaussian beams, on nonlinear parametric interactionshave been examined analytically by Boyd and Kleinman[4] (BK). By analysing the interactions of fundamentaland harmonic beams in nonlinear materials under theassumption of no source depletion, they were able to cal-culate optimal focussing conditions of the laser to givemaximal conversion efficiency. This can be interpreted asa balance between tight focusing to give a high intensitybeam waist and the need to utilise as much interactionlength as possible, with the optimal focussing conditiongiven asξ =Lb=L2zR= 2.84, (1)where L is the nonlinear material length, b is the confocalparameter and zR = πω2on/λ is the Rayleigh range.However, as we will show, this particular condition ap-plies only to a linearly invariant crystal where crystallength and Gouy phase shift are both compensated forby compromises in the beam focus.Here, the Gouy phase shift [5, 6] is an inevitable con-sequence of beam focusing and which limits nonlinearconversion efficiency by preventing perfect phase match-ing. Its influence can be seen experimentally as an alter-ation in phase matching conditions between the plane-wave and optimal focused case, and is most commonlyseen as a small temperature or angle tuning shift andincreasing asymmetry with tighter focus.In this paper we shall theoretically demonstrate thatthe domain reversal techniques of quasi-phase matching(QPM) can totally compensate for the phase error re-sulting from the Gouy phase shift. Our QPM gratingdesigns, which are no longer linearly invariant, allow forhigher efficiencies than can be obtained with standardQPM and furthermore require no shift in phase match-ing under focused conditions when compared to a planewave. By carrying out an analysis based on the BK ap-proach we are thus able to predict an optimal quasi-phasematching structure and calculate its performance.In our initial study we consider the Gouy phase shiftas it relates to Gaussian beams. Under focussing a singlemode Gaussian beam receives a π phase shift to its wavefronts as it travels from −∞ to ∞ through a focussedwaist. The full expression for the phase variation is givenasΦ (z) = (m+ n+ 1) arctan(zzR), (2)where m and n reference the Hermite-Gaussian modes(TEMmn).In the case of SHG both the fundamental and harmonicwaves must propagate in phase to obtain coherent energytransfer into the second harmonic. For the simple planewave case efficient phase matching can be achieved byeither birefringent phase matching in an appropriate ma-terial or by QPM in materials such as periodically poledlithium niobate (PPLN). However, in focussed systemsthe Gouy phase shift changes the linear phase mismatchrelationship of the plane wave to become more complex,containing both the linear (dispersive) part and the Gouyphase for which standard critical and quasi phase match-ing cannot compensate.Figure (1a) shows the relative Gouy phase throughouta device when focussed at the BK optimal value. Alsoshown is the linear phase difference between the coher-ence length of a device operating under plane wave andthe coherence length that provides maximum conversionefficiency at this focus. Device efficiency is maximised byminimising the overall phase error. At very loose focusthe Gouy phase variation is approximately linear over thedevice length and is almost completely compensated forby the linear phase of the device, resulting in minimalphase error as shown in Fig. (1b) curve (i).However, as the focus is tightened the Gouy phase vari-ation becomes more significant along the device length.By altering the temperature or angle of the nonlineardevice a linear phase variation in the coherence length20 L/2 L−100−50050100(a)Crystal LengthPhase, [Degrees]0 L/2 L−40−2002040Crystal Length(i)(ii)(iii)(b)FIG. 1: (a) Gouy phase and corresponding linear phase foroptimal operation at ξ = 2.84, and (b) the resulting phaseerror between Gouy and linear device phase for focussing at(i) ξ = 0.01, (ii) ξ = 2.84, (iii) ξ = 5.0.occurs (compared to the plane wave case), which par-tially compensates for the Gouy phase shift. However itis clear from Fig. (1b) in curves (ii) and (iii) that a signif-icant phase mismatch remains which results in a reduc-tion in maximum conversion efficiency. The focus valueof ξ = 2.84 highlighted by BK provides an operationalcompromise for phase matching of a focussed beam bybalancing the power density advantage against a linearfit to the Gouy phase shift.We shall now show, by extending the analysis of BK,how a modified QPM grating can be used to completelycompensate for the deleterious effects of the Gouy phaseshift, leading to higher conversion efficiencies. In thisanalysis we reproduce the steps of the BK derivation em-phasising the key role of the Gouy phase shift on fre-quency conversion.Starting with the Gaussian wave equation for a singlemode TEM00 beamE1 (r, z, t) =Eo2ωoω (z)e−r2ω2(z) ei(kz−ωt)eiΨ(z)eiΦ(z) + c.c.(3)where ω (z) is position dependent beam diameter givenbyω (z) = ωo[1 +(λzπω2on)2]1/2(4)Ψ is the radius of curvature of phase fronts given byΨ (z) =kr22z[1 +(λzpinω2o)]2 (5)and Φ is defined in Eq. (2) with m = n = 0. SubstitutingEq. (2,4,5) into Eq. (3) we haveE1 (r, z) =Eo21√1 + ζ2eik1ze−r2ω2o(1+ζ2) eiζr2ω2o(1+ζ2) eiΦ(z)=Eo211− iζeik1ze−r2ω2o(1+iζ) (6)where ζ = z/zR. From this we determine the inducedpolarisation of the nonlinear materialP (ω2) = 2ǫodeff |E1 (r, z)|2= 2ǫodeffE2o4(11− iζ)2ei2k1ze−2r2ω2o(1+iζ) (7)where deff is the effective nonlinearity of the material.From this an expression for the resulting harmonic canbe realised, in the formE2 (r, z) = A2 (r, z) eik2z. (8)By considering a slice of material of thickness dz the in-cremental harmonic is given asdA2 (r, z) =−ik22ǫon22eik2zP (ω2)=−ik22ǫon22eik2z2ǫodeffE2o4(11− iζ)×{(11− iζ)ei2k1ze−2r2ω2o(1+iζ)}. (9)We notice that the bracketed expression is simply theform of the Gaussian wave equation. Further, by natureof the second harmonic process the Rayleigh range of theSHG Gaussian is equal to that of the fundamental field,resulting in a matching Gouy phase shift.Following BK, integrating over all sources within thecrystal gives the total harmonic field contribution at apoint (r′, z′) outside the crystal. We further make thesubstitution of ∆k = 2k1 − k2, which provides the phasematching condition for SHG, and define ζ = 2 (z − f) /bto allow for arbitrary focal position. Additionally, withthe change of variables ξ = L/b, µ = (L− 2f) /L andσ = ∆kb/2 we arrive at a solution for SHG conversionefficiency under arbitrary focussing conditions, still as-suming no pump depletion, to giveE2 (r′, z′) =−ik24n22deffE2o11− iζ′e−2r2ω2oζ′(1+iζ′)×b2ei∆kf∫ ξ(1+µ)−ξ(1−µ)eiζσ1− iζdζ. (10)Eq. (10) provides the key result of the BK analysis.The optimal focusing condition ξ = 2.84 is that whichmaximises the integral, firstly by placing the beam waistin the centre of the crystal, but also by selecting a valuefor ∆k that provides the best linear compensation in thephase term of the integral for the Gouy phase shift. Inthe remainder of this paper we will show that by design-ing a QPM structure with Gouy phase compensation wecan promote perfect phase matching for a given focussingcondition. We will do this by first introducing a spatiallyvarying nonlinearity to represent quasi-phase matchingand secondly introducing an additional phase term that3varies the period locally to compensate for the focus in-duced phase shift.If we now replace the constant deff of Eq. (10) with aspatially varying effective nonlinearity, κ (ζ) we can de-fine the equationH (σ, ξ, µ) =∫ ξ(1+µ)−ξ(1−µ)κ (ζ) eiζσ1− iζdζ (11)which is comparable to Eq. (2.16) in [4]. From Eq. (11)an expression for κ(ζ) can be obtained which maximisesH(σ, ξ, µ) and thus the SHG conversion efficiency. In thecase of a plane wave, as assumed for loose focussing, thisexpression simplifies to the integral of the sinusoidallyvarying second harmonic electric field along the crystallength. Using standard QPM this integral can be max-imised by inverting the material nonlinearity at a perioddefined by the material dispersion to provide continuousgrowth of the second harmonic. This can be expressedmathematically asκ(ζ) =cos (∆kz)|cos (∆kz)|(12)where κ(ζ) = ±1 to reflect the fact that the nonlinear-ity can take just two values (+d or −d) with periodicpoling. The structure defined by Eq. (12), which is aconventional first-order QPM grating, is routinely usedin periodically poled materials. Numerical analysis showsit has the same optimal focusing condition as found byBK.For the more complex case with focussing, Eq. (12)does not account for the Gouy phase slippage. We willnow show how a suitable modification to the simple lin-early periodic structure allows for compensation of thiseffect.By re-arranging the integrand of Eq. (11) and notingthat the denominator is in the form of the Gouy phaseshift, we obtainκ(ζ)eiζσ1− iζ= κ(ζ)eiζσ(1 + iζ1 + ζ2)=κ(ζ)eiζσ√1 + ζ2ei arctan(ζ). (13)We now wish to identify the optimal design of periodicstructure κ(ζ) to maximise the second harmonic effi-ciency. If we selectκ(ζ) =cos (ζσ)|cos (ζσ)|e−i arctan(ζ) (14)the unwanted Gouy phase term is thus negated promot-ing perfect phase matching. By considering the complexconjugate term throughout the analysis, as is standard innonlinear optical derivations, Eq. (14) can be simplifiedtoκ(ζ) =cos (ζσ + arctan [ζ])|cos (ζσ + arctan [ζ])|(15)0.01 0.1 1 10 100 100000.40.81.21.62.0Focus value, ξSHG Output Intensity, A.U.FIG. 2: A graph of second harmonic output intensity as afunction of the focus parameter ξ for a standard QPM device(dashed curve) and an equal length of a Gouy phase shiftcompensated QPM device (solid curve).In the focussed case it can be seen from Eq. (15) that anadditional spatially varying phase correction of arctan (ζ)can be introduced as a continuous change in the phasematching period. This addition to the linear case allowsour new grating to completely compensate for the effectsof the Gouy phase shift. Thus, for a QPM device withGouy phase compensation the placement of inverted do-mains is simply described by Eq. (15), with the value ofσ determined by the material dispersion.The most prominent result of removing the Gouy phaseshift in nonlinear devices is the increase in conversionefficiency that can be achieved. Under any focussingcondition, if the QPM grating is suitably compensatedefficiencies greater than that produced using standardphase matching techniques can always be obtained. Thisis especially true as the focus parameter is increased to-wards the value outlined by BK, where a clear increasein efficiency over QPM can be observed. This effect isillustrated in the graph of Fig. (2), where the curve hasbeen calculated by numerical integration. Furthermore,after this adjustment the peak conversion efficiency is nolonger found at the BK focussing value of ξ = 2.84 butat a more tightly focussed ξ = 3.317, resulting in a 3.5%increase in efficiency over the previous best case.Results of our simulation also show that for valueslarger than ξ = 2.84 in a linearly invariant crystal thereduction in efficiency due to the Gouy phase shift out-weighs the gain in average power density obtained bytighter focussing, resulting in maximum efficiency at anon-optimal power density. However, with a Gouy phasecompensated device, the dominant effect on efficiency isnow only the average power density, with the maximumaverage power obtained at the tighter focus of ξ = 3.317.As such, to obtain the maximum possible SHG conver-sion, QPM devices should be fabricated with a gratingpitch variation as described by Eq. (15). For the remain-der of this paper we shall refer to such an a device as49.585 9.595 9.605 9.61500.20.40.60.81.0∆k,  x10−5SHG Output, A.U.FIG. 3: Normalised ∆k tuning curves for a QPM device underplane wave (dotted curve), and focussing at ξ = 3.3 (dashedcurve), and a GQPM device with ξ = 3.3 focussing (solidcurve).Gouy compensated quasi-phase matching (GQPM).Theoretically, further advantages of GQPM device arethe symmetric ∆k tuning response and a peak ∆k iden-tical to a device operating under ideal plane wave con-ditions. This is in contrast to a standard QPM devicewhich, as shown in Fig. (3), has an asymmetric tuningcurve with a peak offset to the ideal plane wave phasematching value. This offset and the oscillations whichoccur at even tighter focusing are entirely due to the spa-tial dependence of the Gouy phase shift along the device.GQPM must be designed for a particular focussingcondition and is always most efficient with the beamwaist at the centre of the crystal. In the following weexamine the effect on operation for focussing at otherthan the design conditions or of offsetting the waist fromthe centre of the device. The first effect is a variationin the peak ∆k phase matching value. As can be seenin Fig.(4), GQPM has a complex relationship betweenfocussing and peak phase matching value, with the ∆kvalue initially moving away from that of the plane wavebefore returning to zero offset at optimal focus. In con-trast the peak ∆k value of a QPM device asymptoticallyapproaches a large offset as the focus is tightenedTranslation of the focussed beam along the propaga-tion axis from a central location results in an asymmetrybeing re-introduced into the ∆k tuning curve as the posi-tion dependent phase adjustment of the grating no longercoincides with the centre of the Gouy phase shift. How-ever, a similar effect is also seen in QPM when the focalposition is not central to the device.From Eq. (2) it is clear that the effects of the Gouyphase shift are further exaggerated for higher order spa-tial modes. The consequence of this is a rapidly evolv-ing phase error between the fundamental wave and thegrating structure, resulting in poor conversion efficiency.Thus, by specifically compensating for the high order0.01 0.1 1 10 100 1000−50050100150200250300Focus value, ξ∆k shift from plane waveFIG. 4: A plot of the shift in peak phase matching ∆k fromthe plane wave case for a range of focussing values in a stan-dard QPM structure (dashed curve) and a GQPM structure(solid).Gouy phase large gains in efficiency should be obtained,possibly opening up the field of harmonic generation ofhigher order spatial modes.In addition, the concept of compensating for a focus in-duced Gouy phase shift ought to be extendable to opticalparametric processes, such as OPOs where the greater de-gree of symmetry may allow for narrower linewidths. Theuse of Gouy phase shift compensation to remove phasemismatch along the crystal may also help prevent para-metric down conversion when attempting to achieve highconversion efficiencies in SHG.In conclusion, we have shown that for a focussed Gaus-sian beam the Gouy phase shift puts a limit on the ob-tainable SHG conversion efficiency by preventing perfectphase matching along the length of a spatially invariantnonlinear material. We have further shown that withthe use of suitable QPM devices (GQPM) the deleteri-ous effects of the Gouy shift can in theory be completelynegated, resulting in an improved efficiency of 3.5% overthat predicted by BK. This higher efficiency is achievedat a tighter focussing parameter of ξ = 3.317 than theξ = 2.84 value previously calculated by Boyd and Klein-man [4]. Theoretically, these devices have perfectly sym-metric tuning profiles and phase matching conditions areidentical to a plane wave. Finally, we have discussed theapplication of this result to higher order beams whichcould result in large efficiency gains above the 3.5% out-lined here.∗hem@orc.soton.ac.uk[1] R. Paschotta, P. Ku¨rz, R. Henking, S. Schiller, andJ. Mlynek, Opt. Lett. 19, 1325 (1994).[2] M. Belmonte, T. Skettrup, and C. Pedersen, J. Opt. A:Pure Appl. Opt. 1, 60 (1999).[3] L. Myers and W. Bosenberg, IEEE J. Quantum Electron.533, 1663 (1997).[4] G. Boyd and D. Kleinman, J. Appl. Phys. 39, 3597 (1968).[5] A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, andH. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).[6] S. Feng and H. G. Winful, Opt. Lett. 26, 485 (2001).

Gouy phase compensation in quasi-phase matching

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Gouy phase compensation in quasi-phase matching

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