numerical simulations of transfer of spatial beam

Research ArticleNumerical Simulations of Transfer of Spatial Beam Aberrationsin Optical Parametric Chirped-Pulse Amplification

Ying Chen 12 Yuan Zhou 1 Guobao Jiang 12 and LuluWang 1

1School of Electronic Information and Electrical Engineering Changsha University Changsha 410003 China2Key Laboratory for Micro-Nano-Optoelectronic Devices of Ministry of Education School of Physics and ElectronicsHunan University Changsha 410082 China

Correspondence should be addressed to Guobao Jiang andguobao163com and Lulu Wang llwccsueducn

Received 21 June 2018 Accepted 30 August 2018 Published 1 October 2018

Academic Editor Qinghua Guo

Copyright copy 2018 Ying Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper the spatial characteristics of the optical parametric chirped-pulse amplification (OPCPA) process were numericallystudied when initial pump beam was aberrated Numerical results showed that the spatial walk-off effect transferred phasemodulation partly to the signal beam as the pump phase wasmodulated Moreover themodulation amplitude became increasinglysevere as the nonlinear length extended In the absence of phase aberration in the initial input signal the induced phase aberrationin the output signal was assumed as the differential form of the pump beam phase As the pump beam intensity was modulatedthe spatial walk-off effect reduced the influence of pump beam noise on beam quality and the angular spectrum but reduced signalgain simultaneously thus it may do more harm than good in the OPCPA process In the case of a non-diffraction-limited pumpbeam the greater the beam quality factor1198722119901 the lower the conversion efficiency of the output signal in the OPCPA processTheseresults have important guiding significance for optimized design of an OPCPA system for high power laser

1 Introduction

High-energy ultrashort solid lasers have comprised a devel-opment trend in laser optics in recent decades In 1986Strikland et al suggested the idea of chirped-pulse ampli-fication (CPA) [1] which dramatically reduced the possi-bility of nonlinear damage to the laser medium and fullyleverages gain bandwidth during amplification [2 3] Opticalparametric amplification (OPA) is an efficient approach ofgenerating tunable pulses from a fixed laser source [4ndash6] including large energy fiber laser sources [7 8] Theintegration of OPA and CPA into optical parametric chirped-pulse amplification (OPCPA) has rapidly extended the peakpower of amplified ultra-short laser pulses to the petawattlevel due to unique features of efficient conversion high gainand broad bandwidth [9ndash13]

In strong-field physics applications laser beam qualityand pulse contrast are key issues in OPCPA laser systems[14] To date the pulse contrast of OPCPA systems hasbeen studied extensively [15 16] and various methods have

been proposed to increase pulse contrast on a picosecondtimescale [17ndash20] However less attention has been paid tobeam quality in the OPCPA process as the OPCPA techniquewas normally considered capable of ensuring the opticalquality of an amplified signal field [16 21] This might notbe a problem for most low-energy OPCPA systems wherethe pump laser sources are diffraction-limited or in a type 0(119890119901 997888rarr 119890119904 + 119890119894) quasi-phase-matching OPA process withoutspatial walk-off between interacting waves [22 23] Howevercircumstances are quite different for high-energy OPCPAsystems where high-energy pump lasers are typically farbelow the diffraction limit [24 25] Meanwhile the spatialwalk-off effect is always present due to the adoption ofbirefringent nonlinear crystals

Several articles have considered beam quality in OPCPAsystems [26 27] mainly by examining the influences ofthe pump beam-profile and dephasing on OPA gain andsignal beam quality A theoretical model was developedfor dephasing effects due to angular deviation from idealphase matching and the impact of the angular content of

HindawiAdvances in Condensed Matter PhysicsVolume 2018 Article ID 5731938 7 pageshttpsdoiorg10115520185731938

2 Advances in Condensed Matter Physics

the beam on small signal gain and conversion efficiencyin a strongly depleted regime was evaluated numerically[26] In this paper the spatial characteristics of the OPCPAprocess pumped by a spatially aberrated beam are studiedtheoretically and numerically Three typical types of spa-tially aberrated pump beams are discussed including thephase-modulated pump beam intensity-modulated pumpbeam and non-diffraction-limited pump beam In the spatialdomain the nonlinear process of OPCPA is identical to thatof OPA thus most parts of this paper do not differentiateexplicitly between OPA and OPCPA

2 Numerical Model

In this paper the type I phase-matching OPA process issimulated by nonlinear coupled-wave equations in the spatialdomain implying that all temporal effects in the time domainare ignored The equations governing the evolution of theenvelopes Ep Es and Ei of the pump signal and idler pulses[28 29] respectively are

120597119864119904 (119911 119909)

120597119911+119871119873119871119871 119904119901

120597119864119904 (119911 119909)

120597119909+ 1198941198711198731198711198712119904

1205972119864119904 (119911 119909)

1205972119909

= minus119894120582119901

120582119904119864119901 (119911 119909) 119864

lowast119894 (119911 119909) 119890

minus119894Δ119896119911

(1)

120597119864119894 (119911 119909)

120597119911+119871119873119871119871 119894119901

120597119864119894 (119911 119909)

120597119909+ 1198941198711198731198711198712119894

1205972119864119894 (119911 119909)

1205972119909

= minus119894120582119901

120582119894119864119901 (119911 119909) 119864

lowast119904 (119911 119909) 119890

minus119894Δ119896119911

(2)

120597119864119901 (119911 119909)

120597119911+ 1198941198711198731198711198712119901

1205972119864119901 (119911 119909)

1205972119909

= minus119894119864119904 (119911 119909) 119864119894 (119911 119909) 119890119894Δ119896119911

(3)

where Ej(z x) is normalized to the input pump field E0 Forthe sake of simplicity a one-dimensional transverse modelis used in simulations diffraction effects are ignored due tothe large beam aperture A Gaussian pump beam is assumedthroughout the paper although different beam shapes maybe involved The spatial variable x is normalized to theradius of the pump beam waist w Δk = ks + ki - kp is thewave-vector mismatch among the three waves where wavevector kj = n120596jc The nonlinear length is defined by 119871119873119871 =119899120582119901(120587120594

(2)1198640) as a measure of the pump intensity The pumpbeam is considered as the reference beam thus walk-offterms only appear in (1) and (2)The signal walk-off length Lspis defined as Lsp = w120588 where 120588 is the walk-off angle and Lip= Lsp in the type-I collinear configuration The ratio of Lsp tocrystal length L indicates the practical walk-off magnitude ofthe OPA process in the nonlinear crystal In the calculationsthe initial signal wave is assumed to be a Gaussian beamwith phase uniformity and Δk is set to zero because itprimarily affects signal gain and conversion efficiency whichhas been thoroughly discussed [22] Investigation of thewalk-off effect in isolation can explicitly clarify its impact on

the output signal beam The standard split-step method andRunge-Kutta algorithm were adopted to solve the nonlinearequations numerically [29ndash31]

3 Analyses and Discussions

31 Pump Beam with Phase Modulation First the OPCPAprocess pumped by a phase-modulated beam is investi-gated For simplicity the pump beam is assumed to includesinusoidal phase modulation as 119864(119909 0) = 1198640exp(minus119909

2 +119894 119886 sin2(119899120587119909)) where parameters a and n correspond to themodulation amplitude and spatial frequency respectivelyThe angular spectrum distributions of the signal beam andidler beam are displayed in Figure 1 under small-signalconditions The insets in the top right corner are the localenlarged figure As shown in Figure 1 the angular spectrumof the signal beam remain basically unchanged as the walk-off effect is not considered (the dotted line in Figure 1(a))The phase modulation of the pump beam leads to angularspectrum aberration of the idler beam (the dotted line inFigure 1(b)) However the angular spectrum of the signalbeam become aberrated as the spatial walk-off effect is takeninto account (the solid and dashed lines in Figure 1(a))Additionally the greater parameter a the worse the angularspectrum aberration in the output signal beam and theaberration of the idler beam is weakened accordingly Withregard to saturation amplification the spatial characteristicsof OPA are similar to those of the small signal

Figure 2 shows the transverse distribution of the signalphase and its evolution in the crystal As the pump phase ismodulated by a sinusoidal phase a sinusoidally modulatedphase is observed in the output signal The modulationamplitude increases in line with crystal length as explainedwhen analyzing the phase transfer mechanism of OPA Onaccount of spatial walk-off the pump beam and idler beamare staggered in space and their phases deviate Their phasedifferences therefore become disordered and the signal beamwill partially undertake the distortion originated from thepump beam Meanwhile the greater the crystal length (ornonlinear length) the clearer the walk-off of the pump beamand idler beam Moreover the modulated amplitude of thesignal phase became increasingly intense (see the phasecurve in the position of z = 5 mm) One consequence ofphasemodulation is angular spectrum aberration which alsodescribes the phenomenon in the insets of Figure 1

Figures 1 and 2 illustrate that the output signal inheritedthe aberration of the pump beam as its phase is modulatedin the presence of spatial walk-offTherefore high-frequencymodulation in the pump phase should be suppressed asmuchas possible to obtain a good-quality signal beam in the OPAprocess

Figure 3 depicts the phase of the output signal whenthe pump phase Φ(x) is equal to x2 and x3 respectivelyThe shape in Figure 3(a) is approximately linear whereasthat in Figure 3(b) is approximately parabolic Therefore theinduced phase of the signal is essentially the differential of thepump phase

Next the physical process of OPA is analyzed to explainthe above phenomena The phases of the signal (Φs) pump

Advances in Condensed Matter Physics 3

10

05

00

Inte

nsity

minus30 minus15 0 15 30

0015

0010

0005

0000

15 20 25 30 35

(061 w)

(a)

10

05

00

Inte

nsity

000

001

002

15 20 25 30 35


(061 w)

(b)

Figure 1 Angular spectrumof output signal beam (a) and idler beam (b) when pump phase is modulated Dotted line spatial walk-off effectnot considered a = 05 Dashed (solid) line spatial walk-off effect considered a = 02 (05) E0 = 10minus6 (small-signal condition) crystal lengthL is 10 mm LNL = 023L Lsp = 5L n = 4

04

02

00

minus02

minus041

2

3

4

5minus2 minus1 0 1 2

x (w)

z (mm)

Phas

e (au

)

Figure 2 Evolution of signal phase in the transverse direction ofnonlinear crystal under saturation amplification conditions (E0 =10minus2 LNL = 023L Lsp = 5L a = 05 n = 4 L = 5 mm)

(Φp) and idler (Φi) are known to be fulfilled by the followingrelationship without walk-off effect and when the phase-matching condition is satisfied

Φ119904 = Φ119904 (0) (4)

Φ119894 =120587

2+ Φ119901 minus Φ119904 (5)

Equations (4) and (5) show that the phase of the outputsignal is independent of the pumpphase whereas the phase ofthe idler beam is related to the difference between Φs andΦpas the walk-off effect is neglectedTherefore phase aberrationin the pump beam will be transferred to the idler beam asthe walk-off effect is ignored as demonstrated by the dottedline in Figure 1(b) Supposing the phase of the initial signalis uniform then the idler phase is consistent with that of the

pump beam denoted as Φp(x)+1205872 and Φp(x) respectivelyHowever the pump beam and idler beamwill no longer coin-cide in space if the walk-off effect occurs resulting in someoffsets Similarly if their phases are assumed to be Φp(x) and1205872+Φp(x+Δx) respectively then the phase expression of thegenerated signal is Φs=1205872+Φp(x)-(1205872+Φp(x+Δx))=Φp(x)-Φp(x+Δx) representing the differential form of the pumpphase and explaining the phenomena in Figure 3 The abovetheoretical results are helpful in precompensating or pre-filtering phase aberration during the pulse-shaping process

32 Pump Beam with Intensity Modulation The spatial char-acteristics of the output signal are then analyzed as the pumpamplitude is modulated Given the amplified spontaneousemission the pump beam usually generates noise Thereforeit is necessary to discuss the effect of pump noise on the beamquality of the output signal

The intensity modulation of the pump beam (namely thenoise expression) is assumed to be 01E0sin

2(4120587x) and thenear-field distribution and angular spectrum of the outputsignal is obtained by numerical calculation as shown inFigures 4(a) and 4(b) respectively Figure 4(a) indicates thatthe disturbance of the pump beam could be partially trans-ferred to the signal beam and the degree of noise transferis weakened when the spatial walk-off effect is considered(the solid line in Figure 4(a)) The OPA gain is known tobe related to pump intensity therefore the output signal ishighly sensitive to the disturbance of pump intensity Whenconsidering the walk-off effect the sidelobes of the pumpspectrum caused by its intensity modulation are not met thephase-matching condition and these spatial frequency com-ponents are accordingly suppressed during the OPA processTherefore the beam quality of the output signal is somewhatbetter than that without walk-off effect However the gainof the signal beam clearly declines as confirmed by thenumerical simulation The spatial walk-off effect is therefore


010

005

000

minus005

minus010

Phas

e (ra

d)


x (w)

(a)

010

005

000

Phas

e (ra

d)


x (w)

015

(b)

Figure 3 Phase curve of output signal when pump phaseΦ(x) equals x2 (a) and x3 (b) (Lsp = 5L LNL = 023L L = 10 mm E0 = 10minus6)

10

05

00

Inte

nsity


x (w)

(a)

10

05

00

Inte

nsity


(061 w)

0030

0015

0000

16 24 32

(b)

10

05

00

Inte

nsity


x (w)

(c)

10

05

00

Inte

nsity


(061 w)

0002

0001

0000

16 24 32

(d)

Figure 4 Near-field distribution (a c) and angular spectrum (b d) of the output signal when pump amplitude is modulated Solid linewalk-off effect considered Dashed line no walk-off effect (a) and (c) are stimulated under small-signal conditions (E0 = 10minus6 L = 10mm) (b)and (d) are calculated under saturation amplification conditions (E0 = 10minus2 L = 5 mm) Other parameters are the same as those in Figure 1

found to reduce the influence of pump noise on the near-field distribution and angular spectrum of the signal beamwhich appears to benefit the OPA process However it alsoreduces the gain of the signal beam which is not desirableTherefore the pump noise does more harm than good to the

OPA process In addition the degree of fluctuation in thesignal beam during saturation amplification (the dashed linein Figure 4(c)) is lower than the performance under small-signal conditions (the dashed line in Figure 4(a)) as the gainis more stable during the saturation amplification process


40

30

20

10

0

Effici

ency

()

00 05 10 15 20

z (cm)

－2０=1

－2０=5

－2０=20

(a)

10

05

00

Inte

nsity

minus20 0 20

(061 w)

(b)

Figure 5 (a) Relationship between the signal gain and crystal length under different1198722119901 factors (b) angular spectrum of pump beam (solidline) as the signal gain reaches the peak value the dashed line is the initial angular spectrum of input pump beam where1198722119901 = 5 LNL = 01L119871 sp = 5L L = 20 mm E0 = 10minus6

33 Pump Beam Is Non-Diffraction-Limited In this sectionthe influence of pump beam equality on signal gain isdiscussed as the pump beam is non-diffraction-limited Thepump beam is assumed to be a one-dimensional Gaussianbeamwith a slow varying phaseEp (0 x) =E0 exp[-x

2+iΦ(x)]where Φ(x) = 120572 exp[-(x25)2]+120573 exp[-(x25)4] and a andb are constants From the perspective of beam quality thisexpression describes the characteristics of typical high powerlasers after spatial filtering [32] Three pump beams areconsidered in this study namely 120572 = 0 120573 = 0 120572 = 940 120573 =947 and120572= 1770120573= 1747 with corresponding beamqualityfactors1198722119901 of 1 5 and 20

Figure 5(a) shows the relationship between signal gainand crystal length under different1198722119901 factorsThe entire pro-cess of small-signal amplification saturation amplificationand back conversion are numerically simulated The signalgain is nearly zero within a large length (zlt10 mm) at firstin the case of small-signal amplification it then rises rapidlyindicating saturation amplification and finally decreasesrapidly after reaching the peak denoting a backconversionprocess In addition the shapes of the gain curves are similarunder different1198722119901 factors and the crystal lengths at whichthe signal gain reaches the peak value are almost the sameThe only difference is that a larger1198722119901 factor leads to a lowersignal gain peak The formation of the three processes isrelated to energy consumption of the pump beam and thesignal gain originates from the pump energy Initially nopump energy loss occurs under small-signal amplificationThen in the saturation amplification process pump energy isquickly consumed As the signal gain peaks the pump energyis mostly consumed as shown by the solid line in Figure 5(b)Next some frequency components do not satisfy the phase-matching condition due to the spatial walk-off effect andbackconversion hence occurs with an increase in crystallength The greater 1198722119901 factor is the more the frequencycomponents fail not to satisfy the phase-matching condition

which is analogous to a reduction in pump intensity As signalgain is closely related to pump intensity the peak value ofsignal gain under the pump beam with a large 1198722119901 factoris small as shown by the dashed line in Figure 5(a) Inaddition the divergence of the pump beam is assumed tobe centrosymmetric so the gain curves are similar underdifferent 1198722119901 factors The poorer the pump beam quality isthe lower the signal gain is Therefore the quality of thepump beam should be optimized to minimize divergence orconvergence before initiating the OPA process

4 Conclusions

The spatial characteristics of the OPCPA process were ana-lyzed in detail In the numerical simulation some effects wereconsidered such as the walk-off effect phase modulationamplitude modulation of the pump beam and pump beamquality The walk-off effect was found to worsen signal beamquality in the OPCPA systemwhen the pump beam exhibitedphase modulation However as the pump intensity wasmodulated the walk-off effect filtered out some aberrationsthe signal beam quality improved but at the cost of reducedgain In addition the divergence or convergence of pumpbeam also compromised OPCPA gain Therefore it is highlyimportant to select appropriate system parameters whendesigning the OPCPA system such as a minimum walk-offangle suitable crystal length and good pump beam quality

Data Availability

Figure 1 is the angular spectrum of output signal beam andidler beam as the pump phase is modulated Figure 2 isthe evolution of signal phase in the transverse direction ofnonlinear crystal Figure 3 is the phase curve of output signalas pump phase Φ(x) equals x2 and x3 Figure 4 is the nearfield distribution and the angular spectrum of the outputsignal as the pump amplitude is modulated Figure 5(a) is the


relationship between the signal gain and crystal length underdifferent factors Figure 5(b) is the angular spectrum of pumpbeam The experimental data used to support the findings ofthis study are all included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was partially supported the Natural Science Foun-dation of Hunan Province China (Grant nos 2018JJ2455and 2018JJ3560) the Scientific Research Fund of HunanProvincial Education Department (Grant nos 16B026 and17A020) and theNatural Science Foundation of China (Grantno 61308005)The authors thank Yunfei Mo for enlighteningdiscussions

References

[1] D Strickland and G Mourou ldquoCompression of amplifiedchirped optical pulsesrdquo Optics Communications vol 55 no 6pp 447ndash449 1985

[2] P Maine D Strickland P Bado M Pessot and G MourouldquoGeneration of Ultrahigh Peak Power Pulses by Chirped PulseAmplificationrdquo IEEE Journal ofQuantumElectronics vol 24 no2 pp 398ndash403 1988

[3] M D Perry and G Mourou ldquoTerawatt to petawatt subpicosecond lasersrdquo Science vol 264 no 5161 pp 917ndash924 1994

[4] D K Negus M K Steinershepard M S Armas et al ldquoMicrojoule-energy ultrafast optical parametric amplifiersrdquo Journal ofthe Optical Society of America B vol 12 no 11 pp 2229ndash22361995

[5] G Cerullo and S De Silvestri ldquoUltrafast optical parametricamplifiersrdquo Review of Scientific Instruments vol 74 no 1 I pp1ndash18 2003

[6] G Andriukaitis T Balciunas S Alisauskas et al ldquo90 GWpeak power few-cycle mid-Infrared pulses from an opticalparametric amplifierrdquo Optics Expresss vol 36 no 15 pp 2755ndash2757 2011

[7] Y Chen C Zhao S Chen et al ldquoLarge Energy WavelengthWidely Tunable Topological Insulator Q-Switched Erbium-Doped Fiber Laserrdquo IEEE Journal of Selected Topics in QuantumElectronics vol 20 no 5 pp 315ndash322 2014

[8] S Lu C Zhao Y Zou et al ldquoThird order nonlinear opticalproperty of Bi2Se3rdquoOptics Express vol 21 no 2 pp 2072ndash20822013

[9] I N Ross P Matousek M Towrie et al ldquoThe prospects forultrashort durationand ultrahigh intensity using optical param-eteric chirped pulse amplificationsrdquo Optics Communicationsvol 144 no 1 pp 125ndash133 1997

[10] L Cai J H Wen D L Yu et al ldquoDesign of the coordinatetransformation function for cylindrical acoustic cliaks with aquatity of discrete layersrdquo Chinese Physics Letter vol 31 no 9Article ID 030201 2014

[11] L Yu X Liang L Xu et al ldquoOptimization for high-energy andhigh-efficiency broadband optical parametric chirped-pulseamplification in LBO near 800 nmrdquo Optics Letters vol 40 no14 pp 3412ndash3415 2015

[12] J H Sung H W Lee J Y Yoo et al ldquo42 PW 20 fs Tisapphirelaser at 01 HzrdquoOptics Letters vol 42 no 11 pp 2058ndash2061 2017

[13] X Zeng K Zhou Y Zuo et al ldquoMulti-petawatt laser facilityfully based on optical parametric chirped-pulse amplificationrdquoOptics Expresss vol 42 no 10 pp 2014ndash2017 2017

[14] R A Snavely M H Key S P Hatchett et al ldquoIntense high-energy proton beams from petawatt-laser irradiation of solidsrdquoPhysical Review Letters vol 85 no 14 pp 2945ndash2948 2000

[15] H Yoshida E Ishii R Kodama et al ldquoHigh-power and high-contrast optical parametric chirped pulse amplification in 120573-BaB2O4 crystalrdquo Optics Expresss vol 28 no 4 pp 257ndash2592003

[16] L Cai H Xiaoyun and W Xisen ldquoBand-structure resultsfor elastic waves interpreted with multiple-scattering theoryrdquoPhysical Review B vol 74 no 15 2006

[17] H Ren L Qian H Zhu D Fan and P Yuan ldquoPulse-contrastdegradation due to pump phasemodulation in optical paramet-ric chirped-pulse amplification systemrdquo Optics Express vol 18no 12 pp 12948ndash12959 2010

[18] J H Sung W Y Jin J Lee et al ldquoGeneration of high-contrast30 fs 15 PW laser pulses from chirped-pulse amplificationTisapphire laserrdquoOptics Express vol 20 no 10 pp 10807ndash108152012

[19] V Bagnoud and F Wagner ldquoUltrahigh temporal contrastperformance of the PHELIX petawatt facilityrdquoHigh Power LaserScience and Engineering vol 4 no 4 2016

[20] X L Wang X M Lu X Y Guo et al ldquoExperimentalinvestigation on pulse-contrast degradation caused by surfacereflection in optical parametric chirped-pulse amplificationrdquoChinese Optics Letters vol 16 no 5 2018

[21] I N Ross P Matousek G H C New and K Osvay ldquoAnalysisand optimization of optical parametric chirped pulse amplifica-tionrdquo Journal of the Optical Society of America B vol 19 no 12pp 2945ndash2956 2002

[22] I Jovanovic J R Schmidt and C A Ebbers ldquoOptical paramet-ric chirped-pulse amplification in periodically poled KTiOPO4at 1053 nmrdquo Applied Physics Letters vol 83 no 20 pp 4125ndash4127 2003

[23] B W Mayer C R Phillips L Gallmann and U Keller ldquoMid-infrared pulse generation via achromatic quasi-phase-matchedOPCPArdquo Optics Express vol 22 no 17 pp 20798ndash20808 2014

[24] T Tajima and G Mourou ldquoZettawatt-exawatt lasers and theirapplications in ultrastrong-field physicsrdquo Review of ModernPhysics vol 5 no 3 pp 419ndash426 2001

[25] O V Chekhlov J L Collier I N Ross et al ldquo35 J broadbandfemtosecond optical parametric chirpedpulse amplification sys-temrdquo Optics Expresss vol 31 no 24 pp 3665ndash3667 2006

[26] I Jovanovic B J Comaskey and D M Pennington ldquoAngulareffects and beam quality in optical parametric amplificationrdquoJournal of Applied Physics vol 90 no 9 pp 4328ndash4337 2001

[27] O Novak H TurcicovaM Divoky et al ldquoMismatch character-istics of optical parametric chirped pulse amplificationrdquo LaserPhysics Letters vol 11 no 2 2014

[28] XWei L Qian P Yuan et al ldquoOptical parametric amplificationpumped by a phase-aberrated beamrdquoOptics Express vol 16 no12 pp 8904ndash8915 2008

[29] L L Wang Y Chen and G C Liu ldquoIncreased temperatureacceptance bandwidth in frequency doubling process using twodifferent crystalsrdquo Chinese Optics Letters vol 12 no 11 2014

[30] S Chen LMiao X Chen et al ldquoFew-layer topological insulatorfor all-optical signal processing using the nonlinear kerr effectrdquoAdvanced Optical Materials vol 3 no 3 pp 1769ndash1778 2016


[31] S Chen Q Wang C Zhao Y Li H Zhang and S WenldquoStable single-longitudinal-mode fiber ring laser using topo-logical insulator-based saturable absorberrdquo Journal of LightwaveTechnology vol 32 no 22 pp 3836ndash3842 2014

[32] T Wang T Zhan H Zhu and L Qian ldquoAnalysis of beam-quality degradation in nonlinear frequency conversionrdquo Journalof the Optical Society of America B vol 19 no 5 pp 1101ndash11062002

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the beam on small signal gain and conversion efficiencyin a strongly depleted regime was evaluated numerically[26] In this paper the spatial characteristics of the OPCPAprocess pumped by a spatially aberrated beam are studiedtheoretically and numerically Three typical types of spa-tially aberrated pump beams are discussed including thephase-modulated pump beam intensity-modulated pumpbeam and non-diffraction-limited pump beam In the spatialdomain the nonlinear process of OPCPA is identical to thatof OPA thus most parts of this paper do not differentiateexplicitly between OPA and OPCPA

2 Numerical Model

In this paper the type I phase-matching OPA process issimulated by nonlinear coupled-wave equations in the spatialdomain implying that all temporal effects in the time domainare ignored The equations governing the evolution of theenvelopes Ep Es and Ei of the pump signal and idler pulses[28 29] respectively are

120597119864119904 (119911 119909)

120597119911+119871119873119871119871 119904119901

120597119864119904 (119911 119909)

120597119909+ 1198941198711198731198711198712119904

1205972119864119904 (119911 119909)

1205972119909

= minus119894120582119901

120582119904119864119901 (119911 119909) 119864

lowast119894 (119911 119909) 119890

minus119894Δ119896119911

(1)

120597119864119894 (119911 119909)

120597119911+119871119873119871119871 119894119901

120597119864119894 (119911 119909)

120597119909+ 1198941198711198731198711198712119894

1205972119864119894 (119911 119909)

1205972119909

= minus119894120582119901

120582119894119864119901 (119911 119909) 119864

lowast119904 (119911 119909) 119890

minus119894Δ119896119911

(2)

120597119864119901 (119911 119909)

120597119911+ 1198941198711198731198711198712119901

1205972119864119901 (119911 119909)

1205972119909

= minus119894119864119904 (119911 119909) 119864119894 (119911 119909) 119890119894Δ119896119911

(3)

where Ej(z x) is normalized to the input pump field E0 Forthe sake of simplicity a one-dimensional transverse modelis used in simulations diffraction effects are ignored due tothe large beam aperture A Gaussian pump beam is assumedthroughout the paper although different beam shapes maybe involved The spatial variable x is normalized to theradius of the pump beam waist w Δk = ks + ki - kp is thewave-vector mismatch among the three waves where wavevector kj = n120596jc The nonlinear length is defined by 119871119873119871 =119899120582119901(120587120594

(2)1198640) as a measure of the pump intensity The pumpbeam is considered as the reference beam thus walk-offterms only appear in (1) and (2)The signal walk-off length Lspis defined as Lsp = w120588 where 120588 is the walk-off angle and Lip= Lsp in the type-I collinear configuration The ratio of Lsp tocrystal length L indicates the practical walk-off magnitude ofthe OPA process in the nonlinear crystal In the calculationsthe initial signal wave is assumed to be a Gaussian beamwith phase uniformity and Δk is set to zero because itprimarily affects signal gain and conversion efficiency whichhas been thoroughly discussed [22] Investigation of thewalk-off effect in isolation can explicitly clarify its impact on

the output signal beam The standard split-step method andRunge-Kutta algorithm were adopted to solve the nonlinearequations numerically [29ndash31]

3 Analyses and Discussions

31 Pump Beam with Phase Modulation First the OPCPAprocess pumped by a phase-modulated beam is investi-gated For simplicity the pump beam is assumed to includesinusoidal phase modulation as 119864(119909 0) = 1198640exp(minus119909

2 +119894 119886 sin2(119899120587119909)) where parameters a and n correspond to themodulation amplitude and spatial frequency respectivelyThe angular spectrum distributions of the signal beam andidler beam are displayed in Figure 1 under small-signalconditions The insets in the top right corner are the localenlarged figure As shown in Figure 1 the angular spectrumof the signal beam remain basically unchanged as the walk-off effect is not considered (the dotted line in Figure 1(a))The phase modulation of the pump beam leads to angularspectrum aberration of the idler beam (the dotted line inFigure 1(b)) However the angular spectrum of the signalbeam become aberrated as the spatial walk-off effect is takeninto account (the solid and dashed lines in Figure 1(a))Additionally the greater parameter a the worse the angularspectrum aberration in the output signal beam and theaberration of the idler beam is weakened accordingly Withregard to saturation amplification the spatial characteristicsof OPA are similar to those of the small signal

Figure 2 shows the transverse distribution of the signalphase and its evolution in the crystal As the pump phase ismodulated by a sinusoidal phase a sinusoidally modulatedphase is observed in the output signal The modulationamplitude increases in line with crystal length as explainedwhen analyzing the phase transfer mechanism of OPA Onaccount of spatial walk-off the pump beam and idler beamare staggered in space and their phases deviate Their phasedifferences therefore become disordered and the signal beamwill partially undertake the distortion originated from thepump beam Meanwhile the greater the crystal length (ornonlinear length) the clearer the walk-off of the pump beamand idler beam Moreover the modulated amplitude of thesignal phase became increasingly intense (see the phasecurve in the position of z = 5 mm) One consequence ofphasemodulation is angular spectrum aberration which alsodescribes the phenomenon in the insets of Figure 1

Figures 1 and 2 illustrate that the output signal inheritedthe aberration of the pump beam as its phase is modulatedin the presence of spatial walk-offTherefore high-frequencymodulation in the pump phase should be suppressed asmuchas possible to obtain a good-quality signal beam in the OPAprocess

Figure 3 depicts the phase of the output signal whenthe pump phase Φ(x) is equal to x2 and x3 respectivelyThe shape in Figure 3(a) is approximately linear whereasthat in Figure 3(b) is approximately parabolic Therefore theinduced phase of the signal is essentially the differential of thepump phase

Next the physical process of OPA is analyzed to explainthe above phenomena The phases of the signal (Φs) pump


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15 20 25 30 35


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x (w)

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Φ119904 = Φ119904 (0) (4)

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2+ Φ119901 minus Φ119904 (5)







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40

30

20

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0

Effici

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00 05 10 15 20

z (cm)

－2０=1

－2０=5

－2０=20

(a)

10

05

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Inte

nsity

minus20 0 20

(061 w)

(b)






4 Conclusions


Data Availability






Acknowledgments


References





































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






10

05

00

Inte

nsity


0015

0010

0005

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15 20 25 30 35

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(a)

10

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3

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Phas

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Φ119904 = Φ119904 (0) (4)

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2+ Φ119901 minus Φ119904 (5)







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40

30

20

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Effici

ency

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00 05 10 15 20

z (cm)

－2０=1

－2０=5

－2０=20

(a)

10

05

00

Inte

nsity

minus20 0 20

(061 w)

(b)






4 Conclusions


Data Availability






Acknowledgments


References





































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






010

005

000

minus005

minus010

Phas

e (ra

d)


x (w)

(a)

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005

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40

30

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ency

()

00 05 10 15 20

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－2０=5

－2０=20

(a)

10

05

00

Inte

nsity

minus20 0 20

(061 w)

(b)






4 Conclusions


Data Availability






Acknowledgments


References





































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






40

30

20

10

0

Effici

ency

()

00 05 10 15 20

z (cm)

－2０=1

－2０=5

－2０=20

(a)

10

05

00

Inte

nsity

minus20 0 20

(061 w)

(b)






4 Conclusions


Data Availability






Acknowledgments


References





































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery









Acknowledgments


References





































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery











Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





numerical simulations of transfer of spatial beam

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