Phase-mismatched localized fields inA-PPLN waveguide devicesDEREK CHANG,1,2,* YU-WEI LIN,1 CARSTEN LANGROCK,1 C. R. PHILLIPS,1,3

C. V. BENNETT,2 AND M. M. FEJER1

1E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA2Lawrence Livermore National Laboratory, Livermore, California 94551, USA3Department of Physics, Institute of Quantum Electronics, ETH Zurich, Zurich 8093, Switzerland*Corresponding author: djychang@stanford.edu

Received 19 October 2015; revised 14 December 2015; accepted 16 December 2015; posted 16 December 2015 (Doc. ID 251392);published 13 January 2016

Highly phase-mismatched nonlinear interactions cangenerate spatially localized optical fields that can affect theperformance of nonlinear optical devices. We present atheoretical description of the generation of such spatiallylocalized optical fields by ultrafast pulses. The effects oftemporal walk-off and pump depletion are discussed, alongwith methods for suppression of the localized field whilemaintaining the performance of the nonlinear device. Themodel is validated by the measurement of the spatial profileof the localized field in a quasi-phase-matched (QPM) ape-riodically poled lithium niobate (A-PPLN) waveguide.Finally, we fabricate and characterize A-PPLN devices witha 33% duty cycle to reduce the locally generated field by90%. © 2016 Optical Society of America

OCIS codes: (190.4360) Nonlinear optics, devices; (320.7110)

Ultrafast nonlinear optics; (190.5330) Photorefractive optics;

(190.4975) Parametric processes.

http://dx.doi.org/10.1364/OL.41.000400

Lithium niobate (LiNbO3) is widely used for nonlinear opticalfrequency conversion [1], and the advent of quasi-phase match-ing [2,3] has led to a wide range of practical applications[4]. Unfortunately, congruent LiNbO3 (CLN) suffers fromoptically induced changes in the refractive index, which causesbeam distortion and alters phase matching of nonlinear opticalinteractions [5]. Photo-refractive damage (PRD) can be re-duced by using magnesium-doped LiNbO3 [6] or by modify-ing the QPM structure [7]. However, CLN is still advantageousin applications where PRD does not limit the performance [4]due to its more advanced waveguide technologies.

In many applications, both the spectral amplitude and phaseimposed by the mixing process, as quantified in the ratio ofoutput field to the appropriate product of input fields, arerelevant performance parameters. Distortions in this complextransfer function (CTF) [8] can be caused by PRD-inducedchanges in the local phase matching, hence compromisingdevice functionality. If the effects of PRD are small and uni-

form along the length of the device, then the phase-matchingerror is uniform, and the CTF simply shifts in spectrum, leav-ing the overall profile unchanged. However, if the effects ofPRD are not uniform, then phase-matching errors vary spatiallyand the CTF will exhibit distortions. Such spatially nonuni-form changes in phase matching can be generated by spatiallylocalized fields resulting from phase-mismatched interactions ina nonuniform QPM grating.

We begin with a discussion of the generation of localizedfields in pulsed interactions, including effects of temporalwalk-off and pump depletion. Next, we present measurementsof the spatial profile of the local field for various wavelengthsand powers in a typical linearly chirped PPLN second-harmonicgeneration (SHG) device pumped with femtosecond pulses.Finally, we demonstrate how this field can be suppressed viaQPM grating design. We focus on the generation and effectsof the local fields, rather than the complex topic of PRD [9,10].

The localized field is illustrated by analyzing SHG in alinearly chirped QPM A-PPLN waveguide device. The QPMgrating profile is chosen to phase match SHG of the first-harmonic pulse A1. However, with enough power, A1 and itssecond-harmonic, A2, can generate noticeable amounts ofthird-harmonic A3, despite this process being far from phasematched, i.e., Δk ≈ 103 cm−1; here, Δk � k3 − k2 − k1 is thewave vector mismatch. A3 is the localized field that can del-eteriously affect device performance. The propagation equationfor A3 is

∂∂z

A3 � −iκ3dA1A2 exp�iΔkz�; (1)

where d �z� is the z-dependence of the nonlinear coefficient. Theenvelope Aj is related to the electric field Ej by Ej�x; y; z� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2∕njε0cp

Aj�z�Ej�x; y� exp�−ikjz� for j ∈ f1; 2; 3g, where Ejis the normalized transverse field amplitude. These functionsare normalized such that

RR jEjj2dxdy � 1 and jAjj2 is equalto the optical power of field j. The coupling coefficient κj is de-fined as κj ≡ θ�8π2d 2

0∕n1n2n3cε0λ2j �12, where c is the speed of

light, λj is the center wavelength of field Aj, nj is the indexof refraction for the jth wave, d 0 � χ�2�∕2 is the amplitudeof the pertinent component of the nonlinear susceptibility

400 Vol. 41, No. 2 / January 15 2016 / Optics Letters Letter

0146-9592/16/020400-04$15/0$15.00 © 2016 Optical Society of America

tensor, and θ � RRdE1E2E3dxdy is the spatial overlap of the

interacting modes, where d �x; y� is the transverse spatial distri-bution of the nonlinear coefficient, but assumed constant in lateranalysis. We restrict our analysis to the fundamental spatialmodes in A1 and A2; our devices were designed to use input/output waveguide tapers to ensure single-mode operation. Fromsimulations of proton diffusion in lithium niobate [11], we com-puted spatial mode indices and concluded that higher-orderspatial modes in A3 are negligible as they contribute less than2% to the total local field.

We consider a nonuniform QPM grating where variationsin the local grating period and duty cycle are slow, comparedwith the grating period itself, such that d can be accuratelyrepresented by a sum of distinct Fourier components Gm, witha z-dependent k-vector. The general form of d �z� is

d�z� �Xm

Gm exp�iK mz � iΦm�z��; (2)

where Gm � 1∕�mπ� sin�mπD� is the amplitude of the mthFourier component of the grating, and D is the grating dutycycle. A linear component K m � 2πm∕Λ is explicitly separatedout of the total grating phase, where Λ is the period of thegrating, and Φm�z� represents the phase of the grating beyondthe K m component. The average coherence length for gener-atingA3 is π∕Δkm, where Δkm � k3 − k2 − k1 − K m. Since thegrating is designed to phase match SHG, the correspondingcoherence length for generating A3 is very short, comparedwith the length of the device and the distances over which A1

and A2 amplitudes change significantly, suggesting the use ofmultiple scales analysis [12] to understand the generation ofA3. This analysis leads to

A3 � κ3Xm

Gm

ΔkmA1A2 exp�iφm�; (3)

where φm � Δkmz �Φm is the total phase mismatch. Cross-terms from combinations of different grating Fourier componentsare of the orderO�1∕Δk2m� and are considered to be insignificant.As a result, A3 is generated from the sum of contributions fromdistinct Fourier components of the grating and is proportional tothe spatial and temporal overlap of A1 and A2.

The time-domain picture of generating A3 from SFG in alinearly chirped QPM grating is illustrated in Fig. 1. Pulse A1

with bandwidth Ω0 propagates through the grating withgroup velocity u1. Frequencies in A1 phase match to generatesecond-harmonic frequencies in A2 over a phase-match lengthLp � δνΩ0∕4Dg , where Dg is the chirp rate of the grating andδν � 1∕u2 − 1∕u1 is the group velocity mismatch parameter.A2 then travels freely with group velocity u2 ≠ u1. WhileA1 and A2 overlap in space and time, A3 is generated, butonce SHG for A1 is no longer phase matched, A2 is no longergenerated, and the overlap betweenA1 andA2 decreases as the

fields walk off each other over a length Lw � 1∕δνΩ0. A3 van-ishes once A1 and A2 no longer overlap and, as a result, it canonly exist locally in a finite spatial region of the grating, whichwe define as w.

If this local field A3 causes PRD, then its spatial extent de-termines the portion of the transfer function that is distorted,and its maximum pulse energy affects the magnitude of thedistortion. These factors depend on parameters such as pulsebandwidth Ω0, pulse chirp C , and grating chirp Dg .

To understand the relationship between these factors andparameters, we must evaluate Eq. (3). It will be convenientto work in the frequency domain so we define a frequencydomain envelope Aj�z;Ωj� � Aj�z;Ωj� expfi�k�ωj � Ωj�−k�ωj��zg, where Aj�z;Ωj� is the Fourier transform of Aj�z; t�and Ωj � ω − ωj is the angular frequency detuning from car-rier frequency ωj. For simplicity, we assume A1 is undepletedand that there is negligible GVD so that A1�z;Ω1� �A1�0;Ω1� and, thus, write A2 in terms of the fundamentalA1 via the transfer function D�z;Ω2� [13], according to

A2�z;Ω2� � D�z;Ω2�cA21�Ω2�, where cA2

1�Ω2� is the convolu-tion A1�Ω1� ⊗ A1�Ω1�. With these representations for thefields, Eq. (3) becomes

A�m�3 �z;Ω3� � κ3

Gm

ΔkmfD�z;Ω2��A1�Ω1� ⊗ A1�Ω1��g

⊗ A1�Ω1� exp�iφm�z��: (4)

To compute A3 and illustrate the basic properties of local fields,we consider a linearly chirped pulse A1 with chirp rate C �C1Ω2

0 and bandwidth Ω0 propagating through a linearlychirped grating with chirp rate Dg as defined in [13]. To sim-plify the analysis, we only consider the first Fourier component,m � 1, and assume an infinitely long grating in which d�x; y� isconstant. By computing A3, we can calculate the spatial extentw and maximum pulse energy U of the localized field.

In Fig. 2, we explore how the normalized length w �w∕L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDgL∕�1∕L�

pvaries with normalized bandwidth Ω0 �

δνΩ0∕ffiffiffiffiffiffiffiffi4Dg

p � ffiffiffiffiffiffiffiffiffiffiffiffiLp∕Lw

pfor different chirp C . The length w

represents the fraction of the grating, of length L, where A3

A1 A2 A3

Fig. 1. Time-domain illustration of local field A3 generation due toSFG in a chirped QPM grating.

0.2 0.5 1 2 5 10

1

2

5

10

¯

Fig. 2. Length w of locally generated field A3 varies with pulsebandwidth Ω0 for different chirp C , where Dg > 0 is assumed.

Letter Vol. 41, No. 2 / January 15 2016 / Optics Letters 401

exists, scaled by the square root of the ratio of the chirpedgrating bandwidth, DgL, to the uniform grating bandwidth1∕L. w grows with increased phase-match length Lp or walk-off length Lw. First, consider A1 to be a transform-limited pulse(C � 0). For small bandwidths (Lw ≫ Lp), A1 is broad in timewhich increases the walk-off length and results in increased w.For large bandwidths (Lp ≫ Lw), A1 phase matches over alonger length which also increases w. However, for bandwidthsnear unity (Lp ≈ Lw), neither effect dominates, and w is at itsminimum.

Now consider a chirped pulse (C ≠ 0). For larger chirp ratesjC j, A1 is broader in time which increases the walk-off lengthand results in increased w. The curves shift toward largerlengths, and the minimum shifts toward larger bandwidths.Pulses with chirp C , such that C∕Dg < 0, have a chirp thatpartially cancels out the grating chirp and temporally com-presses A2. This decreases the walk-off length Lw and, there-fore, w is smaller compared with those generated by pulseswith C∕Dg > 0.

In Fig. 3, we explore how the normalized pulse energy U �U∕�κ22κ23G2

mU 30∕�2DgΔk21�� of A3 depends on Ω0. The nor-

malization factor represents the energy in the third-harmonicgenerated by a CW first-harmonic with power equal to that ofa flat-top pulse of energy U 0 and duration 1∕Ω0, where U 0 isthe pulse energy of A1. For small bandwidths (Lw ≫ Lp), A1 isbroad in time and, therefore, the peak power is less, whichmeans A3 will have lower power U . For large bandwidths(Lp ≫ Lw), A1 is compressed, and its peak power increases, re-sulting in higher U . For larger chirp rates jC j, A1 is broader andU decreases. Pulses with chirp C , such that C∕Dg < 0, gen-erate a temporally compressed A2 and, therefore, increase U .

To the extent that a localized field is causing PRD, it is im-portant to choose parameters so that w >

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDgL∕�1∕L�

p; as a

result, w > L and, therefore, the CTF only shifts in spectrum.Additionally, it is advantageous to choose a smaller bandwidthΩ0 so that the pulse energy is minimized.

The position of the local field within the grating can changedue to changes in the center wavelength or depletion level ofA1. This effect is potentially important as it can alter the spatial

profile of the localized field. We explore these effects experi-mentally in an A-PPLN waveguide device by measuring thespatial profile of the locally generated field, A3, by collectinglight scattered out of the waveguide at the chip surface alongthe length of the waveguide. A 400 μm core fiber, oriented at45° with respect to surface normal, is scanned along the wave-guide, and the collected light is detected with a silicon ava-lanche photodiode; a bandpass filter ensures that only A3 isdetected. We assume the scattered light measured at any givenposition is proportional to jA3j2.

Our A-PPLN device has a QPM grating length of 43 mmand a bandwidth of 22 nm, centered at 1560 nm with a gratingchirp rate of Dg � 0.07 mm−2. The sign of the chirp is suchthat shorter wavelengths phase match later in the grating.Our transform-limited input A1 has a bandwidth of 16 nm,which is equivalent to Ω0 � 7.4; we therefore expect w �4.7, i.e., w � 18 mm based on Fig. 2. We launch 2.5 mWof A1 into the waveguide and measure the local field profilefor λ1 � 1550 nm, 1560 nm, and 1570 nm, shown as the bluelines in Figs. 4(a)–4(c). Theoretical plots, obtained via numeri-cal simulations of Eq. (1) and the corresponding equations forthe evolution of A1 and A2 to enable inclusion of pump-depletion effects, are shown as the red lines. The peak positionshifts toward the end of the grating as λ1 becomes shorter, andthe length matches the expected value of 20 mm, as shown inFig. 4(b). In Figs. 4(d)–4(f ), we fix the center wavelength at1550 nm and change the input power of A1 for each scan(2.5 mW, 10 mW, and 20 mW). As the power is increased,A1 depletes as a larger fraction is converted to A2. As a result,the product A1A2 decays along the waveguide, and the peakposition shifts closer to the beginning of the grating.

Distortions in the CTF due to local-field-induced PRD canbe minimized by reducing the local field intensity through low-ering the input power of A1 at the expense of reduced efficiencyof the main interaction. Another way is to increase the operat-ing temperature of the device such that the space charge fieldcan be screened more effectively by increased electron mobility[14]; however, excessive temperatures can change waveguide

0.2 0.5 1 2 5 10

.01

.1

1

10

Bandwidth |Ω0|

Pul

seE

nerg

y|U

|

C = 5

C = 2

C = 0

C = -2

C = -5

Fig. 3. Maximum pulse energy U of locally generated field A3 varieswith pulse bandwidth Ω0 for different chirp C , where Dg > 0 isassumed.

z position (mm)

0

0.1

0

0.2

0

0.4

0

0.2

0 20 40 0 20 400

10

0

4

Pow

er (

nW)

(a)

(b)

(c)

(d)

(e)

(f)

18 mm

Fig. 4. Measurement (blue) and simulation (red) of local fieldspatial profile using input power of 2.5 mW with center wavelengthsat (a) 1570 nm, (b) 1560 nm, and (c) 1550 nm, and using centerwavelength at 1550 nm with input powers of (d) 2.5 mW, (e) 10 mW,and (f ) 20 mW.

402 Vol. 41, No. 2 / January 15 2016 / Optics Letters Letter

properties. It is also possible to consult Fig. 2 and design thelength of the local field to span the entire length of the gratingsuch that transfer function distortion is uniform, and the over-all effect is a constant frequency shift, though these designs maynot be consistent with the requirements for a given application.Therefore, it is often advantageous to design the QPM gratingto minimize local field generation. This can be achieved byadjusting the grating duty cycle D. The locally generated field,A3, is composed of contributions from different orders of theQPM grating as seen in Eq. (3). Each contribution is weightedby the factor Gm∕Δkm. In Fig. 5(a), we illustrate the contribu-tion of each grating order to the total local field for a gratingwith 50% duty cycle. The third-order Fourier component G3

provides the largest contribution, but can be eliminated bysetting D � 33%. The ratio of jA3j2 for 33% to 50% dutycycles is about 13%; by setting D � 33%, jA3j2 can be re-duced by almost 90%. The trade-off is a 14% reduction inSHG efficiency due to the reduction of the first-order Fouriercomponent G1.

To demonstrate this effect, we test devices with 33% dutycycle QPM gratings. We measured the amount of A3 generatedby scanning a tunable, CW laser at 1550 nm across the band-width of our devices and measured jA3j2 with a lock-in detec-tion scheme. Figure 5(b) illustrates the amount of A3 generatedfor different gratings with duty cycles bracketed around 33% asa percentage of the amount of A3 generated for a 50% dutycycle grating. The values are averages over different gratings ofidentical design. The minimum amount of A3 generated is

about 8%, and the measured results follow the theoreticaltrend. The slight discrepancy at larger duty cycles can be attrib-uted to measurement inaccuracy as the amount of A3 is quitesmall. With this design, the local field power is suppressedby almost 10 dB, and one can expect much improved deviceperformance.

In summary, we have provided a model for understandinga general class of parasitic effects called localized fields inA-PPLN waveguide devices. This theory is supported by strongagreement between simulation and measurement results. Toimprove device performance, we have demonstrated that theaverage power of localized fields can be suppressed by 10 dBby using a 33% duty cycle grating. The analysis and designpresented offers guidance for designing optical frequency con-verters that avoid the effects of localized fields.

Funding. Air Force Office of Scientific Research (AFOSR)(FA9550-12-1-0110); U.S. Department of Energy (DOE)(DE-AC52- 07NA27344).

REFERENCES

1. R. S. Weis and T. K. Gaylord, Appl. Phys. A 37, 191 (1985).2. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J.

Quantum Electron. 28, 2631 (1992).3. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan,

Phys. Rev. 127, 1918 (1962).4. C. Langrock, S. Kumar, J. E. McGeehan, A. E. Willner, and M. M.

Fejer, J. Lightwave Technol. 24, 2579 (2006).5. A. M. Glass, Opt. Eng. 17, 175470 (1978).6. D. A. Bryan, R. Gerson, and H. E. Tomaschke, Appl. Phys. Lett. 44,

847 (1984).7. H. H. Lim, S. Kurimura, and N. E. Yu, Opt. Express 22, 5209 (2014).8. D. Chang, C. Langrock, Y.-W. Lin, C. R. Phillips, C. V. Bennett, and

M. M. Fejer, Opt. Lett. 39, 5106 (2014).9. M. Carrascosa, J. Villarroel, J. Carnicero, and J. M. Cabrera, Opt.

Express 16, 115 (2008).10. J. Villarroel, J. Carnicero, F. Luedtke, M. Carrascosa, A. García-

Cabañes, J. M. Cabrera, A. Alcazar, and B. Ramiro, Opt. Express18, 20852 (2010).

11. R. V. Roussev, C. Langrock, J. R. Kurz, and M. M. Fejer, Opt. Lett.29, 1518 (2004).

12. C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W.Chinaglia, and G. Valiulis, J. Opt. Soc. Am. B 19, 852 (2002).

13. G. Imeshev, M. A. Arbore, and M. M. Fejer, J. Opt. Soc. Am. B 17, 304(2000).

14. G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer,and R. L. Byer, Opt. Lett. 22, 1834 (1997).

Fig. 5. (a) QPM Fourier component contribution to local fieldpower for 50% and 33% duty cycles. (b) Fabricated 33% duty cyclegratings suppress the local field power by an average of 90%, comparedwith standard 50% duty cycle grating.

Letter Vol. 41, No. 2 / January 15 2016 / Optics Letters 403