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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 463­477. Translated from Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, Vol. 125, No. 3, 2004, pp. 527­542. Original Russian Text Copyright © 2004 by Martemyanov, Dolgova, Fedyanin.

ATOMS, SPECTRA, RADIATION

Optical Third-Harmonic Generation in One-Dimensional Photonic Crystals and Microcavities
M. G. Martemyanov*, T. V. Dolgova, and A. A. Fedyanin**
Moscow State University, Moscow, 119992 Russia e-mail: *misha@shg.ru; **URL:http://www.shg.ru
Received August 25, 2003

Abstract--The formalism of nonlinear transfer matrices is used to develop a phenomenological model of a cubic nonlinear-optical response of one-dimensional photonic crystals and microcavities. It is shown that thirdharmonic generation can be resonantly enhanced by frequency-angular tuning of the fundamental wave to the photonic band-gap edges and the microcavity mode. The positions and amplitudes of third-harmonic resonances at the edges of a photonic band gap strongly depend on the value and sign of the dispersion of refractive indexes of the layers that constitute the photonic crystal. Model calculations elucidate the role played by phase matching and spatial localization of the fundamental and third-harmonic fields inside a photonic crystal as the main mechanisms of enhancement of third-harmonic generation. The experimental spectrum of third-harmonic intensity of a porous silicon microcavity agrees with the calculated results. © 2004 MAIK "Nauka / Interperiodica".

1. INTRODUCTION Photonic crystals have been extensively studied in recent years because of their unique dispersion properties and the possibility of modulating the spectral density of optical field modes, which manifests itself by the formation of photonic band gaps [1]. Fundamental interest in photonic crystals, in particular, stems from peculiar nonlinear optical effects, such as bistability [2] and optical switching [3] due to modulation of the refractive index of one-dimensional photonic crystal layers in a high-intensity field. This modulation causes a dynamic or quasi-stationary shift of the photonic band gap in a photonic crystal with a cubic nonlinearity. In such crystals, one can observe four-wave mixing and excitation of the waveguide mode at the anti-Stokes frequency which propagates along interfaces [4]. In media with modulation only of nonlinear susceptibility with a period of the order of the wavelength, nonlinear diffraction effects are observed [5]. The use of photonic crystals for effectively generating radiation at the second harmonic frequency was suggested in [6] and experimentally implemented for the first time in [7]. Phase mismatch between the fundamental and second harmonic waves is minimized by adding the reciprocal lattice vector of the periodic medium to the wave vectors of the interacting waves. When one of the frequencies is tuned to the edge of a photonic band gap, the phase matching condition for pumping and second harmonic waves is satisfied, which results in the enhancement of second harmonic generation in photonic crystals [8­11]. Third-harmonic generation (THG) in a photonic crystal can occur either directly due to cubic susceptibility or in a cascade manner as a result of quadratic

susceptibility. The first process was considered for an infinite photonic crystal in [12], where it was shown that there were structure parameters at which phase matching conditions were simultaneously satisfied for the fundamental and second-harmonic waves and the fundamental and third-harmonic waves. With these parameters, the time evolution of second- and third-harmonic intensities was studied, and it was shown that the pump energy could not be completely transfered to the second or third harmonic. In cascade THG by a onedimensional photonic crystal with quadratic susceptibility [13, 14], simultaneous phase matching of the pump with the second and third harmonics can also be achieved by adjusting the optical thicknesses of photonic crystal layers. The calculations reported in [13] were performed for a photonic crystal with an infinite number of layers, and only phase matching effects were therefore studied. In [14], a photonic crystal with a finite number of layers was considered, and effects of the spatial localization of fields related to its finite dimensions were taken into account. Pumping field localization effects can be enhanced by the introduction of a defect into a photonic crystal. In such a microcavity with distributed mirrors, the electromagnetic field resonant to the microcavity mode is effectively localized, which enhances second [15, 16] or third [17] harmonic generation. The enhancement of harmonic wave generation at the photonic band gap edge and in the microcavity mode is a result of the combined effects of phase matching due to the periodicity of Bragg reflectors and field localization caused by the presence of a microcavity spacer and finite photonic-crystal dimensions [11]. A key parameter that determines the enhancement of third harmonic generation is the dispersion of refractive indexes of layers in a photonic crystal or microcavity.

1063-7761/04/9803-0463$26.00 © 2004 MAIK "Nauka / Interperiodica"


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Its compensation is the essence of the phase matching at the photonic band gap edges in a photonic crystal and in the microcavity mode. However, the dependence of the magnitude and spectral position of third harmonic resonances on the magnitude and sign of the dispersion of layers constituting a photonic crystal or microcavity has not been studied yet. In this work, we study THG in one-dimensional finite photonic crystals and microcavities characterized (3) ^ by a cubic nonlinearity . The formalism of nonlinear transfer matrices is used to elucidate the role played by each of the mechanisms of the enhancement of third harmonic generation, that is, phase matching and pumping and third harmonic field localization when the pumping wave is in resonance with the microcavity mode or photonic band gap edge. The spectra of third harmonic intensity in the spectral range containing a photonic band gap with a microcavity mode and a passband region are calculated in the approximation of a given pumping field. The dependence of third harmonic generation enhancement on the dispersion of the refractive indexes of the layers that constitute a photonic crystal is studied. This dispersion determines the mutual arrangement of photonic band gaps and microcavity modes at the pump and third-harmonic wavelengths. The calculation results are compared with the experimental third harmonic spectrum generated in a microcavity made from mesoporous silicon. 2. NONLINEAR TRANSFER-MATRIX METHODS FOR CALCULATING THIRD-HARMONIC GENERATION IN PHOTONIC CRYSTALS 2.1. Nonlinear Transfer-Matrix Method There are several approaches to calculating optical harmonic generation in one-dimensional photonic crystals. One of these is via solving a system of reduced equations obtained in the method of slowly varying amplitudes [12­14, 18]. This approach can be used to take into account energy transfer from the pump to the generated harmonic; analyze simultaneous generation of the second and third harmonics; and examine the time evolution of the fundamental and the second- and third-harmonic waves, which is important for studying harmonic generation by femtosecond pulses. In the approximation of constant fundamental field, THG is described by a single inhomogeneous equation, which can be directly solved using the Green function of a multilayer structure [4, 19]; the solution is constructed based on linearly independent solutions to the homogeneous wave equation, which can be found using the formalism of transfer matrices [20]. Lastly, a convenient method is the extension of the formalism of transfer matrices to harmonic generation. This method is applicable in the approximation of constant fundamental wave and when the fields are stationary, that is, under the condition that the pumping pulse width is much greater than the time of fundamental-wave propagation

across the photonic crystal. There are two equivalent approaches. The first one, described in [21], uses the formalism of Green functions suggested by Sipe [22]. The second approach given in [23] is based on the formalism of coupled and free harmonic waves introduced by Bloembergen and Pershan [24]. Both rely on direct solution of an inhomogeneous wave equation with the use of the Green functions of a photonic crystal and provide additional information about the nonlinear optical processes under consideration, such as contributions of separate layers to the resulting third-harmonic wave. The method of nonlinear transfer matrices suggested in [23] can conveniently be used to calculate nonlinear-optical effects in one-dimensional photonic crystals because of its simplicity and form optimal for numerical calculations. The problem of THG in photonic crystals can be decomposed into three sequential stages. First, the fundamental wave propagation in a multilayer linear structure is described taking into account multiple-reflection interference, and the spatial pumping field distribution within the photonic crystal is calculated. At the second stage, cubic polarization ^ (3) induced in a medium with nonzero is determined. Lastly, the linear problem of third harmonic wave propagation in a layered structure is solved taking into account coupled and homogeneous waves, and the intensity of the third harmonic wave that emerges from the photonic crystal is found. Let the z axis be perpendicular to the surface of the photonic crystal and xz be the plane of pumping wave incidence (Fig. 1a). A monochromatic linearly polarized fundamental wave with frequency , wave vector + + k 1 , and amplitude E 1 propagates in half-space 1 at angle 1 to the normal to the surface of the photonic crystal. We assume that the photonic crystal is optically inactive and nonmagnetic; s- and p-polarized waves will therefore be considered separately. The electromagnetic field in the jth layer is a superposition of two plane forward waves (propagating in the positive direction along the z axis) and backward, E j ( z, t ) = E j exp [ ik z, j ( z ­ d ij ) + ( ik
+ x, j

x ­ it )]

+ E j exp [ ­ ik z, j ( z ­ d ij ) + ( ik
­



x, j

x ­ it )],

(1)

where k
z, j

= k j cos j ,

k

x, j

= k j sin j ,

dij is the z coordinate of the boundary between the ith and jth layers, kj is the wave vector, and j is the angle of refraction of the pumping wave in the jth layer. The exp( ik x, j x ­ it) term will further be omitted because of the translational invariance of the problem in the xy
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465

plane and its stationary character. The field at the left boundary of the jth layer is represented by the two+ ­ component vector Ej ( E j + E j ); the first component of this vector is the amplitude of the forward wave and the second one, that of the backward wave. The relation between Ej and Ek in the kth layer at its left boundary is given by the two 2 â 2 matrices + E k = 1/ t kj r kj / t kj ­ r kj / t kj 1/ t kj Ek exp ( ik z, j d j ) 0 â exp ( ­ ik z, j d j ) 0 + Ej . ­ Ej (2)

The third stage of solving the problem can be considerably simplified. In the approximation of a given pumping field, cubic polarization in each layer is a third harmonic source independent of the other layers. Thus, we can solve the problem of third-harmonic generation and propagation for a single photonic crystal layer with a cubic susceptibility. The third-harmonic field generated by the photonic crystal can then be obtained by summing such partial third-harmonic outputs of all layers and taking into account their phases. Let the jth layer be nonlinear. The interference of the coupled E(s) and free Ej third harmonic waves is included in the boundary conditions, which, for the ijth and jkth boundaries, have the form F i E i = M ij E j + M i E , M kj F j E j + M k F E
(s) (s) (s) (s) (s)

(5)

Here, the first matrix contains the Fresnel reflectivity rkj and transmissivity tkj for the wave incident from the kth to the jth layer and relates fields to the left and right of the kjth interface. In what follows, this matrix is denoted by Mkj . The second j(dj) matrix describes field propagation in the jth layer of thickness dj from the left to right boundary. The fields in half-spaces 1 and l can therefore be related. Under the assumption that the backward wave is absent in half-space l and that a wave with unit amplitude is incident on the 1­2 boundary, we obtain T t = T 0
11 21

= Ek .

T T

12 22

1 . r

(3)

The amplitude of the inhomogeneous third-harmonic wave in (5) is calculated in the jth nonlinear layer at its left boundary, and all homogeneous waves are thirdharmonic waves. Matrices M with index (s) are constructed similarly to the usual transfer matrices, but the Fresnel coefficients in their elements contain refractive indexes for the inhomogeneous third-harmonic wave in the nonlinear layer and the free third harmonic in the layer whose number equals the lower index of the matrix. The F(s) matrix is similar to j and is obtained from the latter by replacing the wave vectors of the free third harmonic with the wave vectors of the inhomogeneous wave. System (5) yields E k = M kj F j ( M ji F i E i + S j ) , where the vector S j = (F jM j
(s) (s)

Here, r and t are the reflectivity and transmissivity of the photonic crystal as a whole. They are determined from (3), which gives T 21 r = ­ ------ , T 22 T 11 T 22 ­ T 12 T 21 t = ------------------------------------ , T 22

(6)

­ M j )E

(s)

(s)

(7)

where T are the elements of the transfer matrix for the photonic crystal as a whole, T Ml (l ­ 1)F(l ­ 1) ... M21. The fundamental field distribution within the photonic crystal is given by E j (z) = F j(z ­ d
j( j ­ 1)

)M

j( j ­ 1)

F

( j ­ 1)

... M 21 1 . r

(4)

is singled out for convenience. The F j matrix is inverse to Fj . Equation (6) determines the third-harmonic field in the kth layer as a superposition of the waves transmitted from the ith layer and generated in the jth layer by nonlinear sources. Equation (7) contains the contribution of the nonlinear jth layer to the third-harmonic wave; the term in parentheses takes into account the interference of the homogeneous and inhomogeneous waves. Under the assumption that no external field with the third-harmonic wavelength is incident on the photonic crystal, (6) can be rewritten as El ( j ) ­ L 0
+

Spatial distribution (4) can be used to calculate the spatial distribution of the cubic polarization wave and inhomogeneous third harmonic can be calculated; this is done in the next Section.
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

R

0
j1

jl

E ( j)
­ 1

= S j,

(8)

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466 1 2 ... (a) k­ 1 1 k+ 1 E­ 1 E+ 1 k­ i k+ i i (b) x y z II PII s Ex
I II Px

MARTEMYANOV et al. i j E E
(s) y (s)

E(s) || ks

k ...

l­1

l

E+ j

k+ j

E­ k­ j j kII j s
II



II

k
I Px

PI s

I

PII z

I I ks PI z

dipole surface and bulk quadrupole sources; conversely, the generation of the bulk dipole third harmonic is allowed. As a result, the cascade process becomes less effective. A nonlinear photonic crystal layer can be treated as a medium rotationally isotropic in the plane of the layer. Such a medium is characterized by the symmetry groups /mm and 2 (cylinders) and m (a cone with a symmetry axis of an infinite order perpendicular to the surface of the layer). Equations for inhomogeneous third-harmonic waves for other symmetry groups can be obtained similarly. The tensor of dipole cubic sus^ (3) ceptibility invariant with respect to the m, 2, and /mm groups and symmetrical with respect to permutations of the last three indices has four independent nonzero components [25], 1
xxyy

Ep E Ez

s

k

1 = -- 3 2 yyzz = =
yyxx

xxxx xxzz

1 = -- 3

yyyy

, (9)

, .

3 4
zzxx

zzzz

,
zzyy

Fig. 1. (a) Scheme of a one-dimensional photonic crystal (layers 2 ... l ­ 1) (half-spaces 1 and l denote the vacuum and substrate, s-polarized fundamental wave is shown) and (b) THG scheme in the jth nonlinear layer.

=

where the matrices R jl F j M jk F k M
k (k + 1)

...

(l ­ 1)

M
21

(l ­ 1)l

,

Let the jth layer with cubic susceptibility be situated between two linear layers i and k (Fig. 1). Let us determine the amplitude of the inhomogeneous third harmonic E(s) on the jith interface from the fundamental field amplitude Ej on the same boundary. The angle between the fundamental wave vector k = n --- , c where n is the refractive index at the fundamental frequency, and the z axis is . Here and throughout, index j numbering layers is omitted. The dipole cubic polarization is given by the convolution P ...
3

L j 1 M ji F i M

i(i ­ 1)

... F2 M

characterize the propagation of homogeneous thirdharmonic waves from half-spaces 1 and l to the jth layer. It follows that, given Sj , we can find the amplitude and phase of the third-harmonic fields E 1 ( j ) and
­

E l ( j ) generated in the jth photonic crystal layer and emerging from the photonic crystal into the vacuum (half-space 1) and substrate (half-space l). The total third-harmonic field in the substrate or vacuum is the + ­ sum of E l ( j ) or E 1 ( j ) taken over all layers.
+

^ =

(3)

( E exp ( ik z z ) + E exp ( ­ ik z z ) )
+ ­ s, I I­ s, I s, I I z





3

P exp ( ik z z ) + P exp ( ­ ik z z )
I+

(10)

+P


II +

exp ( ik

z) + P

II ­

exp ( ­ ik

s, I I z

z).

2.2. Inhomogeneous Third-Harmonic Waves in a Photonic Crystal Let us obtain equations describing inhomogeneous third-harmonic waves for direct THG by means of a ^ (3) cubic susceptibility . Equations for cascade THG ^ (2) caused by quadratic susceptibility can be obtained similarly. The cascade process can be ignored when the nonlinear medium has an inversion center. The generation of the bulk dipole second harmonic is then forbidden, and the second harmonic is only generated by

Here, k z = k cos is the projection of the fundamental wave vector onto the normal to the interface. In Eq. (10), all cubic polarization terms are divided into two types. The terms of the first type have the z wave s, I vector component k z = 3 k z and are obtained by the convolution of three fundamental waves propagating in the same direction. The other terms have the normal s, I I projection of the wave vector k z = k z and are obtained by the convolution of three fundamental
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467

waves one of which has the z component of the wave vector opposite to the projections of the other two waves. The propagation of cubic polarization and inhomogeneous third-harmonic waves is determined by the effective permittivity calculated as k
s, ( I , I I )

+ 2 ( E p E p E s sin + 2 E s E p E p sin ) ,
2 2

P

II z

= ­ 3 E p sin
3 3 2

­ 4 sin ( 3 E p cos + E s E s E p + 2 E p E s E s )
3

3 = -----c

I, I I

.

for PII. Here, we use the equation E y = Es , E x = E p cos , E z = ­ E p sin .

For the waves of type I, k that is,
I s, I

3 = 3 k = -----c

();

The product of three two-component vectors is given by the rule + + + E E E Ea Eb Ec = a b c ­ ­ ­ Ea Eb Ec for the waves of type I and by the rule + + ­ E E E Ea Eb Ec = a b c ­ ­ + Ea Eb Ec

= ()

(13)

and the polarization wave propagates in the medium collinearly to the fundamental wave, I = . Similar calculations for the polarization waves of type II give
II

= ( ) ( 1 + 8 sin ) /9.
2

(14)

The angle between axis z and the propagation direction of the polarization waves of type II is different from ; it is given by
II

= arctan ( 3 k x / k z ) .





Taking into account nonzero components (9) of the ^ (3) tensor , the projections of the two-component vectors PI = (PI+, PI­) and PII = (PII+, PII­) onto the coordinate axes expressed in terms of the s- and p-polarized fundamental wave components can be written as P = 1 ( E cos + E E p cos )
I x 3 p 3 2 s

for the waves of type II. Cubic polarization can conveniently be decomposed into components with polarization directions normal and parallel to the wave vector of the inhomogeneous wave and the s-polarized component, P P
I, I I || I, I I

=P =P

I, I I x I, I I x

sin (

I, I I

)+P

I, I I z

cos ( sin (

I, I I I, I I

), ), (15)

cos ( P

I, I I I, I I y

)­P .

I, I I z

+ 2 E p cos sin ,
3 2

For such components, the transition to third-harmonic inhomogeneous waves is simple [24]:
2 2

P y = 1 ( E s + E s E p cos ) + 2 E s E p sin ,
I 3 2 2

(11)

E

( s ) I, I I

1 3 I 3 P z = ­ -- 3 E p sin 3 ­ 4 ( E p E s sin + E p sin cos )
2 3 2

4 = ----------------------------- ( P I, I I ­ (3) 4 I, I I ­ -------------- P || . (3)

I, I I y

+P

I, I I

) (16)

for P
II x

PI

and
3 3

= 1 ( 3 E p cos + 2 E p E s E s cos + E s E s E p cos ) +3 2 E p sin cos ,
3 2

P

II y

= 1 ( 3 E s + 2 E s E p E p cos
2 2

+ E p E p E s cos )
2

(12)

Equations (16) for third-harmonic inhomogeneous waves are substituted into (7). The reduction of the general problem of THG in a photonic crystal to THG in a photonic crystal containing a single layer with cubic susceptibility allows partial contributions of each photonic crystal layer to the total third-harmonic wave to be calculated. In combination with the feasibility of calculating the fundamental field at each point in the photonic crystal, this is a convenient tool for analyzing the mechanisms of nonlinear-optical phenomena in one-dimensional photonic crystals. Variation of the iniVol. 98 No. 3 2004

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468 Rs 1.0 0.8 0.6 0.4 0.2 0 1.0 0.8 0.6 0.4 0.2 0 300 500 700 , nm 900
3 2 cavity spacer 10 pairs 9.5 pairs 20 pairs

MARTEMYANOV et al.

()

Si(001)

(b)

Si(001)

1100

Fig. 2. Dependence of the linear reflection coefficient of the s-polarized fundamental wave on its wavelength calculated for (a) a photonic crystal and (b) a microcavity. Photonic crystal and microcavity schemes are shown in the insets.

of porous silicon layers (porous silicon was used to fabricate the microcavity studied experimentally in this work). The microcavity is obtained by doubling the thickness of the central twenty-first layer. It is assumed that the photonic crystal and microcavity are placed between two half-spaces, air and a linear medium with a refractive index equal to the refractive index of silicon. THG from the silicon substrate is ignored. Fundamental radiation comes from air. The mechanisms of THG enhancement are studied in the frequency domain, that is, under fundamental wavelength variations, which allows both photonic band gap edges to be observed for the same sample. This cannot be done by means of angular third-harmonic spectroscopy with a fixed fundamental wavelength and normal component of the fundamental wave vector tuned by changing the angle of incidence. All frequency spectra are calculated at the incidence angle 1 = 45°. The fundamental wavelength is varied from 730 to 1100 nm. The refractive indexes of photonic crystal layers are assumed to be constant in the spectral ranges of the fundamental and third harmonic wavelengths. The spectra of the linear reflection coefficient of the photonic crystal and microcavity are shown in Fig. 2. The same figure contains the spectra of the linear reflection coefficient in the wavelength ranges of the second (360­560 nm, this spectrum is further denoted by R2) and third (240­370 nm, spectrum R3) harmonics. The reflection coefficient R is close to unity from 820 to 940 nm, which is a manifestation of the photonic band gap. The R3 spectrum also contains a wavelength region with the reflection coefficient close to unity (photonic band gap at third-harmonic wavelengths), whereas there is no photonic band gap in the R2 spectrum, because the phase difference between the waves reflected from layers with optical thicknesses /4 or 3/4 is n, where n is an even integer, and the waves are added in phase. If the optical thickness of layers is /2, the phase difference between the waves reflected from such layers is m, where m is an odd integer, and the waves interfere destructively. The presence of a resonant layer manifests itself in the spectrum of the linear reflection coefficient by a dip within the photonic band gap corresponding to the microcavity mode. The microcavity mode is present in both R and R3 spectra. The spatial distributions of the fundamental field amplitude shown in Fig. 3 were calculated at the wavelengths corresponding to the minima in the spectrum of the linear reflection coefficient that are closest to photonic band gap (Figs. 3b, 3c), in the region within the photonic band gap of the photonic crystal (Fig. 3d), and in the microcavity mode (Fig. 3a). The fundamental + ­ + ­ field amplitude is E = | E j + E j |, where E j and E j are the complex components of the E j two-component pumping field vector given by (4). When is tuned across the photonic band gap, the fundamental field
Vol. 98 No. 3 2004


tial calculation parameters, such as the fundamentalradiation wavelength or the corresponding incidence angle, allows us to obtain the frequency and angular spectra of the linear reflection coefficient and the intensities and phases of third harmonic waves. Note that (11), (12), and (15) specify possible polarizations of third-harmonic waves. If the fundamental wave is s-polarized, only an s-polarized third harmonic is generated (the first term of the Py component), and if the fundamental wave is p-polarized, the third harmonic is also p-polarized (the first terms of the Px and Pz components). 3. THIRD-HARMONIC GENERATION IN PHOTONIC CRYSTALS 3.1. The Absence of Dispersion Calculations of the enhancement of third-harmonic generation at the edge of the photonic band gap are performed for a photonic crystal. For a microcavity, we study THG enhancement effects that appear when the fundamental radiation is tuned across the region in the frequency-angle space that corresponds to the microcavity mode. The influence of the microcavity layer is weak at the edge of the photonic band gap, and thirdharmonic enhancement is similar to that characteristic of photonic crystals. The model photonic crystal consists of 20 pairs of alternating layers with refractive H L indexes n = 1.93 and n = 1.61 and optical thicknesses 0/4, where 0 = 960 nm (Fig. 2). The refractive indexes selected are close to the real refractive indexes

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


OPTICAL THIRD-HARMONIC GENERATION IN ONE-DIMENSIONAL PHOTONIC CRYSTALS E 4.0 3.5 (a) 3.0 2.5 2.0 1.5 1.0 0.5 0 3.5 (c) 3.0 2.5 2.0 1.5 1.0 0.5 0 2000 4000 0 2000 4000 1000 3000 5000 1000 3000 5000 d, nm 0 750 800 850 900 950 1000 1050 1100 , nm (d) 2 6 I3 4 8 (b) 10 3, nm 250 275 300 325 350

469

375 1.0 0.8 0.6 0.4 0.2 0 Rs

Fig. 4. The third harmonic intensity spectrum of a photonic crystal calculated for s-polarized fundamental and third harmonic waves (SS geometry) (the thick solid line); the reflection coefficient spectra of an s-polarized wave calculated in the regions of fundamental (solid thin line) and third-harmonic (dashed line) wavelengths. The third-harmonic intensity I3 is given in arbitrary units.

Fig. 3. Fundamental field amplitude distributions inside a photonic crystal and microcavity: (a) for the microcavity mode, fundamental wavelength is = 877 nm; (b) at the short-wavelength edge of the photonic crystal band gap, = 806 nm; (c) at the long-wavelength edge of the photonic band gap, = 966 nm; and (d) within the photonic band gap, = 870 nm. The dashed line is the incident wave amplitude equal to one.

exponentially decays as the depth of penetration into the photonic crystal increases (Fig. 3d). The fundamental wave which is in resonance with the microcavity mode is strongly localized inside the resonant layer. At the chosen microcavity parameters, E increases approximately fourfold. At the short-wavelength and long-wavelength edges of the photonic band gap, the fundamental field is localized less strongly and is enhanced 2.1­2.3 times. This effect is caused by the finite photonic crystal length; with a larger number of layers, the fundamental field is distributed more evenly. The amplitude of the pumping field resonant to the microcavity mode reaches a maximum in the microcavity layer and sharply decreases in several neighboring layers, whereas in a photonic crystal, the fundamental field tuned to photonic band gap edges is more evenly distributed over the photonic crystal. This means that for effective THG in a photonic crystal, phase matching of the homogeneous third harmonic waves generated by various layers and having amplitudes of the same order is important.

The third-harmonic intensity spectrum I3 generated in a photonic crystal in the absence of refractive index dispersion is shown in Fig. 4. Multiple peaks located both within the photonic band gap and near every R spectrum minimum to the left and right outside the photonic band gap are observed in the third-harmonic spectrum. Third-harmonic enhancement is less pronounced at the edge of the photonic band gap, but the amplitude of third-harmonic peaks increases in the next linear reflection coefficient minima. When the fundamental wavelength is tuned across the photonic band gap, the amplitude of the fundamental field decreases exponentially as the depth of penetration into the photonic crystal increases and the source of third-harmonic generation is several photonic crystal layers near surface. Upon tuning to the minimum of the reflection coefficient, the fundamental field effectively penetrates deep into the photonic crystal and all its layers become sources of third-harmonic waves. At the same time, the amplitudes of the peaks of third-harmonic intensity are commensurate no matter whether is tuned across the photonic band gap or the minima of the reflection coefficient spectrum. This is evidence of dephasing of third-harmonic waves coming from different photonic crystal layers; these waves destructively interfere with each other. It can be expected that the inclusion of dispersion into calculations (when the refractive indexes of photonic crystal layers have different values at the fundamental and third-harmonic wavelengths) will change the phases
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470 3, nm 10 250 300

MARTEMYANOV et al. 3, nm 350 () 8 250 300 350 (b)

I3

6

4

2 0 1.0 (c) 0.8 (d)

Rs

0.6

0.4

0.2

0

800

900 , nm

1000

1100

800

900 , nm

1000

1100

Fig. 5. (a, b) Third-harmonic intensity spectra of a photonic crystal calculated for the SS geometry in the presence of dispersion of photonic crystal layers and (c, d) s-polarized wave reflection coefficient spectra calculated in the wavelength ranges of fundamental waves (solid line) and the third harmonic (dashed line). Third-harmonic intensity units in Figs. 5a, 5b and 4 are incommensurate.

of partial third-harmonic waves coming from different layers and the peaks corresponding to the minimum of R will gain in third-harmonic intensity. 3.2. Enhancement of Third-Harmonic Generation in the Presence of Dispersion Characteristic third-harmonic intensity spectra calculated in the presence of dispersion of the refractive indexes of photonic crystal layers are shown in Figs. 5a and 5b. The bottom panels show the linear reflection coefficient spectra R and R3 . Figures 5a and 5c correspond to the dispersion of the refractive indexes of optiL cally less dense photonic crystal layers n3 = n 3 ­

n = -0.045, and Figs 5b and 5d, to n3 = n 3 ­ n = 0.051. The refractive index at the fundamental waveL L L

length is taken to be n = 1.610 + i â 0.00003; that is, absorption is virtually absent. The dispersion of the refractive indexes of optically denser photonic crystal
L

layers is set equal to n3 n / n . The I3() spectra
H L

show that the third-harmonic intensity resonantly increases in the spectral region of photonic band gap edges. At n3 < 0, the intensity of the third harmonic peak at the first R spectrum minimum to the left of the photonic band gap (the short-wavelength edge of the photonic band gap) increases by no less than two orders of magnitude as compared to the intensity of the peaks
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within the photonic band gap. At n3 > 0, the third harmonic intensity at the long-wavelength edge of the photonic band gap increases by approximately an order of magnitude. The correspondence between the sign of dispersion and the spectral position of the third-harmonic intensity peak with respect to the photonic band gap is unambiguous; namely, at n3 < 0 and n3 > 0, third-harmonic intensity resonances are observed only at the short- and long-wavelength photonic band gap edges, respectively. This is clearly seen in Fig. 6, where the dependence of the amplitude of the third-harmonic intensity max peaks I 3 on dispersion n3 are shown at the shortand long-wavelength photonic band gap edges. The main conclusion is that we do not observe simultaneous enhancement of the third harmonic at both photonic max band gap edges. The I 3 (n3) dependences are oscillatory in character. Their maxima that appear, for instance, at dispersion n3 values equal to 0.051, -0.114, and ­0.045 are reached when coincides with the minimum of the spectrum of the reflection coefficient for the fundamental wave R and one of the minima of the R3 spectrum is observed at the third harmonic wavelength (Figs. 5c, 5d). The farther from the photonic band gap minimum of the R3 spectra that coincides with the photonic band gap edge in the R spectrum corresponds to the weaker the third harmonic resonance. It follows that a key factor of third-harmonic enhancement is the coincidence of the pumping wavelength with the photonic band gap edge and of the thirdharmonic wavelength with the minimum of the reflection coefficient R3 . Third-harmonic resonances are observed not only at the photonic band gap edge but also at the fundamental wavelengths at which the R spectrum has the second, third, etc., minima. These resonances are enhanced to a lesser degree than the peaks at the edge of the photonic band gap. This is clearly seen from Fig. 7, where the whole set of all third harmonic frequency spectra obtained as n3 changed from ­0.21 to 0.19 in steps of 0.002 in wavelength steps of 1 nm is shown. Peaks at the edges of the photonic band gap are quite pronounced. The enhancement at the short-wavelength edge is more intense, and the enhancement at the other R spectrum minima is much weaker. The resonance enhancement of third-harmonic generation at the edge of the photonic band gap is determined by multiple-reflection interference of both fundamental and third-harmonic waves, because the strongest enhancement of the third harmonic is observed when the third-harmonic wavelength occurs at an R3 spectrum minimum. To elucidate the role played by interference effects at the fundamental wavelength we must use (4) to calculate the spatial distribution of the amplitude of the fundamental wave within the photonic

I3 10

max

­ 0.114 8
max I3

0.5 0.4 0.3 0.2 0.1

6

4

­ 0.045 0

0.05

0.10 n3

0.15

0.20

2

­ 0.131 0.049 0.138 0.1 0.2 ­ 0.1 0 n3

0 ­ 0.2

Fig. 6. Dispersion n3 dependences of maximum thirdharmonic peak intensity at the right (thin line) and left (thick line) edges of the photonic band gap. Arrows show the dispersion values that correspond to the characteristic third-harmonic intensity maxima. The region with positive n3 values is shown in the inset on an enlarged scale.

crystal (Fig. 3) and the amplitude of inhomogeneous third harmonic waves at the boundary of each layer [Eq. (16)]. The multiple-reflection interference effects at the third-harmonic wavelength are determined by comparing the amplitude and phase of the output ­ homogeneous third-harmonic waves E 1 ( j ) generated separately by each jth layer [see Eq. (8)] and the amplitude and phase of inhomogeneous third harmonic waves at the boundary of each layer. The spatial distribution of the amplitude of the inhomogeneous third-harmonic wave inside the photonic crystal when corresponds to the edge of the photonic band gap is shown in Fig. 8a. The shape of the E(s)( j ) dependence reproduces that of the distribution of the fundamental wave amplitude. When the dispersion n3 is introduced into calculations, the shape of the E(s)( j ) dependence does not change and enhancement in the middle of the photonic crystal caused by fundamental field localization is retained, while the amplitude of the inhomogeneous wave decreases as n3 increases. The phase distribution of the inhomogeneous third-harmonic wave at the boundary of each layer is determined by the fundamental field phase at the same boundary. The phase difference between third-harmonic fields in neighboring layers with equal refractive indexes is close to . For this reason, each point of the polar diagram in which inhomogeneous third harmonics are shown with their phases (Fig. 8b) has a correVol. 98 No. 3 2004

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MARTEMYANOV et al. 3, nm 10 250 275 300 325 350 375 1.0

8

0.8

6 I3

0.6 Rs

4

0.4

2

0.2

0

750

800

850

900

950 1000 1050 1100

0

, nm
Fig. 7. Solid circles are the whole set of the third-harmonic intensity spectra obtained with n3 varied from ­0.21 to 0.19, and the solid line is the linear reflection coefficient spectrum.

E(s) 10 () 8

120

60

(b)

6 180 4 0

2 240 0 10 20 j 30 40 300

Fig. 8. (a) Dependence of the amplitude of the inhomogeneous third-harmonic wave determined by (16) on the number of the photonic crystal layer and (b) amplitudes and phases of inhomogeneous third-harmonic waves. The calculations were performed for n3 = 0.

sponding point with a comparable amplitude and an almost opposite phase. The phase of inhomogeneous waves does not change at n3 0. The dependences of the amplitudes and phases of ­ the third-harmonic partial waves E 1 emerging from the photonic crystal on the layer number j are shown in Fig. 9. These dependences were calculated at the max-

ima of the I 3 (n3) dependence (Figs. 9a­9d) and at its minimum (Figs. 9e, 9f). The amplitudes of inhomogeneous third-harmonic waves at the minimum and max maximum of the I 3 (n3) dependence are of the same order of magnitude (Fig. 8), and the amplitude of emerging free third harmonic waves at the minimum of max the I 3 (n3) dependence is by a factor no less than 3
max

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473

() 120

(b) 60

240 200 160 120 80 40 0 160 (c) 240 120 (d) 300 60 180 0

120 180 0

80

40 240 120 300 60

0 (e)

(f)

40

180

0

240 0 10 20 j 30 40
­

300

Fig. 9. (a, c, e) Dependences of the partial third-harmonic E 1 amplitude on the layer number j calculated when the fundamental wavelength is tuned to the edge of the photonic band gap for dispersion n3 values of (a) ­0.114, (c) ­0.131, and (e) ­0.085 and (b, d, f) partial third-harmonic waves with their phases. The insets show schematically the I The points at which the calculations were made are marked by arrows.
max 3

(n3) dependence shown in Fig. 6.

smaller than at its maximum. This implies additional amplitude enhancement of the emerging third harmonic by constructive multiple-reflection interference of third-harmonic waves; namely, the partial third harmonic-wave generated in the jth nonlinear layer reaches an interference maximum outside the photonic

crystal. This is accompanied by phase matching of partial third-harmonic fields outside the photonic crystal, ­ and the phases of the E 1 waves become localized in a narrow angular interval. The weakest phase matching of partial third-harmonic waves is observed at the minVol. 98 No. 3 2004

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Rs

1

0.8 0.6 0.4

0.1

0.2 0 800 900 1000 , nm 1100

the R3 spectra closest to the photonic band gap coincide; namely, the larger the n3 value, the larger phase changes of partial third-harmonic waves and the stronger their destructive interference. As a result, the highest intensity of the total emerging third harmonic is attained because, first, the amplitude of the emerging partial third-harmonic waves increases and, second, their phases are matched (Figs. 9a, 9b). The smallest max I 3 value is observed when the photonic band gap in the R spectrum corresponds to the edge of the photonic band gap in the R3 spectrum (Figs. 9e, 9f). The maximum of the E 1 ( j ) dependence for the selected photonic crystal parameters is then shifted by six layers toward the photonic crystal surface with respect to the maximum of the spatial distribution of the inhomogeneous third-harmonic wave amplitude. This is explained by the third-harmonic wavelength falling within the photonic band gap in the R3 spectrum. The contributions of deeper photonic crystal layers to the emerging third harmonic then exponentially decrease. The plots in Fig. 9 were constructed for n3 < 0, when the third harmonic resonance is observed at the short-wavelength edge of the photonic band gap. The dependences for n3 > 0 are similar.
­

0.01

250

275

300 3, nm

325

350

375

Fig. 10. The intensity spectrum of the third harmonic generated in a microcavity and calculated for the SS geometry. Shown in the inset are the reflection coefficient spectra of an s-polarized wave calculated in the pumping (solid line) and third harmonic (dashed line) wavelength ranges. The R3 spectrum is plotted on a triply enlarged wavelength scale.

I3 0.8

Rs 0.05 0.04

4. THIRD HARMONIC GENERATION IN MICROCAVITIES The dependence of the intensity of the third harmonic generated in a microcavity on its wavelength is shown in Fig. 10. The third-harmonic spectrum has a peak corresponding to the resonance between the fundamental radiation and the microcavity mode. Its intensity is more than three orders of magnitude higher than the intensity of the third harmonic in other spectral regions. No enhancement is observed at the edge of the photonic band gap. If n3 0, the resonance peak amplitude substantially changes and THG enhancement at the edge of the photonic band gap arises. The max I 3 (n3) dependences of the maxima of third-harmonic intensity peaks at the resonance between the fundamental radiation and the mode and between the fundamental radiation and the short- and long-wavelength edges of the photonic band gap are shown in Fig. 11. If n3 < 0, the resonance amplitudes in the microcavity mode and at the left edge of the photonic band gap increase. If n3 > 0, we observe enhancement in the mode and at the right edge of the photonic band gap. THG enhancement in the microcavity mode is maximum at zero dispersion. If dispersion is nonzero, its amplitude decreases by approximately an order of magnitude and oscillates as a function of n3 . The localization of the fundamental field resonant to the mode results in a very substantial increase in the amplitude of partial third-harmonic waves, which
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0.6

0.03 0.02 0.01

0.4
0 0.05 0.10 0.15 , nm

0.2

0 ­ 0.2

­ 0.1

0 n

0.1
3

0.2

Fig. 11. Maximum intensity of third-harmonic peaks generated in the microcavity mode (thin line) and at the longwavelength (thick line) and short-wavelength (dot-and-dash line) edges of the photonic band gap as a function of dispersion n3; the dependences in the n3 > 0 region are shown in the inset on an enlarged scale.

imum of the I 3 (n3) dependence. The phase jump through 2/3 in Fig. 9d explains the reason of a maximum third-harmonic enhancement when the photonic band gap edges in the R spectrum and the minima in
max

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475

() 120 60

­ E1 10

(b) 120 60

12 5 6 0 180 ­6 ­5 ­12 ­18 240 300 ­10
­

0 0 180

0

240

300
max 3

Fig. 12. Polar diagrams of complex partial third harmonic waves E 1 ( j ) corresponding to (a) the maximum of the I dependence, n3 = 0 and (b) one of the minima of the I
max 3

(n3)

(n3) dependence, n3 = ­0.04.

depend on the third power of the fundamental field amplitude. Irrespective of the phase difference between these waves, the minimum intensity of the emerging third-harmonic wave is then no less than the maximum amplitude of the third-harmonic wave generated at the edge of the photonic band gap. The partial third-harmonic wave amplitudes in the microcavity mode remain almost invariant as n3 is varied, whereas their phases change substantially. The E 1 ( j ) phases at the
­

the dispersion value, whereas the peaks at the shortwavelength edge of the photonic band gap appear at a negative dispersion, and those at the long-wavelength edge, when dispersion is positive. 5. AN EXPERIMENTAL STUDY OF THIRD HARMONIC GENERATION IN MICROCAVITIES We experimentally studied THG in microcavities. The sample was a one-dimensional microcavity made from mesoporous silicon by electrochemical etching of a heavily doped p-type silicon plate with a (100) crystallographic orientation in a solution of hydrofluoric acid following the procedure described in [11]. The sample was two one-dimensional photonic crystals consisting of five pairs of quarter-wave (0 = 1300 nm) porous silicon layers separated by a half-wave cavity layer. The 0 value is the spectral position of the microcavity mode at the normal fundamental wave incidence. H The refractive index and thickness were n 1.93 and dH 170 nm for optically denser layers and n 1.61 and dL 200 nm for less dense ones. The porous silicon microcavity layer has the refractive index nL = 1.61 and thickness dres = 400 nm. The spectral dependences of the third harmonic intensity were measured using a tunable optical parametric generator. Its output linearly polarized radiation had the following characteristics: pulse width of 4 ns, pulse energy of about 10 mJ at a 800 nm wavelength, and tuning range of 410­690 nm for the signal wave and 735­2200 nm for the idle wave. We used the idler tuning range because the photonic
L

I 3 (n3) peaks are nearly equal (see Fig. 12a). At the minimum reached at n3 = ­0.04 (Fig. 12b), the partial
max

waves E 1 ( j ) are out of phase, and their interference is destructive.
­

It follows that, when the fundamental wavelength is tuned to the photonic band gap edges and the microcavity mode, the fundamental wave is localized in the neighborhood of the cavity layer. The fundamental field enhancement is stronger when the fundamental wavelength is in resonance with the microcavity mode. As a result, the amplitude of inhomogeneous partial thirdharmonic waves increases. At certain dispersion n3 values, the amplitudes of partial homogeneous thirdharmonic waves become maximum because of multiple-reflection interference and their phases are close to each other. This results in a resonant increase in the third-harmonic intensity. The third-harmonic intensity peak in the microcavity mode is less sensitive to phase matching of partial waves because of strong fundamental field amplitude enhancement. THG enhancement in the microcavity mode is observed almost irrespective of

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6. CONCLUSIONS We developed a phenomenological model of optical THG in one-dimensional photonic crystals and microcavities. The model is based on the formalism of transfer matrices. It describes the generation and propagation of third-harmonic waves through a photonic crystal by taking into account multiple reflections of fundamental and third-harmonic wave at interfaces and interference between the homogeneous and inhomogeneous third-harmonic waves. The spectra of third-harmonic intensity in photonic crystals and microcavities were studied for the example of nonlinear optical media with the limiting point symmetry groups 2, m, and /mm of layers constituting photonic crystals. The spectra were studied in the spectral range of wavelengths and incidence angles of fundamental waves containing the photonic band gap and regions near its edges. At the resonance between the fundamental wave and the microcavity mode, the third-harmonic intensity increases by more than three orders of magnitude. When the fundamental wavelength coincides with photonic band gap edges, we observe resonance enhancement of third-harmonic intensity, which depends on the magnitude and sign of dispersion. If the refractive indexes of photonic crystal layers at the third-harmonic wavelength n3 are smaller than the refractive indexes at the fundamental wavelength, we observe resonant thirdharmonic enhancement by a factor exceeding 100 at the short-wavelength edge of the photonic band gap. If n3 > n , THG is enhanced at the long-wavelength edge of the photonic band gap. The main mechanism of resonant THG enhancement in the microcavity mode is fundamental field localization, which manifests itself by a four- to tenfold increase in the third-harmonic amplitude. An additional enhancement factor is phase matching of partial thirdharmonic waves from each photonic crystal layer when they leave the microcavity. THG enhancement at the edges of the photonic band gap is caused to a greater extent by an increase in the amplitude of emerging partial third-harmonic waves and their phase matching as a result of multiple-reflection interference of third-harmonic waves and, to a lesser extent, by fundamental wave localization within the photonic crystal. In agreement with the calculated third-harmonic spectra, the experimental spectrum of the intensity of the third harmonic generated in a one-dimensional microcavity fabricated from porous silicon shows that the intensity of the third harmonic increases approximately 1000 times in the microcavity mode. Additional third-harmonic peaks are observed at the left edge of the photonic band gap and outside the gap. The spectral positions of the resonances coincide with the minima of the linear fundamental wave reflection coefficient to within several nanometers.
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()

(b)

0.04

0 750 800 850 900 950 1000 1050 1100 1150 , nm
Fig. 13. (a) Spectrum of the reflection coefficient of the s-polarized pumping wave and (b) dependence of the intensity of the s-polarized third harmonic wave on the wavelength of s-polarized pumping. The sample is a microcavity made from porous silicon with 0 = 1300 nm.

band gap and the allowed mode of the microcavity were within it. Third-harmonic radiation reflected from the microcavity sample was separated by UV filters and detected by a photoelectron multiplier. The frequency spectra of third-harmonic intensity were measured for fundamental and third-harmonic waves polarized in the plane of the sample (SS geometry) at a 60° angle of fundamental wave incidence. The third-harmonic intensity spectrum is shown in Fig. 13b, and the spectrum of the linear reflection coefficient for the fundamental wave is shown in Fig. 13a. The I3() spectral dependence has a resonance at a fundamental wavelength of 1075 nm, which coincides with the microcavity mode wavelength shifted to shorter waves with respect to 0 at an oblique angle of incidence. We also observe resonant third-harmonic features when is tuned to the spectral regions of the photonic band gap edge and outside the gap (inset to Fig. 13). The third-harmonic intensity in these resonance peaks is three orders of magnitude lower than in the microcavity mode. The positions of third-harmonic intensity resonances correlate with the minima of the spectrum of the linear reflection coefficient for the fundamental wave.

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ACKNOWLEDGMENTS The authors are deeply indebted to O.A. Aktsipetrov for the statement of the problem and numerous useful discussions. REFERENCES
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13. V. V. Konotop and V. Kuzmiak, J. Opt. Soc. Am. B 17, 1874 (2000). 14. M. Centini, G. D'Aguanno, M. Scalora, et al., Phys. Rev. E 64, 46 606 (2001). 15. T. V. Dolgova, A. I. Maoedykovskioe, M. G. Martem'yanov, et al., Pis'ma Zh. èksp. Teor. Fiz. 73, 8 (2001) [JETP Lett. 73, 6 (2001)]. 16. T. V. Dolgova, A. I. Maidykovski, M. G. Martemyanov, et al., Appl. Phys. Lett. 81, 2725 (2002). 17. T. V. Dolgova, A. I. Maoedykovskioe, M. G. Martem'yanov, et al., Pis'ma Zh. èksp. Teor. Fiz. 75, 17 (2002) [JETP Lett. 75, 15 (2002)]. 18. G. D'Aguanno, M. Centini, M. Scalora, et al., J. Opt. Soc. Am. B 19, 2111 (2002). 19. A. V. Andreev, A. V. Balakin, A. B. Kozlov, et al., J. Opt. Soc. Am. B 19, 2083 (2002). 20. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964; Nauka, Moscow, 1970). 21. N. Hashizume, M. Ohashi, T. Kondo, and R. Ito, J. Opt. Soc. Am. B 12, 1894 (1995). 22. J. E. Sipe, J. Opt. Soc. Am. B 4, 481 (1987). 23. D. S. Bethune, J. Opt. Soc. Am. B 6, 910 (1989). 24. N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962). 25. Yu. I. Sirotin and M. P. Shaskolskaya, Fundamentals of Crystal Physics, 2nd ed. (Nauka, Moscow, 1979; Mir, Moscow, 1982).

Translated by V. Sipachev

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