nonlinear process in photonic crystals under the noncollinear interaction

Andreev et al. Vol. 19, No. 9 /September 2002 /J. Opt. Soc. Am. B 2083

Nonlinear process in photonic crystalsunder the noncollinear interaction

Anatoli V. Andreev, Alexei V. Balakin, Alexander B. Kozlov, Ilya A. Ozheredov, Ilya R. Prudnikov,and Alexander P. Shkurinov

Department of Physics and International Laser Center, M. V. Lomonosov Moscow State University,Moscow 119899, Russia

Pascal Masselin and Gael Mouret

Laboratoire de PhysicoChimie de l’Atmosphere, Centre Nationale de la Recherche Scientifique, Unite Mixte deRecherche, 8101, Universite du Littoral, 145 Avenue Maurice Schumann, Dunkerque 59140, France

Received November 30, 2001; revised manuscript received March 20, 2002; accepted March 22, 2002

We analyzed theoretically and studied experimentally the sum-frequency generation and four-wave-mixingprocesses under the noncollinear geometry of interaction in finite-length periodic photonic crystals. The effi-ciency of the surface and the bulk mechanisms of sum-frequency generation are compared. It is shown thatsurface and bulk mechanisms cannot be separated on the polarization of the sum-frequency signal only butthat the angular dependencies of the sum-frequency signal intensity enable us to separate them. The exci-tation of inhomogeneous waves at the four-wave-mixing frequency of v3 5 2v1 2 v2 is discussed and demon-strated. © 2002 Optical Society of America

OCIS codes: 050.0050, 190.2620, 190.4380, 130.2790, 190.0190, 260.0260.

1. INTRODUCTIONAlthough more than 40 years have passed since Frankenand Bloembergen with co-workers first demonstratednonlinear optical effects in the bulk1 and on theboundaries2 of nonlinear media, studies of second-ordernonlinear optical processes still attract much attention.Generally, for a material to have high-efficiency conver-sion, it must have a sufficient second-order nonlinear op-tical response,3 and the so-called phase-matching condi-tions should be fulfilled.4

In this paper, we present the results of theoretical andexperimental studies of sum-frequency generation (SFG)in one-dimensional photonic crystals5 (PCs) that are mul-tidimensional periodic dielectric structures characterizedby the presence of a frequency range [photonic bandgaps(PBGs)] for which the propagation of light inside the PC isforbidden.6 Here we consider a one-dimensional PC thatis a multilayer stack of two materials that have the peri-odic x (1) function with the period comparable with thewavelength of light.

Nonlinear optical phenomena in multilayer structureshave been studied since the earlier development of non-linear optics when Armstrong et al.7 and Franken andWard8 suggested a new mechanism of phase matchingthat takes into account the reciprocal lattice vector of pe-riodic media. This mechanism is in addition to the tra-ditional dispersive one that also has specific behavior inPCs in which, near the PBG, the essential changes (bothincrease and decrease) of the effective refraction index ofthe composite media6 take place.

Specific features of the nonlinear process in PCs aredue to the strong influence of dynamic diffraction effectson the field distribution inside the PC structure. At the

0740-3224/2002/092083-11$15.00 ©

edge of the PBG the intensity of the field oscillates withthe spatial period of the structure.9–11 As a result therelative role of spatially nonlocal interactions increasessignificantly.12 The importance of the interference fielddistribution inside the PC structure leads to the appear-ance of a non-phase-matching mechanism of enhance-ment of the nonlinear optical signal, which is concernedwith increasing the energy density of the field inside thestructure near the PBG’s edge. This mechanism was re-cently studied experimentally13 for multilayer structureswith deep modulation of the refractive index.

One of the intriguing tasks in studies of nonlinear op-tical phenomena in dielectric multilayer PC structures isthe source of nonlinear optical polarization in each dielec-tric layer. This source may result from several main rea-sons: First, the origin of the nonlinear polarizationmight come from the bulk of the layer’s material. Sec-ond, the nonlinear optical polarization might be causedmainly by the surface contribution of each layer of themultilayer structure. Because of the symmetry proper-ties of the second-order susceptibility tensor for materialspossessing inversion symmetry, the second-harmonic gen-eration (SHG) and the SFG in the bulk are forbidden inthe electric-dipole approximation.14 But these processesbecome permissable at the interface of the centrosymmet-ric media at which the symmetry is broken15 and in thebulk of the media as a result of a high-order multipolarityresponse (if quadrupole and magnetodipole nonlinear sus-ceptibilities are taken into account). To estimate thenonlinear optical response at the sum frequency (SF) inthe bulk of the isotropic medium, we use the generalizedmodel of a two-level atom, which was developedrecently.16 This model describes the dynamics of spa-

2002 Optical Society of America

2084 J. Opt. Soc. Am. B/Vol. 19, No. 9 /September 2002 Andreev et al.

tially nonlocal interactions (magnetic-dipole and quadru-pole interactions, as well as the interactions that are dueto the gradient of the pondermotive potential) of a laserfield with centrosymmetric atoms in the two-level ap-proximation. Here we calculate and compare the bulkand the surface contributions to the SF signal generatedin a multilayer structure.

2. ONE-DIMENSIONAL PHOTONIC-CRYSTAL SAMPLEIn our experiments, we used a one-dimensional PC struc-ture consisting of ZnS and SrF2 layers deposited on aquartz substrate with a 3-mm thickness by using stan-dard electron-beam-production technology. The PC iscomposed of 15 layers: eight alternating layers of high(ZnS, n1 5 2.29, Ref. 17) and seven of low (SrF2 , n25 1.46, Ref. 18) refractive indices. The layers have athickness of di 5 3l0/4ni (i 5 1, 2) for the referencewavelength of l0 5 800 nm to form the quarter-wavestack for l0 at normal incidence. The total thickness ofthe PC sample is roughly equal to L 5 4.8 mm for the ref-erence wavelength l0 . This PC structure strongly re-flects light at the normal incidence within the wavelengthrange of 755–840 nm. In the theoretical part of the pa-per all the simulations are made with the parameters ofthe structure described above taken into account.

The electronography image shown in Fig. 1 was madefrom the ZnS/SrF2 multilayer sample to determine thestructure. It was found that in the sample there is b-ZnS(cubic phase) in polycrystalline form and a weak dominat-ing orientation of its crystals parallel to the plane of thesubstrate (that is, [111]). The SrF2 layers are closer tothe amorphous state, and the layers do not exhibit anycrystalline or ordered symmetries. This characterizationwas also indirectly confirmed by studies of the PCsamples with the polarization coherent Raman technique,which allowed us to assume that, in our model, both ma-terials of the structure are distributed uniformly through-out the PC structure.

3. THEORYA. Wave EquationThe propagation of an s-polarized electromagnetic wavein a one-dimensional PC is determined by the equationfor the x component of the electric field as

Fig. 1. Electronogram of the ZnS/SrF2 multilayer structure.The acceleration voltage is 75 kV, and the angle of incidence ofthe beam on the sample is 0.5°.

DzEx 1 ~k2e 2 ky2!Ex 5 2iv

4p

c2 Jx , (1)

where e is the dielectric permittivity, ky and k are the tan-gential projection and the absolute value of the wave vec-tor in vacuum, respectively, J is the density of the atomiccurrent, and we assume that the z axis is normal to theinterfaces of the layers and that the electromagnetic wavepropagates in the y –z plane.

It is convenient to describe the propagation of thep-polarized electromagnetic wave in a multilayer struc-ture by the introduction of the variable

Gx 5 Hx /n, (2)

where Hx is the x component of the magnetic field andn 5 Ae is the refractive index. The equation for Gx is

DzGx 1 ~k2e 2 ky2!Gx 1 n¹zS ¹zn

n2 DGx

5 24p

cn F ikyJz 2 e¹zS Jy

eD G . (3)

In the approximation of the given incident wave the solu-tion of the inhomogeneous equations (1) and (3) can beperformed with the help of the Green’s functions. In thecase of an s-polarized electromagnetic wave the solutionis

Ex~z ! 5 2iv4p

c2

1

wsFEx

~1 !~z !E0

z

Ex~2 !~z8!Jx~z8!dz8

1 Ex~2 !~z !E

z

L

Ex~1 !~z8!Jx~z8!dz8G , (4)

where L is the thickness of the multilayer structure, Ex(1)

and Ex(2) are two linear-independent solutions of the ho-

mogeneous wave equation, and ws is a Wronskian. Notethat, as the linear-independent solutions, we can choosethe wave fields generated in the structure by the wavesthat are incident upon the structure from the top and thebottom layers, respectively. In the case of a p-polarizedelectromagnetic wave the solution is

Hx~z ! 5 24p

c

1

wpikyFHx

~1 !~z !E0

z Hx~2 !~z8!Jz~z8!

e~z8!dz8

1 Hx~2 !~z !E

z

L Hx~1 !~z8!Jz~z8!

e~z8!dz8G

14p

c

1

wp

iv

c FHx~1 !~z !E

0

z

Ey~2 !~z8!Jy~z8!dz8

1 Hx~2 !~z !E

z

L

Ey~1 !~z8!Jy~z8!dz8G , (5)

where wp is the Wronskian that is constructed from thelinear-independent solutions Gx

(1) and Gx(2) . Linear-

independent solutions of the homogeneous Maxwell equa-tion can be determined with the help of the recurrenceprocedure or the matrix-transfer method. So the nonlin-ear optical response of a multilayer structure can easily


be calculated if we know relations between the atomic-current density and the amplitude of the incident waves.

B. Sum-Frequency GenerationIn this subsection, we consider the process of SFG v35 v1 1 v2 in multilayer structures that are preparedfrom the layers of centrosymmetric materials. In suchstructures SFG is forbidden in the electric-dipole approxi-mation in accord with symmetry properties. However, itcan arise in multilayer structures either at the interfacesof layers in which the central symmetry is broken or inthe bulk as a result of spatially nonlocal interactions,which take into account distortion of atomic-wave func-tions by the external field with an amplitude that variesstrongly within the period of the PC. According to thediscussion in Subsection 3.A the density of the atomic cur-rent can be represented as a sum of two terms

Ja~v3! 5 Jas~v3! 1 Ja

b~v3! (6)

that are responsible for the surface and the bulk effects,respectively. It is well known14 that the surface contri-bution to the atomic current can be represented as

Jas~v3! 5 (

p~x1pnaE1bE2b 1 x2pE1anbE2b

1 x3pE2anbE1b!d ~z 2 zp!, (7)

where the sum over p is the sum over the interfaces of thelayers, x1p , x2p , and x3p are the effective nonlinear sus-ceptibilities, n is the normal to the interfaces of the lay-ers, and E1 and E2 are the electric fields of the pumpingwaves. The tensor structure of Eq. (7) is explained by thesymmetry properties of the interface between the two cen-trosymmetric materials, where only the normal is thepreferential direction. Equation (7) shows how the polar-ization of the SF wave depends on the polarizations of thepumping waves. The possible combinations are shown inTable 1. It can be seen that, if the pump waves have thesame linear polarization, the wave at the SF is p polar-ized. The pumping waves with orthogonal linear polar-izations generate the s-polarized SF wave.

In Ref. 16, we developed the generalized model of atwo-level atom. The model describes the dynamics ofspatially nonlocal interaction (magnetic-dipole and quad-rupole interactions, as well as the interactions that aredue to the gradient of the pondermotive potential) of a la-ser field with two-level centrosymmetric atoms. Withinthe framework of perturbation theory for this model theatomic current on the SF is

Table 1. Polarization of the Wave at the SF as aFunction of the Polarizations of the Pump Waves

Polarization of theFirst Pump Wave

Polarization of theSecond Pump Wave

Polarization of theWave at the SF

s s ps p sp s sp p p

Jab~v3!

5 xS v12

v02 2 v1

2 E1b¹aE2b 1v2

2

v02 2 v2

2 E2b¹aE1b

2v1v3

v02 2 v1

2 E1b¹bE2a 2v2v3

v02 2 v2

2 E2b¹bE1aD ,

(8)where

x 5 iN

V

2eudu2v0v3

m\v1v2~v02 2 v3

2!, (9)

N/V is the atomic density, and v0 and udu are the fre-quency and the absolute value of the dipole matrix ele-ment. To show how the polarization of the wave at theSF depends on the polarizations of the pump waves, it isuseful to rewrite Eq. (8) in a component-type form

Jxb~v3! 5 ixS v3

c D F v1v2

v02 2 v1

2 ~E1yH2z 2 E1zH2y!

1v1v2

v02 2 v2

2 ~E2yH1z 2 E2zH1y!G , (10a)

Jyb~v3! 5 ixFv3

c S v1v2

v02 2 v1

2 E1zH2x

1v1v2

v02 2 v2

2 E2zH1xD1 S k2yv1

2

v02 2 v1

2 1k1yv2

2

v02 2 v2

2D E1xE2x

2 S k2yv1v2

v02 2 v1

2 1k1yv1v2

v02 2 v2

2D ~E1yE2y

1 E1zE2z!G , (10b)

Jzb~v3! 5 ixFv2

c

v12

v02 2 v1

2 ~E1xH2y 2 E1yH2x!

1v1

c

v22

v02 2 v2

2 ~E2xH1y 2 E2yH1x!

2 S k2yv1v2

v02 2 v1

2 2k1yv1v2

v02 2 v2

2D ~E1yE2z

2 E1zE2y!G 1 xS v1v2

v02 2 v1

2

¹ze2

e2

1v1v2

v02 2 v2

2

¹ze1

e1D E1zE2z , (10c)

where k1y and k2y are the tangential projections of thewave vectors of the pump waves. Only the last term inEq. (10c) is associated only with the surface effects. Notethat it can be omitted here because in Eq. (7) for the sur-face atomic current at the SF the similar term is taken


into account. From Eqs. (10) it can be seen that the po-larization of the wave at the SF depends on the polariza-tions of the pump waves in the same way as in the case ofSFG at the interfaces of the layers (see Table 1). Sowithin the framework of the presented approach the bulkand the surface contributions to the process of SFG can-not be separated by polarization dependencies alone.The same result was previously shown in the case of sur-face SHG.19

C. Numerical ComputationsIn this subsection, we present the numerical results ofSFG in a one-dimensional PC for the case of noncollinearinteraction when the incidence angles of the pump wavesare different. The noncollinear geometry of interactionhas a significant advantage over the collinear one becausethe existence of a nonzero angle between wave vectors ofthe pump waves gives an additional degree of freedomthat can be used to achieve the optimal conditions for ef-fective generation of the SF.

In numerical simulations, we can study the interfaceand the bulk mechanisms of SFG separately. At first, we

Fig. 2. Transmission coefficients (a) T1 at l1 5 736 nm, (b) T2at l2 5 813 nm, (c) T3 of the wave at the SF, and (d) the inten-sity ISFG of the transmitted wave at the SF plotted as functions ofthe angle of incidence u1 of the first pump wave. The pumpwaves and the wave at the SF are s-, p-, and s-polarized, respec-tively. The angle between the wave vectors of the pump wavesis f 5 36.2°. The first-order transmission resonances coincidefor all three interacting waves.

consider the case when the SF is generated in the bulk asa result of light–atom spatially nonlocal interactions.We assume that the SF is generated in only the ZnS lay-ers (layers with the larger index of refraction) and thatthe wavelengths of two pump waves and the wave at theSF are l1 5 736 nm, l2 5 813 nm, and l3 5 386 nm, re-spectively. The analysis shows that the optimal condi-tion for the SFG is in the coincidence of transmissionresonances at the edges of PBGs for all three interactingwaves. This coincidence is due to the dispersive proper-ties of PCs, and triple coincidence can be accomplished asfollows: By varying the angle f 5 u2 2 u1 between thewave vectors of the pump waves, one can easily achievecoincidence of the transmission resonances at the edges ofPBGs for pump waves. The triple coincidence can thenbe accomplished by variation of the index of refraction ofthe wave at the SF. Note that in the experiments triplecoincidence can be accomplished by simultaneous varia-tion of the wavelengths and the incidence angles of thepump waves.

In Fig. 2, we plot the transmission coefficients of twopump waves, the wave at the SF, and the intensity of thetransmitted wave at the SF as functions of the angle ofincidence of the first pump wave. Note that the outletangle of the wave at the SF can easily be calculated fromthe conservation law of tangential projection of the wavevector: v3 sin u3 5 v1 sin u1 1 v2 sin u2 . The pumpwaves are s- and p-polarized, and the wave at the SF iss-polarized. The first-order transmission resonancesof the pump waves coincide at the angles f 5 36.2° andu1 5 256.5°. The triple coincidence of first-order trans-mission resonances at the edges of the PBGs is accom-plished by slight variation in refractive index for the waveat the SF (approximately 5% relative to an approximateformula for the index of refraction of ZnS). It can be seenthat there is significant enhancement of the intensity ofthe SF wave when all three first-order transmission reso-nances almost coincide. Note that, really, the maximumenhancement occurs at conditions under which there is asmall but distinctive difference in the positions of thetransmission resonances. The intensity of the wave atthe SF at u1 5 256.5° is 3 orders of magnitude largerthan for the cases when the edges of the bandgaps do notcoincide.

In Fig. 3, we plot the transmission coefficients of inter-acting waves and the intensity of the transmitted wave atthe SF as functions of the angle of incidence of the firstpump wave. The polarizations of the waves are the sameas in the previous case, but the angle between wave vec-tors of the pump waves is f 5 45°. In this case thesecond- and the first-order transmission resonances atthe edges of the bandgaps of the first and the secondpump waves, respectively, coincide at the angle of inci-dence u1 5 265.2°. The second-order transmission reso-nance of the wave at the SF is then tuned to the pumpwaves in the above-mentioned manner. The intensity ofthe wave at the SF at u1 5 265.2° is 2 orders of magni-tude higher than for the cases when the edges of thebandgaps do not coincide.

The following results of computer simulations can bementioned without illustration. The enhancement of theintensity of the transmitted SF wave is significantly


smaller when there is triple coincidence of the second-,the first-, and the first-order transmission resonances atthe edges of the bandgaps of the first and the secondpump waves and the SF wave, respectively. This resultdoes not depend on the polarization of the interactingwaves. So, for example, there is a significant increase inthe intensity of the p-polarized SF wave when it is gener-ated by the two p-polarized pump waves under conditionsof coincidence of the first-order transmission resonancesat the edges of PBGs for all three interacting waves.

As we mentioned in Section 1, the SF can be generatedat the interfaces of layers because the central symmetryis broken here. This contribution to SFG in multilayerstructures has also been investigated. For numericalcomputations, we assumed that the nonlinear suscepti-bilities x1p , x2p , and x3p are equal in value but have op-posite signs for the two interfaces of the same layer. Theanalysis shows that both mechanisms of SFG have thesame maximum enhancement condition, namely, thetriple coincidence of transmission resonances at the edgesof the PBGs of the interacting waves.

Fig. 3. Transmission coefficients (a) T1 at l1 5 736 nm, (b) T2at l2 5 813 nm, (c) T3 of the wave at the SF, and (d) the inten-sity ISFG of the transmitted wave at the SF plotted as functions ofthe angle of incidence u1 of the first pump wave. The pumpwaves and the wave at the SF are s-, p-, and s-polarized, respec-tively. The angle between the wave vectors of the pump wavesis f 5 45°, and there is coincidence of the second-, the first-, andthe second-order transmission resonances of the first and the sec-ond pump waves and the wave at the SF, respectively.

We mentioned in Subsection 3.B that the dependency ofthe SF signal polarization on the polarization of the pumpwaves is the same for the interface and the bulk mecha-nisms of SFG. Therefore we could not recognize thesetwo mechanisms with the help of the polarization of theSF signal. However, the angular dependency of the SFsignal intensity could be significantly different in the casein which we account for the interface or the bulk mecha-nisms separately. In Fig. 4 the transmission coefficientsfor all three waves and the intensity of the SF signal gen-erated in the multilayer structure are shown as functionsof the first pump wave’s incidence angle. It can be seenthat there is a drastic difference in the intensities of theSF signals generated at the interfaces or in the bulk ofthe layers. Thus the significant advantage of the noncol-linear scheme of interaction lies in the fact that it enablesus to determine the mechanism of nonlinear response ofthe multilayer structure.

4. EXPERIMENTAL TECHNIQUEThe setup for nonlinear optical experiments is depicted inFig. 5. We used a laser system consisting of a femtosec-

Fig. 4. Transmission coefficients (a) T1 at l1 5 736 nm, (b) T2at l2 5 813 nm, (c) T3 of the wave at the SF, and (d) the inten-sity ISFG of the transmitted wave at the SF, which is generated atthe interfaces (solid curve) and in the bulk (dotted curve) of thelayers, plotted as functions of the angle of incidence u1 of the firstpump wave. The pump waves and the wave at the SF are s-, p-,and s-polarized, respectively. The angle between the wave vec-tors of the pump waves is f 5 28°.


ond mode-locked Ti:sapphire laser (Mira 900) to seed a re-generative amplifier (RegA 9000) [both pumped by an ar-gon laser (Innova 420)] and an optical parametricamplifier (OPA) Model 400. All laser equipment was pro-duced by Coherent, Inc. The output pulses of the Mira900 are regeneratively amplified to as high as 4 mJ/pulseat a 200-kHz repetition rate with the RegA 9000. A partof the RegA 9000 laser radiation is seeded into the OPA toproduce tunable light, which is used as the v1 beam inthe three-wave-mixing (v3 5 v1 1 v1) and the four-wave-mixing (FWM) (v3 5 2v1 2 v1) processes. OurOPA produces tunable light in the range of 480–740 nm.For the rejection in the v1 beam from the nonamplifiedpart of the white-light continuum that is generated insidethe OPA, we used a double-pass heavy-flint dispersingitem denoted as the spatial filter in Fig. 5 and describedin detail previously.20 This item was also used for group-velocity dispersion minimization. After the spatial filterthe typical average power of the v1 beam was as high as10 mW. Another part of the RegA 9000 output forms thesecond beam v2 that is necessary for the FWM signal gen-eration. The central wavelength of the spectrum of thispulse is tunable in the range of 810–825 nm with 10-mWaverage power and a spectral width of approximately 10nm. The temporal overlapping of v1 and v2 pulses isachieved by use of a delay line.

The temporal-pulse profile is determined by thebackground-free autocorrelation of a SHG (this equip-ment is not depicted in the figure) and was well fitted by aGaussian line-shape function. We obtain v1 and v2cross-pulse durations of t3 5 200 6 10 fs, which do notdepend significantly on the wavelength of the v1 pulse.There are two pairs of Glan–Taylor polarizers in both thev1 and the v2 beams of the fundamental frequency (FF)to manage the average power of radiation coming to thePC sample. The average power in front of the PC samplewas measured with a photoelectric detector and in eachbeam was adjusted to as high as 3 mW. A double Fresnelrhombus provides the possibility of rotating the angle ofthe polarization plane of the v2 beam. In the experi-

ments, we used p- and s-polarized v2 beams, whereas thepolarization of the v1 beam is fixed as p polarization.Both beams are focused on the sample with a 76-mm lens.The focused-beam diameters at the sample are not morethan 100 mm. For small angles f (6.5°) the beams are fo-cused by use of lens L1 so that both beams are incidentsymmetrically upon the lens’s optical axis. The sample ismounted on the rotary part of the goniometer to vary andalign the incident angles. The incident angle u1 iscounted from the position of normal incidence of the v1beam on the sample. Positive and negative directions ofthe angle u1 of rotation of the PC sample are shown inFig. 5. In addition to a collimating lens the receivingpart of the experimental setup consists of a diaphragm, aGlan–Taylor polarizer to select the p- or the s-polarizationcomponent of the FWM signal, and a set of glass filters toreject scattered light coming from the v1 and the v2beams. The signal registration is realized with aHamamatsu Model R 4220P photomultiplier tube that isconnected to a lock-in amplifier (EG&G, Model EEG-5110)for synchronous selective detection. There is a chopperon both beams’ paths. It chops the v1 and the v2 beamswith frequencies of f1 ' 628 Hz and f2 ' 383 Hz, respec-tively. The frequency of detection is chosen as f1 , f2 , f11 f2 5 1011 Hz.

The receiving part is mounted on another rotary part ofthe goniometer. In our experiments, we used twoschemes for the signal registration: in-transmission ge-ometry and in-reflection geometry. In the first case thereceiving setup was rotated to follow the transmitted sig-nal constantly. Because of the finite thickness of thesample substrate ('3 mm), there is continuous change inthe FWM beam path during sample rotation. In the sec-ond case the receiving setup was rotated to an angle of 2Vwhen the incident angle u was changed to u 1 V to followthe reflected signal.

To control the wavelengths of the v1 and the v2 beamsand the v3 signal, we used a spectrograph (Chromex,Model 500IS) with a liquid-nitrogen-cooled back-illuminated CCD detector (Princeton Instruments, Inc.).

Fig. 5. Experimental setup: C, a mechanical chopper; M, dichroic mirrors; L, lenses; S, the PC sample; GSF, set of glass filters; GP, theGlan–Taylor polarizer; DFR, the double Fresnel rhombus; D, the diaphragm; and PMT, the photomultiplier tube.


Fig. 6. Transmission coefficients (a) T1 at l1 5 736 nm of the pump wave, (b) T2 at l2 5 813 nm of the pump wave, (c) T3 and thereflection coefficient R3 of the wave at the SF, and (d) the intensity ISFG of the transmitted wave at the SF plotted as functions of theangle of incidence u1 of the first pump wave. The graphs on the left-hand side represent the experimental results, and those on theright-hand side, the theoretical calculations. The signal at the SF is generated at the interfaces (solid curve) and in the bulk (dottedcurve) of the layers. All interacting waves are p polarized. The angle between the wave vectors of the pump waves is f 5 29°.

In the figures that show the experimental data, we donot indicate error bars because they were not, in fact, suf-ficient. Moreover, the angular scans are fully reversibleand reproducible, especially in the case of detection of thehighest intensities of the FWM process in the PC struc-tures. No irreversible changes in the PC sample were ob-served in the PC structure even with intensities of eachbeam to as high as 20 MW/cm2, which is much higherthan that which we used in the experiments.

5. EXPERIMENTAL RESULTS ANDDISCUSSIONA. Sum-Frequency GenerationThe collinear scheme of the SFG process in PCs was real-ized in the study reported in Ref. 21 in which it wasshown that the efficiency of nonlinear processes like SFGand SHG in multilayered structures is largely enhancedwhen the conditions of field localization and quasi-phasematching are satisfied simultaneously. In this paper, weconcentrate on studies of the roles of the surface and thebulk contributions to the second-order nonlinear opticalsignal. Here we show the advantages of the noncollinearscheme for the SFG process.

In this subsection, we discuss and analyze the experi-mental results. We studied the properties of transmittedsignals on SFG versus the angle of incidence u on the PBGstructure. These measurements were performed for thesame set of wavelengths of the v1 beam, l1 5 736 nm,and the wavelength of the v2 beam was fixed as l25 813 nm. The polarization directions of both FF beamswere set to be p. As shown in Table 1 the output polar-ization of the SF signal in this case is also p. We carriedout three sets of experiments with different angles be-tween the v1 and the v2 beams of f 5 29°, 38°, 45°.

Each figure shown in Section 5 (Figs. 6–8) consists of twocolumns. The left column shows the experimental data,and the right one shows the corresponding theoreticalsimulation. To show the real conversion efficiencies forboth SFG and FWM, we indicate all the experimentaldata in the same measurement scale. Figures 6(a), 7(a),and 8(a) show the angular behavior of the FF-pulse tran-sitions in the linear regime wavelengths of l2 5 736, 813nm. In Figs. 6(c), 7(c), and 8(c), we show the linearreflection for the beam at the SF signal wavelengthl2 5 386 nm. The nonlinear signal of the SF is gener-ated in transmission. In Figs. 6(d), 7(d), and 8(d) of thetheoretical column of the figures the solid curves repre-sent the signal that is caused by the surface nonlinearsusceptibility and the dashed curves those by the bulkone. All linear reflection curves are normalized at theirmaximum values, and the intensity of the SF signal isgiven in arbitrary units. The units in Figs. 6–8 are thesame. Thus we can compare the efficiency of the SF sig-nal for three sets of angles between the FF beams. Incomparison with the theoretical curves, the experimentalones look smoother, and not all the typical line featuresthat are caused by the wide spectra of those used in theexperimental femtosecond laser pulses are visible.

We recognized that the resulting relative shift of angu-lar positions of the band edges for all three differentwavelengths of l1 5 736, 813 nm and l3 5 386 nm leadsto the dramatic changes in the properties of the SF signal.The efficiency of SFG depends strictly on the relative po-sitions of the edges of the PBGs for all three interactingwaves. For the femtosecond laser pulses used it couldnot be an exact coincidence. For f 5 29° the fallingedges of the PBG for all three interacting laser beams arelocated near 50°, and on average the intensity of the SF is1 order of magnitude higher than for f 5 38°, 45°, where


Fig. 7. Transmission coefficients (a) T1 at l1 5 736 nm of the pump wave, (b) T2 at l2 5 813 nm of the pump wave, (c) T3 and thereflection coefficient R3 of the wave at the SF, and (d) the intensity ISFG of the transmitted wave at the SF plotted as functions of theangle of incidence u1 of the first pump wave. The graphs on the left-hand side represent the experimental results, and those on theright-hand side, the theoretical calculations. The signal at the SF is generated at the interfaces (solid curve) and in the bulk (dottedcurve) of the layers. The angle between the wave vectors of the pump waves is f 5 38°.

Fig. 8. Transmission coefficients (a) T1 at l1 5 736 nm of the pump wave, (b) T2 at l2 5 813 nm of the pump wave, (c) T3 and thereflection coefficient R3 of the wave at the SF, and (d) the intensity ISFG of the transmitted wave at the SF plotted as functions of theangle of incidence u1 of the first pump wave. The graphs on the left-hand side represent the experimental results, and those on theright-hand side, the theoretical calculations. The signal at the SF is generated at the interfaces (solid curve) and in the bulk (dottedcurve) of the layers. The angle between the wave vectors of the pump waves is f 5 45°.

the overlapping of the linear transition curves is less ob-vious. This result is in agreement with the results of Ref.21 in which we reported the properties of the collinearSFG and SHG in the PBG and with the theoretical analy-sis given in Subsection 3.C and demonstrated in Figs. 3and 4 of this paper.

Similar to the changes in the intensities of the SF sig-nal, with the variation of the angle of incidence the pro-files of the signal change synchronously. This situationprovides the possibility of analyzing the surface and thebulk contributions to the total nonlinear optical signal.

It is obvious that in the experiments we observe the con-structive interference of the surface and the bulk nonlin-earities. We can assume that the maxima on theu 5 250°, 115° under f 5 29° is caused mainly by thesurface contribution, whereas the maxima on u 5 110°and the shoulder on u 5 260° have a primarily bulk ori-gin. The profile of the SFG signal under f 5 45° showsthe bulk and the surface SFG features, specific for both.The isolated maxima near the u 5 220° and the lower sig-nal level on the angles u 5 230° up to u 5 260° can beassigned to the dominating surface nonlinear signal.


B. Four-Wave Mixing

1. Intensity DependenciesIn the first experimental set, we measured the intensityof the coherent signal at the v3 5 2v1 2 v2 frequency asa function of u for a small angle f between the v1 and thev2 pump beams. The angle f was chosen to be '6.5° tofulfill the phase-matching conditions, according to the ef-ficient (common for the whole structure) linear refractionindex neff(u). This efficient refractive index is calculatedfrom the dispersion equation, modified for the arbitraryangle of incidence u for laser beams with certain polariza-tion sets and wavelengths of the FWM process ofl1 5 690 nm, l2 5 817 nm, and l3 5 597 nm.

Figure 9 demonstrates the experimental data of thelinear transmission curves at l1 5 690 nm and l25 817 nm wavelengths and the intensity of the transmit-ted coherent signal at the l3 frequency as functions of theangle of incidence u on the PC structure. We would liketo remind the reader that the numerical values of theangle u are measured relative to the normal incidence ofthe v1 beam on the PC structure and that the pointf 5 26.5° corresponds to the normal incidence of the v2beam on the PC structure.

In Fig. 9(c), we compare the experimental data of theangular dependencies of the intensity of the signal at thev3 frequency with the theoretical fit based on the theoret-

Fig. 9. Transmission coefficients (a) T1 at l1 5 690 nm of thepump wave, (b) T2 at l2 5 817 nm of the pump wave, and (c) theintensity IFWM of the transmitted wave at the FWM frequencyplotted as functions of the angle of incidence u1 of the first pumpwave. In (c), the curve of open circles represents the experimen-tal results, and the solid curve, the theoretical fitting. Theangle between the wave vectors of the pump waves is f5 6.5°.

ical formalism presented in Refs. 22 and 23. For the ex-perimental conditions of Fig. 9, we observe four mainmaxima of the FWM intensity in angular range u from250° to 180°: u 5 233°, 23°, 36°, 64°. The first threeangles have arbitrarily the same intensity, but the lastone located at u 5 64° demonstrates an '10 times higherintensity than the others. Comparison of the angular po-sitions of the maxima of the FWM signal with the relativepositions of the linear transmission curves for the wave-lengths corresponding to the v1 and the v2 beams [Figs.9(a) and 9(b)] allows us to assume that we observe twomechanisms of enhancement of the FWM nonlinear opti-cal signal. We observed and described closer behavior al-ready for noncollinear SFG in this paper and for collinearSHG and SFG in Refs. 12, 13, and 21. The maxima at u5 233°, 23°, 36° are located near the PBG edges for thev1 or the v2 beams, where the localization of the field’senergy is optimal for the corresponding frequencies.Moreover, we observed in experiments with other wave-lengths that the intensity and the curve topology arestrictly dependent on the mutual position and coverage ofthe bandgaps.

In fact, in the angular range of u 5 240° throughu 5 225°, the localization of the field energy for the wave-lengths of both the v1 and the v2 beams is optimal for thesame angle of u 5 233°, and we observed only one angu-lar maximum. For the angular range from 110° to 150°the localization of energy for the v1 and the v2 beams isoptimal for two different angles of incidence, u 5 23°,36°, and consequently two separate maxima were ob-served.

The difference between the amplitudes of the nonlinearoptical signal in these two cases (u 5 233° and u 5 23°,36°) is concerned with the different phase-matching con-ditions for the FWM interaction. It is actually true be-cause the dispersion of the PC has anomalous behavior onthe PBG edges. This assumption also describes the spe-cific behavior of the FWM signal intensity for u 5 64°.In this case, both frequencies v1 and v2 lie outside thePBG area, but frequency v1 will be on the PBG edge (it isnot shown in Fig. 9) and phase-matching conditions forthe FWM process at u 5 64° are fulfilled.

2. Inhomogeneous Wave Excitation in aOne-Dimensional Photonic Crystal at the Four-Wave-Mixing FrequencyCoherent waveguide Raman experimental structures areknown from the publications of Stegeman andco-workers24 who showed that, in the coherent anti–Stokes Raman scattering process v3 5 2v1 2 v2 , thebeams at the v1 and the v2 FFs are propagating as guid-ing modes inside a planar or a channel dielectric wave-guide. The field at the coherent Raman frequency v3 ,which is generated in the general case, can be radiatedinto all the possible guided and radiated modes. In thecoherent waveguide Raman technique only the light atthe v3 frequency that is generated into guided modes canbe phase matched and will, therefore, lead to high signallevels. For this reason in the coherent Raman waveguideonly the light at the v3 frequency that propagates asguiding modes is considered. The use of PCs in the co-


herent Raman waveguide technique enables one to widenthe field of its application considerably and predicts newphenomena.

In the experiments described hereafter the so-called de-generate or two-color FWM process is used. Two beamswith different frequencies v1 and v2 are incident uponthe medium. If the frequency relation (2v1 2 v2) corre-sponds to or is equal to the energy of the Raman activevibration of the medium, a vibrationally resonant coher-ent Raman signal is generated at the frequency v35 2v1 2 v2 . In this paper, we study the general case ofthe v3 5 2v1 2 v2 interaction when the frequency differ-ence may be both vibrationally and electronically reso-nant.

We now assume that two waves with wavelengths ofl1 5 690 nm and l2 5 817 nm corresponding to v1 andv2 , respectively, are incident under different angles uponthe periodic media described above. The excitation of theinhomogeneous wave at the FWM frequency is character-ized by the value of the tangential component of the wavevector at the v3 frequency that is determined by the ex-pression ky(v3) 5 2(v1 /c)sin(u1) 2 (v2 /c)sin(u2), whereu1,2 are the angles of incidence of the beams at the v1 andthe v2 fundamental frequencies. Varying the angle be-tween the directions of the pump beams by f 5 u12 u2 , we can potentially satisfy the condition ky(v3). (v3 /c), and thus the FWM wave becomes inhomoge-neous in vacuum. Therefore it will run along the surfaceof the PC structure because its averaged Poynting vectoris parallel to the layers of the structure.

For the PC that we used with the angle between twobeams with f 5 6.3° under our experimental conditionsthe regime of inhomogeneous wave excitation cannot beachieved. To observe the effect concerned with inhomo-geneous wave excitation, we performed the same experi-ment with an angle f between the v1 (l1 5 690 nm) andthe v2 (l2 5 817 nm) pump beams to be set at 25°. Thepolarization of the v1 beam was parallel to the plane ofincidence ( p polarized), and the v2 beam was perpendicu-lar to this plane (s polarized). The FWM signal analyzer

Fig. 10. Experimental (triangles) and theoretical (solid curve)transmitted FWM signals with an angle of f 5 25° betweenl1 5 690 nm and l2 5 817 nm beams. The inset shows themeasured (circles) and the calculated (solid curve) angles uFWMbetween the FWM transmitted beam and the normal of thesample plotted versus the angle of incidence u on the PC struc-ture.

was set to transmit the s-polarized v3 light. The powerof both input pump beams was fixed at 2.5 mW.

The results of the measurements of the intensity of thetransmitted FWM signal versus the angle u are shown inFig. 10. The triangles represent the measured FWM sig-nal, and theoretical data are symbolized by the solidcurves. The main observed maxima around 15° and115° are well described by the theory. These maximacorrespond to the angles for which the v2 beam is con-comitant with the gap edge of its transmission curve andits first oscillation.

The inset in Fig. 10 displays the value of the angle u3between the FWM beam and the normal of the structureversus the angle of incidence u. From the theoretical(solid) curve it can clearly be seen that the guidingoccurs at angles larger than 242° and is complete foru . 248°, i.e., u3 5 290°. The offset between the ex-perimental data (circle) and theory is within the accuracyof the angle measurements.

6. CONCLUSIONSWe have investigated nonlinear process in PCs by usingthe noncollinear scheme SFG and FWM processes in amultilayer structure consisting of the bilayers of isotropicmaterials. The efficiency and the possible interference ofthe surface and the bulk mechanisms of SFG have beencompared. It has been shown that in both cases the po-larization of the SF wave depends on the polarization ofthe pump waves in the same manner. So, within theframework of the developed approach, the bulk and thesurface mechanisms of SFG cannot be separated by polar-ization alone. It has been shown that there is significantenhancement in the response at the SF and in FWM un-der the condition that the transmission resonances at theedges of PBGs coincide for all three interacting waves.

Results of both theory and experiment suggest that, inthe case of noncollinear beam interaction near the PBGedge, the enhancement of the SF and the FWM signalsis due to a simultaneous action of field localizationand the satisfaction of a phase-matching condition thatalso occurs near the edges of PBGs. The phase-matchingcondition for SFG in a multilayer structure is (q1 1 q22 q3)d 5 2pn, where qi are the Bloch vectors of inter-acting waves, d is the period of the multilayer structure,and n is an integer.

We have demonstrated the possibility of exciting inho-mogeneous and guided waves at the FWM frequency inthe PC sample. This technique could be of great interestfor the development of PC-based waveguide devices25 forsolving beam-coupling problems.

ACKNOWLEDGMENTSWe gratefully acknowledge the support of this study bythe German Ministry for Research and Technology(BMBF) under grant 0081/99 13N7516. We are gratefulto the group of V. A. Bushuev for help in sample charac-terization. This study was supported in part by the Rus-sian Foundation for Basic Research (grants 01-02-17314and 02-02-17138). The Laboratoire de Physico Chimie del’Atmosphere participates with the Center d’Etude et de


Recherche Lasers et Applications, supported by the Min-istere de la Recherche, the Region Nord Pas de Calais,and the Fond Europeen de Developpement Economiquedes Regions.

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nonlinear process in photonic crystals under the noncollinear interaction

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