## Abstract

The second-harmonic generation (SHG) in finite Bragg stacks of alternating linear and nonlinear optical films is studied with the exact Green function under the assumption of perturbation theory. Various mechanisms of enhanced SHG are investigated analytically, and the scaling law with respect to the number *N* of stacking layers is derived for each mechanism. In particular, it is shown that there is a simple mechanism of the enhanced SHG having *N*
^{4} scaling, in which both the enhancement of the Green function and the phase matching condition peculiar to finite Bragg stacks are fulfilled simultaneously.

©2005 Optical Society of America

## 1. Introduction

In artificial periodic structure composed of linear and nonlinear optical substances, which is known as nonlinear photonic crystal (PhC), various nonlinear optical processes can be enhanced [1, 2]. They include second-harmonic generation (SHG), parametric downconversion, Kerr effect, third-harmonic generation, four-wave mixing, and so on. By using nonlinear PhC, it is possible to attain very high enhancement in these processes, in such a way that homogeneous nonlinear substance can never achieve. From the point of view of the perturbation theory, the enhancement is caused by novel properties of linear optics in nonlinear PhC. By neglecting the nonlinear susceptibility of the nonlinear substance, nonlinear PhC can be regarded as linear PhC, so that linear optics is also modulated. Linear PhC can have a peculiar frequency region, known as the photonic band gap, in which propagation of light is completely forbidden. The gap is bounded by the band edges, or in other words, the mobility edges where the group velocity of light becomes extremely small. This yields an enhancement of the effective interaction between light and matter and thus enhances also nonlinear optical processes. In addition, owing to the formation of photonic bands in linear PhC, whose dispersion relation is compactified in the first Brillouin zone, the (quasi) phase-matching conditions for various nonlinear optical processes are easily fulfilled without using the anisotropy of the linear susceptibility in the nonlinear substance. Moreover, by introducing a structural defect in nonlinear PhC, very high local-field intensity around the defect can be obtained within linear optics. Therefore, nonlinear optical processes through the defect are strongly enhanced.

From now on, we focus on the simplest nonlinear optical process, namely SHG, in the simplest nonlinear photonic crystal, namely Bragg stack of alternating linear and nonlinear optical films. The SHG in Bragg stacks can reveal all of the above properties in a rather controllable manner owing to the simplicity of the system. Thus, there is a vast literature on this system ranging from theory to experiment [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. However, it seems that there is little solid understanding of the scaling law with respect to the number of stacking layers. In this paper we study analytically this scaling law and compare various mechanisms of the enhanced SHG regarding the scaling. We employ the perturbation theory by using the exact Green function of linear optics in finite Bragg stacks [14, 15, 16]. With this formalism combined with the diagonal form of the transfer matrix given below, we can extract the scaling law analytically. It should be noted that we do not rely on the coupled-mode theory with the slowly-varying field approximation [9], which provides an analytic treatment of the SHG, but rely on the perturbation theory. Though our approach is limited for small nonlinear susceptibility, it is applicable to fast-varying fields and to other nonlinear optical processes, e.g., third-harmonic generation.

This paper is organized as follows. In Sec. 2 as well as in Appendix we derive analytically various quantities relevant to the SHG in regular but finite Bragg stacks from the point of view of the perturbation theory with the exact Green function. Section 3 is devoted to investigate various mechanisms of the enhanced SHG with emphasis on the scaling law peculiar to each mechanism. The enhanced SHG in Bragg stacks with a structural defect is studied in Sec. 4. Some numerical results on the enhanced SHG in Bragg stacks with and without a structural defect are presented in Sec. 5, in order to confirm the analytical properties obtained in this paper. Finally, we summarize the results and discuss the other nonlinear optical processes than the SHG.

## 2. Formalism

In a nonlinear optical substance without the inversion symmetry the nonlinear polarization due to the second-order susceptibility *χ*
^{(2)} is generally quite small compared to the linear polarization. This enables us to study the SHG using the perturbation theory taking *χ*
^{(2)} as the expansion parameter. In the first order approximation, the electric field is given by the sum of the linear and nonlinear fields:

where *c.c.* stands for complex conjugate. The linear field *E*^{L}
is the solution of Maxwell’s equation with vanishing nonlinear polarization. The nonlinear field *E*^{NL}
is induced by the nonlinear polarization, which acts as an external dipole source of frequency 2*ω*:

Therefore, the nonlinear field can be written as

where *Gô* is the retarded Green function in finite Bragg stacks, its definition being

Here, *q* = *ω*/*c* and (î)_{ij} = *δ*_{ij}
is the Kronecker delta. The Green function can be obtained by solving the radiation problem of a point dipole placed in Bragg stacks. A basic formalism to calculate the SHG spectrum in finite multilayers was developed by Hashizume et al [16]. We adapt the formalism to deal with Bragg stacks with and without a structural defect.

The system under consideration is the Bragg stack which consists of alternating linear and nonlinear optical films as shown in Fig. 1. The linear film has (linear) susceptibility *ε*_{A}
and thickness *d*_{A}
, whereas the nonlinear film has linear susceptibility *ε*_{B}
, second-order susceptibility χ^{(2)}, and thickness *d*_{B}
. There are *N* layers in total and *N* is taken to be odd. The edge layers were assumed to be the linear substance. The total thickness of the Bragg stack is equal to *D* = *a*(*N* - 1)/2 + *d*_{A}
, where *a* = *d*_{A}
+ *d*_{B}
is the lattice constant. In Sec. 4 we also deal with the defective Bragg stack in which a defect layer with thickness *d*_{D}
(≠ *d*_{B}
) of the same nonlinear substance is introduced in the middle of the structure. The fundamental harmonic (FH) wave is incident from the left of the Bragg stack, and the transmitted and reflected radiations with higher harmonics are generated. We restrict ourselves to those of the FH and second-harmonic (SH) for simplicity.

Here we present the perturbation theory of the SHG in the finite Bragg stack by employing the transfer matrix formalism. To derive analytically the scaling law with respect to *N*, we use the diagonal form of the transfer matrix for one period. This procedure elucidates the role of band edge, Fabry-Perot resonance, and phase matching in the SHG.

#### 2.1. Transmission and reflection of fundamental harmonic wave

In the perturbation theory of the SHG we must first solve the linear problem for the FH wave of frequency *ω*. Suppose that the incident wave is coming from left with wave vector (**k**
_{∥},*k*),*k* being $\sqrt{{q}^{2}-{k}_{\parallel}}$. The induced electric field in the *i*-th layer is written as

where *z*_{i}
is the center coordinate of the *i*-th layer and the factor *e*
^{ik∥·x∥} was omitted in Eq. (7). Outside the Bragg stack, the field consists of the incident wave and reflected wave for *z* < 0 and solely of the transmitted wave for *z* > *D*:

The transmission and reflection coefficients (*t*_{N}
and *r*_{N}
, respectively) are given by

where *T*_{l}
and *T*_{u}
are the transfer matrices of the front and rear surfaces, respectively and *T*(= *T*_{BA}*T*_{AB}
) is the transfer matrix of one period, which relates

where *i* is an odd integer. By using these quantities *a*
^{(i)} and *b*
^{(i)} (for odd *i*) are given by

Then *a*
^{(i+1)} and *b*
^{(i+1)} becomes

In order to see the dependence on *i* and *N* in the following arguments, it is convenient to diagonalize the transfer matrix *T* as

So far, we have treated various quantities as scalar ones. This is because the polarization of the incident wave, namely either S (the electric field is parallel to the interface) or P (the magnetic field is parallel to the interface), is preserved. That is, the S and P components of the vectorial electric field are scattered independently. Thus, the above scalar quantities are of either S or P components. To recover the vectorial form of the electric field, it is enough just to multiply the corresponding unit vector **s** or ${\mathbf{p}}_{i}^{+}$ to *a*
^{(i)} and **s** or ${\mathbf{p}}_{i}^{-}$ to *b*
^{(i)}, where **s** and ${\mathbf{p}}_{i}^{\pm}$ are given by

This prescription must be accompanied in constructing the nonlinear polarization Eq. (4).

#### 2.2. Green function of second-harmonic wave

The linear field given above induces the SH wave through the nonlinear polarization which oscillates with 2*ω*. The SH wave is given by Eqs. (4) and (5), so that the Green function of 2*ω* is necessary to derive the SHG spectrum. To obtain the Green function, we need to solve the dipole radiation problem.

An oscillating point dipole with polarization vector **p**
_{0} and frequency *ω* placed at **x** = **x**
_{d} in the *j*-th layer generates the intrinsic electric field given by

where the subscript of *ε* stands for the sign of *z* - *z*_{d}
. Again, the polarization as well as the parallel momentum **k**
_{∥} is preserved in the Bragg stack, so that the S- and P-polarization components with different **k**
_{∥} are scattered independently. Thus, we follow the scalar notation as in the previous subsection. By taking the intrinsic field as the incident wave, the forward- and backward-transmitted waves outside the Bragg stack, as well as the induced field inside the *j*-th layer are generated. They become

at given **k**
_{∥} and polarization. Using the transfer matrix formalism we can find

where

Thus, the analytical expression of the Green function in the finite Bragg stack is given by

$$\phantom{\rule{.2em}{0ex}}+{\left({t}_{++}^{\left(j\right)}\right)}_{p}{\mathbf{p}}^{+}\otimes {\mathbf{p}}_{j}^{+}{e}^{-{\mathit{ik}}_{j}\left(z\prime -{z}_{j}\right)}+{\left({t}_{+-}^{\left(j\right)}\right)}_{p}{\mathbf{p}}^{+}\otimes {\mathbf{p}}_{j}^{-}{e}^{-{\mathit{ik}}_{j}\left(z\prime -{z}_{j}\right)}),$$

for *z*(> *D*) being outside the Bragg stack and *z*′ being inside the *j*-the layer. Here, **p**
^{±} is the unit vector of the P-polarization in vacuum. A similar expression can be easily obtained for other combinations of *z* and *z*′. The absence of the mixing term such as **s** ⊗ **p**
^{+} in the Green function indicates the absence of the polarization mixing in the Bragg stack.

#### 2.3. Second-harmonic generation

The results given in the last two subsections are used to derive the SHG intensity induced by the FH wave. According to Eqs. (4) and (5), the SH wave outside the Bragg stack is shown to be

where

Again, the factor *e*
^{2ik∥·x∥} was omitted in Eq. (40). The factor *Z*
_{1} is relevant to the phase-matching condition in a homogeneous nonlinear slab with thickness *d*_{B}
. In fact, *Z*
_{1} reaches the maximum value *d*_{B}
when the phase-matching condition *k*_{B}
(2*ω*) = 2*k*_{B}
(*ω*) of the homogeneous slab is fulfilled. The factors ${T}_{i}^{\alpha \beta}$
are peculiar to the Bragg stack. Roles of ${T}_{i}^{\alpha \beta}$
in the enhanced SHG are given in the next section. By using the diagonal form of the transfer matrix, we can further expand ${T}_{i}^{\alpha \beta}$
exhaustively. The results are given in Appendix. In the final expression of ${T}_{i}^{\alpha \beta}$
the summation over the nonlinear layers (∑_{j:even}) is reduced to the following quantities:

As is described later, these factors are responsible for the quasi-phase-matching (QPM) condition peculiar to the Bragg stack.

We should remind that the quantities ${T}_{i}^{\alpha \beta}$
,*I*_{ab}
, and *J*_{a}
depends on the polarizations of the FH and SH waves. As for ${T}_{i}^{\alpha \beta}$
, the polarization dependence is given in Appendix and Table I. Concerning *I*_{ab}
and *J*_{a}
, for instance, when these quantities appear in the amplitude of the S-polarized FH wave to the P-polarized SH wave, *λ*_{ω}
and *λ*
_{2ω} must be taken as the eigenvalues of the transfer matrix of the S-polarized FH wave and of the P-polarized SH wave, respectively.

## 3. Enhanced SHG in regular Bragg stacks

The SHG in Bragg stacks is supposed to be enhanced when either one of the following conditions is satisfied:

- the quasi-phase-matching condition,
- band-edge condition:
*ω*,2*ω*≃ {*ω*_{c}}, - resonant tunneling condition:
*ω*, 2*ω*= {*ω*_{d}}.

The former two conditions are of regular Bragg stacks which have a perfect periodicity in the dielectric function except for the sample edge. The latter condition is of Bragg stacks with a structural defect. Here, {*ω*_{c}
} and {*ω*_{d}
} are band edge frequencies and eigenfrequencies of localized defect mode, respectively. These conditions are known well and the enhanced SHG through them has been verified both in experiment and in numerical simulation. However, there is little solid understanding of the scaling law with respect to number of the layers in Bragg stacks. The purpose of this section is to derive analytically the scaling law with respect to *N*.

In order to extract the scaling law peculiar to the above mechanisms, we assume lossless Bragg stacks with the perfect translational invariance except at the sample edge. In actual Bragg stacks dielectric loss and disorder are inevitable, resulting in blurred scalings with respect to *N*. However, we believe that our approach is still valuable because the solid understanding of such scaling laws is lacking and the blurred scalings still have the features of the ideal scalings without absorption loss.

In a lossless Bragg stack, the eigenvalues of the transfer matrix for one period, *λ*
_{1} and *λ*
_{2}, are related to the Bloch wave number *k*_{z}
(*ω*) of the corresponding infinite Bragg stack. That is,

provided that the frequency lies outside the band gaps. Inside the band gaps *k*_{z}
(*ω*) becomes pure imaginary, so that

In any case det*T* = *λ*
_{1}
*λ*
_{2} = 1 for the lossless Bragg stack irrespective of *ω*.

#### 3.1. Quasi-phase-matching

The phase-matching condition plays a crucial role in the enhanced SHG in optically homogeneous substances, where the SHG intensity scales as *L*
^{2}, *L* being the thickness of the substance. This scaling is essentially unchanged even in the Bragg stack. The quantities relevant to the QPM in the finite Bragg stack are *I*_{ab}
and *J*_{a}
. In the lossless Bragg stack the *I*_{ab}
becomes

provided that both *ω* and 2*ω* are outside the band gaps. Equation (52) reaches the maximum value (*N* - 1)/2 only if

Similarly, Eq. (53) reaches the same maximum value only if

The above conditions are nothing but the QPM conditions of the bulk Bragg stack, and yields a *N*
^{2} scaling of the SHG intensity at large *N*. This is evident from Eqs. (40), (41) and the results given in Appendix. For instance, when Eq. (54) is satisfied, all *F*_{i}
’s given in Appendix and thus all ${T}_{i}^{\alpha \beta}$
become *O*(*N*). As a result, the coefficient of the SH wave *T*
_{±} becomes *O*(*N*), yielding the *N*
^{2} scaling of the SHG intensity.

On the other hand, *J*_{a}
becomes

provided that 2*ω* is outside the band gaps. Equation (56) reaches the maximum value (*N*- 1)/2 only if

This is the QPM condition peculiar to the finite Bragg stack, because it stems from the interference between the forward- and backward-propagating components in the FH wave and the interference is absent if we neglect the boundary of the sample. This QPM condition is satisfied when the frequency of the SH wave is equal to an eigenfrequency of the bulk Bragg stack at the Γ point (i.e. *k*_{z}
= 0) in the first Brillouin zone. In this condition, the Blochwave number *k*_{z}
(*ω*) of the FH wave is completely absent, so that it is rather simple compared with the standard QPM conditions (Eqs. (54) and (55)) involving both *k*_{z}
(*ω*) and *k*_{z}
(2*ω*). Obviously, to attain the enhanced SHG via this condition, the frequency of the FH wave must be outside the band gaps. Otherwise, the nonlinear polarization as the source of the SHG becomes extremely small. We should remark that in a finite homogeneous nonlinear slab, a similar phase-matching condition certainly exists, but is completely negligible. In such a system, the enhanced SHG stems from the factors *Z*_{i}
(*i* = 1,2,3). Among them the factor *Z*
_{2} reaches the maximal value only if *k*_{B}
(2*ω*) = 0. This equation is satisfied only at *ω* = 0, so that the relevant radiation field does not oscillate in time.

#### 3.2. Band edge enhancement of fundamental harmonic wave

The complex transmission coefficient *t*_{N}
is related to the optical density of states *ρ*_{N}
in the finite Bragg stack as

For sufficiently large *N* the Fabry-Perot (FP) resonance of the Bragg stack, which is defined by |*t*_{N}
| = 1, corresponds to a peak of *ρ*_{N}
.

At the FP resonance near the band edge, |*t*_{N}
| = 1 and |*r*_{N}
| = 0. Thus, its frequency *ω*
_{FP} is determined through

On the other hand, near the band edge of the Γ point, *k*_{z}
satisfies

Here, *ω*_{c}
is the band edge frequency. Since *ω*
_{FP} is near the band edge, it satisfies

Thus, *ω*
_{FP} approaches to *ω*_{c}
with a 1/*N*
^{2} scaling. A similar scaling is obtained for *ω*
_{FP} near the band edge at *k*_{z}
= ±*π*/*a*.

The factor 1/det*U* in *U*
^{-1} plays a crucial role in the field enhancement at the FP resonance. Actually, we can prove det*U* scales as |*ω* - *ω*_{c}
|^{γ} near the band edge. The determinant is given by

where cos(*k*_{z}*a*)=Re*T*
_{--}. At the band edge, *k*_{z}*a*/2*π* = 0 or 0.5, so that Re*T*
_{--} = ±1. The Taylor expansion of Re*T*
_{--} around *ω* = *ω*_{c}
is thereby

Since the other factors in Eq. (62) have modest dependence on *ω* at *ω*_{c}
, we can prove that the exponent *γ* is equal to 1/2. By putting *ω* = *ω*
_{FP}, we can obtain (see Eqs. (15) and (16))

Therefore, the induced field inside the Bragg stack is *O*(*N*). That is why the field enhancement at the FP resonance near the band edge is proportional to *N*.

This enhancement is preserved in the SHG intensity as follows. As shown in appendix, the factors *f*_{i}
given in Appendix include linear terms in *U*
^{-1}, and thus is *O*(*N*). Since *F*_{i}
is a quadratic form of *f*_{i}
, it is *O*(*N*
^{2}). Besides, the SHG coefficient *T*
_{±} is linear in ${T}_{i}^{\alpha \beta}$
, which is also linear in *F*_{i}
. Thus, *T*
_{±} is *O*(*N*
^{2}) and the SHG intensity |*T*
_{+}|^{2} + |*T*
_{-}|^{2} is *O*(*N*
^{4}) at the FP resonance nearest to the band edge of the FH wave.

#### 3.3. Band edge enhancement of the Green function

The Green function of the finite Bragg stack is also enhanced at the FP resonance near the band edge. At this resonant frequency the common denominator of Eqs. (30)–(37) can be written as

Since the extra *N*-dependence arises through det*U* ≃ 1/*N*, the combination det*UU*
^{-1} vanishes the dependence. As a result, the denominator is *O*(*N*
^{0}). On the other hand, no cancellation
${t}_{\pm \pm}^{\left(j\right)}$ is *O*(*N*) and ${c}_{\pm \pm}^{\left(j\right)}$ is *O*(*N*
^{2}). This scaling is consistent with the sum rule of the dipole radiation

$$\phantom{\rule{.2em}{0ex}}={q}^{3}\mathrm{Im}\left[{\mathbf{p}}_{0}^{*}\hat{G}({z}_{d},{z}_{d}){\mathbf{p}}_{0}\right],$$

in which both the left and right hand sides are *O*(*N*
^{2}). The above enhancement is preserved in the SHG intensity as follows. Since *G*_{i}
given in Appendix is linear in *U*
^{-1}, it is *O*(*N*). Besides, the SHG coefficient *T*
_{±} is linear in ${T}_{i}^{\alpha \beta}$
and thus linear in *G*_{i}
. Therefore, the SHG intensity scales as *O*(*N*
^{2}) if 2*ω* coincides the frequency of the FP resonance nearest to the band edge.

#### 3.4. Combination among the three mechanisms

The scaling laws obtained for the three mechanisms of the enhanced SHG can be combined. For instance, when *ω* and 2*ω* are at the FP resonant frequencies nearest to the band edges, the SHG intensity scales in principle with *N*
^{6}. This can be realized by an oblique incidence of the FH wave with a suitable incident angle. Otherwise, appropriate band engineering makes it possible to satisfy this condition even in the case of normal incidence. Similarly, the QPM and the band edge enhancement of *ω* and/or 2*ω* can be simultaneously satisfied. In principle, we will obtained *O*(*N*
^{4}) (QPM + band edge enhancement of 2*ω*), *O*(*N*
^{6}) (QPM + band edge enhancement of *ω*) [12], and *O*(*N*
^{8}) (QPM + band edge enhancements of *ω* and 2*ω*) for the scaling law of the SHG intensity. A numerical evidence of the *N*
^{8} scaling was reported by Kiehne *et al*. [6], though its analytical proof was absent.

We should point out that the above combinations require a fine-tuning of the frequency and the incident angle as well as an appropriate design of the Bragg stack. On the other hand, the combination of the QPM peculiar to the finite Bragg stack and the band edge enhancement of the Green function is easily realized. This can be done just by tuning 2*ω* to the frequency of the FP resonance nearest to the band edge at the Γ point. This results in a *O*(*N*
^{4}) scaling, which is the same as the scaling due to the band edge enhancement of the FH wave. However, as was discussed later, the latter scaling is fragile owing to a destructive interference among the terms that contribute to the SHG coefficients. Therefore, the simplest and naive way to realize a *O*(*N*
^{4}) scaling will be to utilize the QPM condition peculiar to the finite Bragg stack.

## 4. System with a structural defect

So far, we have concentrated on the regular Bragg stack with a perfect translational invariance except for the sample edge. In this system the enhanced SHG is caused by the QPM and the field enhancement at the band edge. As a result, the enhanced SHG obeys a power law with respect to *N* and its analytical proof is rather complicated. On the other hand, in a Bragg stack with a structural defect, the enhanced SHG is dominated by the local field enhancement involving the defect modes. That is, a strong enhancement of the SHG can be achieved when either the frequency of the FH wave or that of the SH wave coincides with the eigenfrequency of the localized defect mode. In this case the enhanced SHG is shown to scale exponentially with respect to *N*, so that the additional power of *N* to the exponential scaling is fairly negligible. As we will see, the exponential scaling can be easily obtained with the diagonal form of the transfer matrix. On the trade-off, the huge enhancement due to the exponential scaling is accompanied by an exponential narrowing of the SHG spectrum around the resonant frequency. Therefore, a fine tuning of frequency is necessary in order to attain the huge enhancement. In this section we again assume that the defective Bragg stack concerned is lossless for simplicity.

#### 4.1. Local field enhancement of fundamental wave

Let us first consider the local field enhancement of the FH wave in the Bragg stack with a structural defect. Using the transfer matrix formalism the field coefficients in the defect layer are given by

where *r*_{N}
satisfies the same equation as Eq. (12) except for

Suppose that the frequency of the FH wave is inside a band gap of the corresponding bulk Bragg stack. At frequency *ω*_{d}
of the resonance tunneling due to the localized defect mode, the reflectance |*r*_{N}
|^{2} is equal to 0. Therefore,

where the convention Im*k*_{z}
(*ω*_{d}
) > 0 was taken. Therefore, in the defect layer, the local field of the FH wave grows exponentially with increasing *N*. If we further assume that the frequency of the SH wave is outside the band gaps of the bulk Bragg stack, this local-field enhancement yields the enhanced SHG which scales with exp(Im*k*_{z}
(*ω*_{d}
)*aN*) solely by the huge nonlinear polarization in the defect layer. Here, it should be noted that the effects of the nonlinear polarization other than in the defect layer is merely a minor correction to the above scaling. That is, the possible correction to the exponential scaling is only a power of *N*.

#### 4.2. Local field enhancement of the Green function

Similarly, the Green function for the second-harmonic wave is also enhanced when 2*ω* coincides the eigenfrequency of the localized defect mode. Let us consider the case that *z*′ in *Gô*(*z*,*z*′;2*ω*) is in the defect layer. Using the transfer matrix formalism, we can obtain

Under the condition of the resonant tunneling, namely, (*T*_{N}
)_{21} = 0, the common denominator in the above equations can be written as

The right hand side of the above equation has solely an implicit dependence on *N* through the resonant tunneling frequency. Therefore, it has a moderate dependence on *N*. Taking into account that detT = 1, ${t}_{\pm \pm}^{\left(D\right)}$ behaves as exp(Im*k*_{z}
(2*ω*)*aN*/4). If we further assume that the frequency of the FH wave is outside the band gaps of the bulk Bragg stack, the nonlinear polarization in the defect layer has a moderate dependence on *N*. Thus, the SHG intensity scales as exp(Im*k*_{z}
(2*ω*)*aN*/2) for large *N*.

According to the sum rule Eq. (66) along with Eqs. (26) and (27), the exponential enhancement of ${t}_{\pm \pm}^{\left(D\right)}$ with respect to *N* implies the similar enhancement of the Green function whose two spatial arguments, *z* and *z*′ are equal and in the defect layer. It also implies the enhancement of the local density of states (LDOS) of photon in the defect layer, because the LDOS is defined by

which is proportional to the right hand side of Eq. (66) after averaging over polarization angle of **p**
_{0}.

Obviously, the summation over the nonlinear layers in the SHG is not limited in the defect layer. However, effects of the other nonlinear layers again yield merely a minor correction (power of *N*).

The above two mechanisms of the enhanced SHG in the Bragg stack with a structural defect can be combined. This can be achieved by imposing both *ω* and 2*ω* coincide with the eigenfrequencies of the localized defect modes [13]. In that case the SHG intensity scales with exp(Im*k*_{z}
(*ω*)*aN* + Im*k*_{z}
(2*ω*)*aN*/2).

## 5. Numerical results

To confirm the analytical results obtained in the previous sections, numerical calculation was performed for the Bragg stacks containing a typical nonlinear substance, namely, LiNbO_{3}. The c-axis of LiNbO_{3} was taken to be parallel to the z-direction. To model LiNbO_{3}, we used a Sellmeier-fitting function of the refractive index for the ordinary wave:

where *λ* is the wavelength in units of [*μ*m]. Here, we restrict ourselves to the case of normal incidence, so that the anisotropy of LiNbO_{3} in the dielectric function can be neglected. Since the point group of LiNbO_{3} is *C*
_{3v}, the nonzero components of the second-order susceptibility are [17]

As a linear substance, we assume a dispersion-free dielectric constant *ε*_{A}
= 2.1 which corresponds to that of SiO_{2} in visible frequency region. The lattice constant was taken to *a* = 1 [*μ*m] and *d*_{A}
= *d*_{B}
= 0.5*a*.

As was explained in this paper, much information on the SHG in a finite Bragg stack can be attained by considering the photonic band structure of the corresponding infinite Bragg stack. Figure 2 shows the photonic band structure for normal incidence (*k*
_{∥} = 0). In the figure both the real and imaginary parts of *k*_{z}
are plotted. The imaginary part of *k*_{z}
becomes non-zero only in the photonic band gaps. The standard QPM condition (Eqs. (54) and (55)) is satisfied at *ω* = 0.4185,0.6785, and 0.9447 in units of 2*πc*/*a*, in the frequency range concerned. At the former two frequencies *k*_{z}
(2*ω*) = 2*k*_{z}*(*(*ω*), whereas at the latter frequency *k*_{z}
(2*ω*) + 2*k*_{z}
(*ω*) = 2*π*/*a*. In all the cases *k*_{z}
(2*ω*) is close to 0.5 in units of 2*π*/*a*. Therefore, the enhanced SHG due to the QPM is more or less affected by band edge effects.

#### 5.1. Regular Bragg stack

We first consider the regular Bragg stacks, changing the number of layers. Figure 3 shows the transmittance as well as the SHG intensity at *N* = 21 as a function of frequency of the FH wave. In the frequency region 0.58 < *ω* < 0.78 of Fig. 3, the SHG intensity oscillates with the transmittance of the FH wave. That is, the peak frequencies of the SHG intensity in this region coincide with those of the FP resonance of the FH wave. This feature remains unchanged even for large *N*, but is absent in the calculation which assumes infinite Bragg stack. The SHG intensity drops considerably if either the frequency of the FH wave or that of the SH wave is inside the band gaps. The former drop is obvious in Fig. 3. The latter drop can be found, for instance, around *ω* = 0.4 where 2*ω* lies in the third band gap. The SHG spectrum has two marked peaks at *ω* = 0.5196 and 0.9487 in the frequency range concerned. They do not correspond to the FP resonances of the FH wave. By increasing *N* up to 101, these peaks and another marked peaks, which are hidden at *N* = 21, grow substantially.

Figure 4 shows the SHG intensity at *N* = 101 as a function of the FH frequency. The spectrum reveals five marked peaks indicated by arrows in Fig. 4. The dominant peak is found at *ω* = 0.5240, and its precursor appeared already in Fig. 3 at *ω* = 0.5196. Concerning this peak, the ratio of the SHG intensities between *N* = 21 and *N* = 101 is about 20000. We should point out that twice the frequency *ω* = 0.5240 is very close to the band edge frequency *ω* = 1.0484 at *k*_{z}
= 0. Thus, this peak is caused by the simultaneous satisfaction of the band edge enhancement of the Green function and the QPM peculiar to finite systems. The second dominant peak is at *ω* = 0.9446, where the standard QPM condition is satisfied. The peaks at *ω* = 0.4182 and 0.6791 are also attributed to the QPM, because their frequencies are very close to the phase-matched frequencies obtained from the photonic band structure. The band edge enhancement of the FH wave can be found at *ω* = 0.7956, though the peak value is very small compared to the dominant peak. This is because 2*ω* is inside the band gaps of the corresponding photonic band structure.

To convince the physical origins of the two dominant peaks in the SHG spectrum, we evaluated the *N*-dependence of the peaks. The results are shown in Fig. 5. In Fig. 5(a) the peak frequency in the SHG spectrum is plotted as a function of *N* for the dominant SHG peak. The peak frequency coincides with the FP resonance of 2*ω* nearest to the band edge frequency *ω* = 1.0484, whose half value is indicated by solid line in the figure. In accordance with Eq. (61) the peak frequency approaches to this value with the 1/*N*
^{2} scaling. On the other hand, in Fig. 5(b) the *N*-dependence of the peak SHG intensity is shown. As is clearly seen, the peak intensity increases with *N*
^{4}, though it oscillates with *N*. This oscillation is caused by the FP oscillation of the FH wave from which the nonlinear polarization arises. A local maximum in the peak intensity as a function of *N* corresponds to a FP resonance *ω* = *ω*
_{FP}. Besides, Fig. 5(c) shows the *N*-dependence of the peak frequencies relevant to the QPM condition around *ω* = 0.9447. At a given *N*, the SHG spectrum reveals a few peaks around *ω* = 0.9447 as a function of frequency. Each peak corresponds to the FP resonance of the FH wave and reveals a red shift as we increase *N*. Near the intersection point between the curve of each peak and the line of *ω* = 0.9447, the peak intensity gets maximum along the curve. We found that this maximum is caused by the FP resonance of 2*ω*. In Fig. 5(d) the peak SHG intensity was plotted for each peak as a function of *N*. We can find that the *N*-dependence of the peak intensity is consistent with the *N*
^{2} scaling of the QPM. In all the cases of Fig. 5 both the frequencies of the FH and SH waves must be on the FP resonance to get a clear evidence of the scaling laws derived in the previous section.

Finally, it should be noted that in the frequency range concerned, namely, 0.2 ≤ *ω* ≤ 1.0, we could not find a clear evidence of the *N*
^{4} scaling of the SHG intensity due to the band edge enhancement of the FH wave, though the intensity of the FH wave is certainly enhanced with *N*
^{2}. Also we could not find a clear evidence of the band edge enhancement of the Green function, other than the dominant peak that stems from a combination of the band edge enhancement of the Green function and the QPM peculiar to the finite Bragg stacks. They may be caused by a destructive interference among the terms which contribute to the final expression of the SHG coefficient *T*
_{±}. However, outside the frequency range of interest, we found a peak at about *ω* = 1.0480 which reveals the band edge enhancement of the FH wave combined, to some extent, with that of the Green function. The detailed analysis of the peak is beyond the scope of the present paper.

#### 5.2. Bragg stack with a structural defect

Next, we show some numerical results on the SHG in the Bragg stacks with a structural defect. The same parameters as in the previous section were assumed, except that the Bragg stacks have a defect layer of NiLbO_{3} with thickness *d*_{D}
= 1.2*a* in the middle of the structure. Within linear optics, the defect induces the localized defect modes in the band gaps. Figure 6 shows the SHG intensity of the defective Bragg stack having *N* = 23, along with the transmission spectrum of the FH wave. The resonant transmissions of the FH wave in the band gaps are found at *ω* =0.2538,0.5406,0.8096,etc. They are caused by the localized defect modes and the resonant frequencies are denoted by *ω*_{Di}
(*i* = 1,2,3,… count from below). On the other hand, the SHG spectrum reveals several marked peaks in the frequency range concerned. They are attributed to the field enhancement due to localized defect mode of either *ω* or 2*ω*. For instance, the peak of *ω* = 0.2538 is caused by the enhancement of the FH wave at *ω* = *ω*
_{D1}, whereas the peak at *ω* = 0.5465 is caused by the enhancement of the Green function at 2*ω* = *ω*
_{D4}. As in the case of the regular Bragg stacks, the FP oscillation results in the oscillation of the SHG spectrum.

Next, we focus on the scaling of the peak intensity with respect to *N*. Figure 7 shows the N-dependence of the following two intensity peaks of the SHG in the defective Bragg stack: *ω* = *ω*
_{D1} and 2*ω* = *ω*
_{D3}. Both the intensities grow exponentially with *N* with a slight oscillation. Again, this oscillation reflects the FP one. In both cases the *N*-dependences are consistent with the scaling law derived in the previous section. We also confirmed that other marked peaks in the SHG spectrum of Fig. 6 have the *N*-dependence given by the scaling law. We should note one of the two dominant peak found at *ω* = 0.5465, which satisfies 2*ω* = *ω*
_{D4} is exceptional, because *ω* is inside the second band gap. Therefore, the peak does not behave in accordance with the scaling law, but saturates to a constant value.

To convince ourselves about the exponential scaling due to the local field enhancement, it is valuable to plot the field profile of the FH wave |*E*(*z*;*ω*)|^{2} for the case *ω* = *ω*
_{D1} and the LDOS of photon *ρ*(*z*;2*ω*) for the case 2*ω* = *ω*
_{D3}. Figure 8 shows these profiles at *N* = 39, which reveal remarkable enhancement around the defect layer. Taking account that the intensity of the incident FH wave is normalized to be 1 (see Eq. (10)), in Fig. 8(a) the intensity of the FH wave around the defect layer is more than 100 times larger than that of the incident wave. Similarly, in Fig. 8(b) the LDOS around the defect layer can be more than 50 times larger than the LDOS in vacuum denoted by *ρ*
_{0}(*z*;2*ω*). At *N* = 99, the intensity of the FH wave and the normalized LDOS reach 1.5 × 10^{5} and 0.5 × 10^{5}, respectively. It is also remarkable that these quantities decay exponentially in *z* away from the defect layer. We should note that the upper bound of the intensity of the FH wave is 1 for *z* > *D* and 4 for *z* < 0. As for the normalized LDOS, it must approach 1 far away from the Bragg stack. The decaying profiles are consistent with these conditions and can be derived analytically.

## 6. Summary and discussion

We have presented the perturbation theory of the SHG in finite Bragg stacks with and without a structural defect. With the help of the diagonal form of the transfer matrix for one period, the analytical expression of the SHG intensity is obtained. Using this expression, various mechanisms of the enhanced SHG are compared from the point of view of the scaling law with respect to *N*, the number of the layers. In particular, we found that in a regular Bragg stack a *N*
^{4} scaling can be easily obtained just by putting the frequency of the SH wave at the nearest Fabry-Perot resonance to the eigenfrequency of the Γ point. This enhancement stems from a simultaneous satisfaction of the band-edge enhancement and the QPM peculiar to finite Bragg stacks. On the other hand, the band-edge enhancement of the FH wave alone is rather fragile because of interference effects in the SHG.

Finally, we should mention that the method employed in this paper can be easily generalized to the other nonlinear optical processes. For instance, the third-harmonic generation (THG) is described as

The phase-matching conditions of the THG for finite Bragg stacks are given by

The latter condition is peculiar to finite Bragg stacks. The intensity of the phase-matched THG is proportional to *N*
^{2}.

When *ω* coincides to the FP resonance at a band edge, *E*_{ω}
(*z*) scales as *N*, so that *P*
_{3ω}(*z*) and |*E*
_{3ω}(*z*)|^{2} scale as *N*
^{3} and *N*
^{6}, respectively. On the other hand when 3*ω* coincides to the FP resonance at a band edge, |*E*
_{3ω}(*z*)|^{2} scales as *N*
^{2} Thus, for an optimized structure it is possible to obtain a *N*
^{10} scaling for the THG intensity.

## Appendix

The final expression of *T*
_{±} by using the diagonal form of the transfer matrix is given here. The ingredients of *T*
_{±} given in Sec. 2.2 become

Here, *$\tilde{\chi}$*
^{(2)} is not the tensor second-order susceptibility, but the polarization-dependent scalar second-order susceptibility given in Table I. The factor *C*,*C*′ and *G*_{i}
stem from the Green function, and thus are the quantities of 2*ω*. On the other hand, the factor *F*_{i}
depends on both *ω* and 2*ω*:

The dependence on 2*ω* in *F*_{i}
’s is accumulated in the factor *I*_{ab}
and *J*_{a}
, and the factors *f*_{i}
depend solely on *ω*.

P(FH)-P(SH) | |
---|---|

${\tilde{\chi}}_{1\pm +}^{\left(2\right)}$ | ${\chi}_{\mathit{ijk}}^{\left(2\right)}{s}_{i}\frac{1}{2}{\left(\right({\mathbf{p}}_{2\omega}^{+})}_{j}{\left({\mathbf{p}}_{\omega}^{-}\right)}_{k}+{\left({\mathbf{p}}_{\omega}^{-}\right)}_{j}{\left({\mathbf{p}}_{\omega}^{+}\right)}_{k})$ |

${\tilde{\chi}}_{2\pm +}^{\left(2\right)}$ | ${\chi}_{\mathit{ijk}}^{\left(2\right)}{\left({\mathbf{p}}_{2\omega}^{+}\right)}_{k}i\frac{1}{2}+\left({\mathbf{p}}_{\omega}^{-}\right)j{({\left({\mathbf{p}}_{\omega}^{+}\right)}_{j}\left({\mathbf{p}}_{\omega}^{-}\right)}_{k}+\left({\mathbf{p}}_{\omega}^{-}\right)j{\left({\mathbf{p}}_{\omega}^{+}\right)}_{k})$ |

${\tilde{\chi}}_{3\pm +}^{\left(2\right)}$ | ${\chi}_{\mathit{ijk}}^{\left(2\right)}{s}_{i}\frac{1}{2}{\left(\right({\mathbf{p}}_{2\omega}^{+})}_{j}{\left({\mathbf{p}}_{\omega}^{-}\right)}_{k}-{\left({\mathbf{p}}_{\omega}^{-}\right)}_{j}{\left({\mathbf{p}}_{\omega}^{-}\right)}_{k})$ |

${\tilde{\chi}}_{1\pm -}^{\left(2\right)}$ | ${\chi}_{\mathit{ijk}}^{\left(2\right)}{s}_{i}\frac{1}{2}{\left(\right({\mathbf{p}}_{2\omega}^{-})}_{j}{\left({\mathbf{p}}_{\omega}^{-}\right)}_{k}+{\left({\mathbf{p}}_{\omega}^{-}\right)}_{j}{\left({\mathbf{p}}_{\omega}^{+}\right)}_{k})$ |

${\tilde{\chi}}_{2\pm -}^{\left(2\right)}$ | ${\chi}_{\mathit{ijk}}^{\left(2\right)}{\left({\mathbf{p}}_{2\omega}^{-}\right)}_{k}i\frac{1}{2}+\left({\mathbf{p}}_{\omega}^{-}\right)j{({\left({\mathbf{p}}_{\omega}^{+}\right)}_{j}\left({\mathbf{p}}_{\omega}^{-}\right)}_{k}+\left({\mathbf{p}}_{\omega}^{-}\right)j{\left({\mathbf{p}}_{\omega}^{+}\right)}_{k})$ |

${\tilde{\chi}}_{3\pm -}^{\left(2\right)}$ | ${\chi}_{\mathit{ijk}}^{\left(2\right)}{s}_{i}\frac{1}{2}{\left(\right({\mathbf{p}}_{2\omega}^{-})}_{j}{\left({\mathbf{p}}_{\omega}^{-}\right)}_{k}-{\left({\mathbf{p}}_{\omega}^{-}\right)}_{j}{\left({\mathbf{p}}_{\omega}^{-}\right)}_{k})$ |

## Acknowledgments

The authors would like to thank Dr. J. Inoue of NIMS for valuable discussions. They would also like to thank Prof. C. Sibilia for useful comments.

## References and links

**
1
. **
M.
Bertolotti
,
C. M.
Bowden
, and
C.
Sibilia
, eds.,
*
Nanoscale Linear and Nonlinear Optics
*
,
vol. 560
of AIP conference proceedings (
American Institute of Physics, New York
,
2001
).

**
2
. **
K.
Sakoda
,
*
Optical Properties of Photonic Crystals
*
(
Springer, Berlin
,
2001
).

**
3
. **
J.
Trull
,
R.
Vilaseca
,
J.
Martorell
, and
R.
Corbalan
, “
Second-harmonic generation in local modes of a truncated periodic structure
,”
Opt. Lett.
**
20
**
,
1746
–
1748
(
1995
). [CrossRef] [PubMed]

**
4
. **
M.
Scalora
,
M. J.
Bloemer
,
A. S.
Manka
,
J. P.
Dowling
,
C. M.
Bowden
,
R.
Viswanathan
, and
J. W.
Haus
, “
Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures
,”
Phys. Rev. A
**
56
**
,
3166
–
3174
(
1997
). [CrossRef]

**
5
. **
J. W.
Haus
,
R.
Viswanathan
,
M.
Scalora
,
A. G.
Kalocsai
,
J. D.
Cole
, and
J.
Theimer
, “
Enhanced second-harmonic generation in media with a weak periodicity
,”
Phys. Rev. A
**
57
**
,
2120
–
2128
(
1998
). [CrossRef]

**
6
. **
G. T.
Kiehne
,
A. E.
Kryukov
, and
J. B.
Ketterson
, “
A numerical study of optical second-harmonic generation in a one-dimensional photonic structure
,”
Appl. Phys. Lett.
**
75
**
,
1676
–
1678
(
1999
). [CrossRef]

**
7
. **
G.
D’Aguanno
,
M.
Centini
,
C.
Sibilia
,
M.
Bertolotti
,
M.
Scalora
,
M. J.
Bloemer
, and
C. M.
Bowden
, “
Enhancement of
*
χ
*^{
(2)
}
cascading processes in one-dimensional photonic bandgap structures
,”
Opt. Lett.
**
24
**
,
1663
–
1665
(
1999
). [CrossRef]

**
8
. **
M.
Centini
,
C.
Sibilia
,
M.
Scalora
,
G.
D’Aguanno
,
M.
Bertolotti
,
M. J.
Bloemer
,
C. M.
Bowden
, and
I.
Nefe-dov
, “
Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions
,”
Phys. Rev. E
**
60
**
,
4891
–
4898
(
1999
). [CrossRef]

**
9
. **
C. D.
Angelis
,
F.
Gringoli
,
M.
Midrio
,
D.
Modotto
,
J. S.
Aitchison
, and
G. F.
Nalesso
, “
Conversion efficiency for second-harmonic generation in photonic crystals
,”
J. Opt. Soc. Am. B
**
18
**
,
348
–
351
(
2001
). [CrossRef]

**
10
. **
Y.
Dumeige
,
P.
Vidakovic
,
S.
Sauvage
,
I.
Sagnes
,
J. A.
Levenson
,
C.
Sibilia
,
M.
Centini
,
G.
D’Aguanno
, and
M.
Scalora
, “
Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap
,”
Appl. Phys. Lett.
**
78
**
,
3021
–
3023
(
2001
). [CrossRef]

**
11
. **
B.
Shi
,
Z. M.
Jiang
, and
X.
Wang
, “
Defective photonic crystals with greatly enhanced second-harmonic generation
,”
Opt. Lett.
**
26
**
,
1194
–
1196
(
2001
). [CrossRef]

**
12
. **
Y.
Dumeige
,
I.
Sagnes
,
P.
Monnier
,
P.
Vidakovic
,
I.
Abram
,
C.
Meriadec
, and
A.
Levenson
, “
Phase-matched frequency doubling at photonic band edges: Efficiency scaling as the fifth power of the length
,”
Phys. Rev. Lett.
**
89
**
,
043901
(
2002
). [CrossRef] [PubMed]

**
13
. **
F. F.
Ren
,
R.
Li
,
C.
Cheng
,
H. T.
Wang
,
J. R.
Qiu
,
J. H.
Si
, and
K.
Hirao
, “
Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes
,”
Phys. Rev. B
**
70
**
,
245109
(
2004
). [CrossRef]

**
14
. **
J. E.
Sipe
, “
New green-function formalism for surface optics
,”
J. Opt. Soc. Am. B
**
4
**
,
481
–
489
(
1987
). [CrossRef]

**
15
. **
M. S.
Tomas
, “
Green-function for multilayers - light-scattering in planar cavities
,”
Phys. Rev. A
**
51
**
,
2545
–
2559
(
1995
). [CrossRef] [PubMed]

**
16
. **
N.
Hashizume
,
M.
Ohashi
,
T.
Kondo
, and
R.
Ito
, “
Optical harmonic-generation in multilayered structures - a comprehensive analysis
,”
J. Opt. Soc. Am. B
**
12
**
,
1894
–
1904
(
1995
). [CrossRef]

**
17
. **
S. A.
Akhmanov
and
R. V.
Khohlov
,
*
Problems of Nonlinear Optics
*
(
Gordon and Breach, New York
,
1973
).