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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 12 — Jun. 11, 2007
  • pp: 7303–7318
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χ (2) and χ (3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities

Alejandro Rodriguez, Marin Soljačić, J. D. Joannopoulos, and Steven G. Johnson  »View Author Affiliations


Optics Express, Vol. 15, Issue 12, pp. 7303-7318 (2007)
http://dx.doi.org/10.1364/OE.15.007303


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Abstract

We derive general conditions for 100% frequency conversion in any doubly resonant nonlinear cavity, for both second- and third-harmonic generation via χ(2) and χ(3) nonlinearities. We find that conversion efficiency is optimized for a certain “critical” power depending on the cavity parameters, and assuming reasonable parameters we predict 100% conversion using milliwatts of power or less. These results follow from a semi-analytical coupled-mode theory framework which is generalized from previous work to include both χ(2) and χ(3) media as well as inhomogeneous (fully vectorial) cavities, analyzed in the high-efficiency limit where down-conversion processes lead to a maximum efficiency at the critical power, and which is verified by direct finite-difference time-domain (FDTD) simulations of the nonlinear Maxwell equations. Explicit formulas for the nonlinear coupling coefficients are derived in terms of the linear cavity eigenmodes, which can be used to design and evaluate cavities in arbitrary geometries.

© 2007 Optical Society of America

1. Introduction

Nonlinear frequency conversion has been commonly realized in the context of waveguides [15–18

15. G. S. Dutta and J. Jolly, “Third harmonic generation in layered media in presence of optical bistability of the fundamental,” Pramana J. Phys. 50, 239 (1988).

], or even for free propagation in the nonlinear materials, in which light at one frequency co-propagates with the generated light at the harmonic frequency [19–22

19. S. Pearl, H. Lotem, and Y. Shimony, “Optimization of laser intracavity second-harmonic generation by a linear dispersion element,” J. Opt. Soc. Am. B 16, 1705 (1999). [CrossRef]

]. A phase-matching condition between the two frequencies must be satisfied in this case in order to obtain efficient conversion [5

5. V. Berger, “Second-harmonic generation in monolithic cavities,” J. Opt. Soc. Am. B 14, 1351 (1997). [CrossRef]

, 9

9. Y. Dumeige and P. Feron, “Wispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” PRA 74, 063,804 (2006).

]. Moreover, as the input power is increased, the frequency conversion eventually saturates due to competition between up and down conversion. Frequency conversion in a doubly resonant cavity has three fundamental differences from this familiar case of propagating modes. First, light in a cavity can be much more intense for the same input power, because of the spatial (modal volume V) and temporal (lifetime Q) confinement. We show that this enhances second-harmonic (χ (2)) conversion by a factor of Q 3/V and enhances third-harmonic (χ (3)) conversion by a factor of Q 2/V. Second, there is no phase-matching condition per se for 100% conversion; the only absolute requirement is that the cavity support two modes of the requisite frequencies. However, there is a constant factor in the power that is determined by an overlap integral between the mode field patterns; in the limit of a very large cavity, this overlap integral recovers the phase-matching condition for χ (2) processes. Third, the frequency conversion no longer saturates—instead, it peaks (at 100%, with proper design) for a certain critical input power satisfying a resonant condition, and goes to zero if the power is either too small or too large.

Fig. 1. Top: Schematic diagram of waveguide-cavity system. Input light from a waveguide (left) at one frequency (amplitude s 1+) is coupled to a cavity mode (amplitude a 1), converted to a cavity mode at another frequency (amplitude a 2) by a nonlinear process, and radiated back into the waveguide (amplitude s 2-). Reflections at the first frequency (s 1-) may also occur. Bottom: 1d example, formed by quarter-wave defect in a quarter-wave dielectric stack. Dielectric material is yellow, and electric field Ez of third-harmonic mode is shown as blue/white/red for positive/zero/negative amplitude.

In particular, we consider the general situation depicted schematically in Fig. 1: a two-mode nonlinear cavity coupled to an input/output channel. For example, a one-dimensional realization of this is shown in Fig. 1: a Fabry-Perot cavity between two quarter-wave stacks [45

45. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

], where the stack has fewer layers on one side so that light can enter/escape. For a nonlinear effect, we consider specifically a χ (ℓ) nonlinearity, corresponding essentially to a shift in the refractive index proportional to the nonlinear susceptibility χ (ℓ) multiplied by electric field E to the (ℓ – 1)th power. Most commonly, one would have either a χ (2) (Pockels) or χ (3) (Kerr) effect. Such a nonlinearity results in harmonic generation [46

46. R. W. Boyd, Nonlinear Optics (Academic Press, California, 1992).

]: light with frequency ω is coupled to light with frequency ℓω. Therefore, if we design the cavity so that it supports two modes, one at ω and one at ℓω, then input power at ω can be converted, at least partially, to output power at ℓω.

2. Temporal coupled-mode theory

We let ak denote the time-dependent complex amplitude of the kth mode, normalized so that |ak|2 is the electromagnetic energy stored in this mode. We let s ± denote the time-dependent amplitude of the incoming (+) or outgoing (-) wave, normalized so that |s ±|2 is the power. (More precisely, s ±(t) is normalized so that its Fourier transform |s̃±(ω)|2 is the power at ω. Later, we will let s k± denote the input/output power at ωk.) [In 1d, the units of |ak|2 and |s ±|2 are those of energy and power per unit area, respectively. More generally, in d dimensions, the units of |ak|2 and |s ±|2 are those of energy and power per length 3-d.] By itself, a linear cavity mode decaying with a lifetime τk would be described by dak/dt = (k-1/τk)ak. [Technically, such a decaying mode is not a true eigenmode, but is rather a “leaky mode” [52

52. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

], corresponding to a “quasi-bound state” in the Breit-Wigner scattering theory [53

53. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Butterworth-Heinemann, Oxford, 1977).

].] The decay rate 1/τk can be decomposed into 1/τk=1/τe,k + 1/τs,k where 1/τe,k is the “external” loss rate (absorption etc.) and 1/τs,k is the decay rate into s -. When the weak coupling (ωkτk ≫ 1) to s ± is included, energy conservation and similar fundamental constraints lead to equations of the form [54

54. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).

]:

dakdt=(iωk1τk)ak+2τs,ks+
(1)
s=s++2τs,kak
(2)

This can be generalized to incorporate multiple input/output ports, direct coupling between the ports, and so on [51

51. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004). [CrossRef]

]. The only unknown parameters in this model are then the frequencies ωk and the decay rates 1/τk, which can be determined by any numerical method to solve for the cavity modes (e.g. FDTD, below). Instead of τk, one commonly uses the quality factor Qk = ωkτk/2.

Nonlinearity modifies this picture with two new amplitude-dependent effects: a shift in the frequency (and decay rate) of the cavity, and a coupling of one cavity mode to another. We neglect nonlinear effects on the input/output ports, under the assumption that intense fields are only present in the cavity (due to spatial and temporal confinement). We will also make two standard assumptions of nonlinear systems. First, that the nonlinearities are weak, in the sense that we can neglect terms of order (χ (ℓ))2 or higher; this is true in practice because nonlinear index shifts are always under 1% lest material breakdown occur. Second, we make the rotating wave approximation: since the coupling is weak, we only include terms for ak that have frequency near ωk. In particular, we suppose that ωk 1, so that ωk is the kth harmonic. The result is that, for a given order nonlinearity, there are only a few possible new terms that can appear in the coupled-mode equations. In particular, for a χ (2) nonlinearity with two modes ω 1 and its second harmonic ω 2, the coupled-mode equations must take the form:

da1dt=(iω11τ1)a1iω1β1a1*a2+2τs,1s+
(3)
da2dt=(iω21τ2)a2iω2β2a12+2τs,2s+
(4)

Similarly, for a χ (3) nonlinearity with two modes ω 1 and its third harmonic ω 3, the coupled-mode equations must take the form:

da1dt=(iω1(1α11a12α13a32)1τ1)a1iω1β1(a1*)2a3+2τs,1s+
(5)
da3dt=(iω3(1α33a32α31a12)1τ3)a3iω3β3a13+2τs,3s+
(6)

In equations 5–6, one sees two kinds of terms. The first are frequency-shifting terms, with coefficients αij, dependent on one of the field amplitudes. For χ (3), this effect is known as self-phase and cross-phase modulation, which is absent for χ (2) (under the first-order rotating-wave approximation). The second kind of term transfers energy between the modes, with coupling coefficients βi, corresponding to four-wave mixing for χ (3). Furthermore, we can constrain the coupling terms βi, by energy conservation: ddt(a12+a22)=0 . For χ (2), the constraint that follows is: ω 1 β 1 = ω 2 β 2 *; for χ (3), the constraint is ω 1 β 1 = ω 3 β 3 *. (This constraint holds even in cavities with external loss as discussed in Sec. 6: energy is still conserved in the sense that the input power must equal the output power plus the loss power, and so the harmonic conversion term must lead to an equal energy loss and gain at ω 1 and ω 2,3, respectively.)

The general process for construction of these coupled-mode equations is as follows. The underlying nonlinearity must depend on the physical, real part of the fields, corresponding to (ak + ak *)/2. It then follows that the χ (ℓ) term will have ℓ powers of this real part, giving various product terms like a 1 * a 2 (for χ (2)) and a 1 * a 1 a 1 (for χ (3)). Most of these terms, however, can be eliminated by the rotating-wave approximation. In particular, we assume that each ak term is proportional to ekiω multiplied by a slowly varying envelope, and we discard any product term whose total frequency differs from for the dak/dt equation. Thus, a term like a 1 * a 3 a 3 would be proportional to e5iω, and would only appear in a da 5/dt equation. (We focus on the simpler case of doubly resonant cavities in this paper.)

At this point, the equations are already useful in order to reason about what types of qualitative behaviors are possible in general. In fact, they are not even specific to electromagnetism and would also apply to other situations such as acoustic resonators. However, in order to make quantitative predictions, one needs to know the nonlinear coefficients αij and βi (as well as the linear frequencies and decay rates). The evaluation of these coefficients requires a more detailed analysis of Maxwell’s equations as described below.

3. Perturbation theory and coupling coefficients

When a dielectric structure is perturbed by a small δε, a well-known result of perturbation theory states that the corresponding change δω in an eigenfrequency ω is, to first order [45

45. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

]:

δωω=12d3xδεE2d3xεE2=12d3xE*δPd3xεE2
(7)

where E is the unperturbed electric field and δ P = δε E is the change in polarization density due to δε. In fact, Eq. 7 is general enough to be used with any δ P, including the polarization that arises from a nonlinear susceptibility. In particular, we can use it to obtain the coupling coefficients of the CMT.

To do so, we first compute the nonlinear first-order frequency perturbation due to the total field E present from all of the modes. Once the frequency perturbations δωk are known, we can re-introduce these into the coupled-mode theory by simply setting ωkωk + δωk in Eq. 1. By comparison with Eqs. 3–6, we can then identify the α and β coefficients.

We consider first a χ (2) nonlinearity, with the nonlinear polarization δ P given by δPi = ∑ijkεχijk (2) EjEk, in a cavity with two modes E 1 and E 2. As before, we require that the modes oscillate with frequency ω 1 and ω 2 ≈ 2ω 1, respectively. Taking E = Re[E 1 e 1t + E 2 e 2t] and using the rotating-wave approximation, we can separate the contribution of δ P to each δωk, to obtain the following frequency perturbations:

δω1ω1=14d3xijkεχijk(2)[E1i*(E2jE1k*+E1j*E2k)]d3xεE12
(8)
δω2ω2=14d3xijkεχijk(2)E2i*E1jE1kd3xεE22
(9)

Similarly, for a centro-symmetric χ (3) medium, δ P is given by δ P = εχ (3)|E|2 E, with E = Re[E 1 e 1t + E 3 e 3t]. We obtain the following frequency perturbations:

δω1ω1=18[d3xεχ(3)(E1E12+2E1E1*2+2(E1E1*)(E3E3*)+2E1E32+2E1E3*2+3(E1*E1*)(E1*E3))d3xεE12]
(10)
δω3ω3=18[d3xεχ(3)(E3E32+2E3E3*2+2(E1E1*)(E3E3*)+2E1E32+2E1E3*2+(E1E1)(E1E3*))d3xεE32]
(11)

There is a subtlety in the application of perturbation theory to decaying modes, such as those of a cavity coupled to output ports. In this case, the modes are not truly eigenmodes, but are rather “leaky modes” [52

52. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

], and are not normalizable. Perturbative methods in this context are discussed in more detail by [51

51. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004). [CrossRef]

, 52

52. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

], but for a tightly confined cavity mode it is sufficient to simply ignore the small radiating field far away from the cavity. The field in the cavity is very nearly that of a true eigenmode of an isolated cavity.

As stated above, we can arrive at the coupling coefficients by setting ωkωk + δωk in Eq. 1. However, the frequency perturbations δωk are time-independent quantities, and we need to connect them to the time-dependent ak amplitudes. Therefore, to re-introduce the time dependence, one can use the slowly varying envelope approximation: a slowly varying, time-dependent amplitude ak(t) is introduced into the unperturbed fields E kE kak(t). The eigenmode must be normalized so that |ak|2 is the energy, as assumed for the coupled-mode theory. Thus, we divide each E k by 12εEk2.

First, we consider the χ (2) medium. Carrying out the above substitutions in Eq. 1 and grouping terms proportional ak yields Eqs. 3–4 with αij and βi given by:

αij=0
(12)
β1=14d3xijkεχijk(2)[E1i*(E2jE1k*+E1j*E2k)][d3xεE12][d3xεE22]12
(13)
β2=14d3xijkεχijk(2)E2i*E1jE1k[d3xεE12][d3xεE22]12
(14)

A similar calculation yields the χ (3) coupled-mode equations with coefficients given by:

αii=18d3xεχ(3)EiEi2+EiEi*2[d3xεEi2]2
(15)
α31=14d3xεχ(3)E12E32+E1E32+E1E3*2[d3xεE12][d3xεE32]
(16)
α13=α31
β1=38d3xεχ(3)(E1*E1*)(E1*E3)[d3xεE12]32[d3xεE32]12
(17)
β3=18d3xεχ(3)(E1E1)(E1E3*)[d3xεE12]32[d3xεE32]12
(18)

4. Numerical validation

To check the predictions of the χ (3) coupled-mode equations, we performed an FDTD simulation of the one-dimensional waveguide-cavity system shown in Fig 1, whose analytical properties are uniquely suited to third-harmonic generation. (The FDTD method, including techniques to simulate nonlinear media, is described in Ref. 56.) This geometry consists of a semi-infinite photonic-crystal structure made of alternating layers of dielectric (ε 1 = 13 and ε 2 = 1) with period a and thicknesses given by the quarter-wave condition (d 1 = √ε 2/(√ε 1 + √ε 2) and d 2 = a - d 1, respectively). Such a quarter-wave stack possesses a periodic sequence of photonic band gaps centered on frequencies ω 1 = (√ε 1 + √ε 2)/4√ε 1 ε 2 (in units of 2πc/a) for the lowest gap, and higher-order gaps centered on odd multiples of ω 1. Moreover, a defect formed by doubling the thickness of a ε 1 layer creates cavity modes at exactly the middle of every one of these gaps. Therefore, it automatically satisfies the frequency-matching condition for third-harmonic generation. In fact, it is too good: there will also be “ninth harmonic” generation from ω 3 to ω 9. This unwanted process is removed, however, by the discretization error of the FDTD simulation, which introduces numerical dispersion that shifts the higher-frequency modes. To ensure the ω 3 = 3ω 1 condition in the face of this dispersion, we slightly perturbed the structure (increasing the dielectric constant slightly at the nodes of the third-harmonic eigenfield) to tune the frequencies. The simulated crystal was effectively semi-infinite, with many more layers on the right than on the left of the cavity. On the left of the cavity, after two period of the crystal the material is simply air (ε = 1), terminated by a perfectly matched layer (PML) absorbing boundary region.

We excite the cavity with an incident plane wave of frequency ω 1, and compute the resulting reflection spectrum. The reflected power at ω 3, the third-harmonic generation, was then compared with the prediction of the coupled-mode theory. The frequencies, decay rates, and α and β coefficients in the coupled-mode theory were computed from a linear FDTD simulation in which the eigenmodes were excited by narrow-band pulses. The freely available FDTD code of [57

57. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. Burr, J. D. Joannopoulos, and S. G. Johnson, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972–2974 (2006). [CrossRef] [PubMed]

] was employed.

The results are shown in Fig. 2, in which the output power at ω 1 and ω 3 = 3ω 1 is denoted by |s 1-|2 and |s 3_ |2, respectively, while the input power at ω 1 is denoted by |s 1+|2. In particular, we plot convenient dimensionless quantities: the third-harmonic conversion efficiency |s 3-|2 /|s 1+| 2 as a function of the dimensionless products n 2 |s 1+|2 in terms of the standard Kerr coefficient n 2 = 3χ (3)/4. There is clear agreement between the FDTD and CMT for small values of n 2 |s 1+|2 (in which limit the conversion goes quadratically with n 2 |s 1+|2). However, as the input power increases, they eventually begin to disagree, marking the point where second-order corrections are required. This disagreement is not a practical concern, however, because the onset of second-order effects coincides with the limits of typical materials, which usually break down for Δn/nχ (3) max|E|2 /2ε > 1%. This is why we also plot the maximum index shift Δn/n in the same figure.

Also shown in Fig. 2 is a plot of Δω 1/ω 1 = Re[δω 1/ω 1]. As expected, when Δω 1 is of the order of 1/Q 1 ~ 10-3, the frequency shift begins to destroy the frequency matching condition, substantially degrading the third-harmonic conversion. (It might seem that Δn/n and Δω 1/ω 1 should be comparable, but this is not the case because Δn/n is the maximum index shift while Δω 1/ω 1 is due to an average index shift.)

Fig. 2. Log-log plot of |s 3-|2/|s 1+|2 vs. n 2|s 1+|2 for the coupled-mode theory (grey) and FDTD (black squares), where n 2 is being varied. Also shown are the corresponding Δn/n (dashed blue) and Δω 1/ω 1 (solid red) curves.

More specifically, the details of our simulation are as follows. To simulate a continuous wave (CW) source spectrum in FDTD, we employ a narrow-bandwidth gaussian pulse incident from the air region, which approximates a CW source in the limit of narrow bandwidth. This pulse is carefully normalized so that the peak intensity is unity, to match the CMT. The field in the air region is Fourier transformed and subtracted from the incident field to yield the reflected flux. Using only two periods of quarter-wave stack on the left of the cavity we obtained two cavity modes with real frequencies ω 1 = 0.31818 (2πc/a), ω 2 = 0.95454 (2πc/a) and quality factors Q 1 = 1286 and Q 3 = 3726, respectively. Given these field patterns, we computed the αij and βi, coefficients. We obtained the following coupling coefficients, in units of χ (3)/a: α 11 = 4.7531 × 10-4, α 22 = 5.3306 × 10-4, α 12 = α 21 = 2.7847 × 10-4,β 1 = (4.55985 - 0.7244i) × 10-5.

5. Complete frequency conversion

We now consider the conditions under which one may achieve complete frequency conversion: 100% of the incident power converted to output at the second or third harmonic frequency. As we shall see, this is easiest to achieve in the χ (2) case, and requires additional design criteria in the χ (3) case.

The key fact in a χ (2) medium is that there are no frequency-shifting terms (α = 0), so the resonance condition ω 2 = 2ω 1 is not spoiled as one increases the power. The only requirement that we must impose is that external losses such as absorption are negligible (τe,kτs,k). In this case, 100% conversion corresponds to setting s 1- = 0 in the steady-state. Using this fact, Eqs. 3-4 for an input source s +(t) = s 1+ exp( 1 t) yields the following condition on the input power for 100% conversion:

s1+2=2ω12β12τs,2τs,12=ω12β12Q2Q12
(19)

Fig. 3. Plot of first and second harmonic efficiency, |s 1-|2 /|s 1+|2 (black) and |s 2-|2 /|s 1+|2 (red), vs. χ (2)|s 1+|. 100% power transfer from ω 1 to ω 2 = 2ω 1 is achieved at χ (2)|s 1+| = 1.8 × 10-3.

A χ (3) medium, on the other hand, does suffer from nonlinear frequency shifts. For example, Fig. 2, which is by no means the optimal geometry, exhibits a maximal efficiency of |s 3-|2/|s 1+|2 ≈ 4 × 10-3, almost three orders of magnitude away from complete frequency conversion. On the other hand, we can again achieve 100% conversion if we can force αij = 0, which can be done in two ways. First, one could employ two χ (3) materials with opposite-sign χ (3) values (e.g., as in Ref. 58). For example, if the χ (3) is an odd function around the cavity center, then the integrals for αij will vanish while the β integrals will not. (In practice, a αβ should suffice.) Second, one could pre-compensate for the nonlinear frequency shifts: design the cavity so that the shifted frequencies, at the critical power below, satisfy the resonant condition ω 3 + Δω 3 = 3(ω 1 + Δω 1). Equivalently, design the device for αij = 0 and then adjust the linear cavity frequencies a posteriori to compensate for the frequency shift at the critical power. (This is closely analogous to the cavity detuning used for optical bistability [55

55. M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Rapid Commun. 66, 055,601(R) (2002).

], in which one operates off-resonance in the linear regime so that resonance occurs from the nonlinear shift.)

If αij is thereby forced to be zero, and we can also neglect external losses (absorption, etc.) as above, then 100% third-harmonic conversion (s 1- = 0) is obtained when:

s1+2=[4ω12β12τs,13τs,3]12=[ω1ω34β12Q13Q3]12
(20)

Fig. 4. Plot of first and third harmonic efficiency, |s 1-|2 /|s 1+|2 (black) and |s 3-|2 /|s 1+|2 (red), vs. n 2|s 1+|2. 100% power transfer from ω 1 to ω 3 = 3ω 1 is achieved at n 2|s 1+|2 = 2.8 × 10-4.

We demonstrate the third-harmonic conversion for αij = 0 by plotting the solution to the coupled-mode equations as a function of input power in Fig. 4. Again, 100% conversion is only obtained at a single critical power. Here, we used the same parameters as in the FDTD calculation, but with α = 0. In this case, comparing with Fig. 2, we observe that complete frequency conversion occurs at a power corresponding to Δn/n ≈ 10-2. This is close to the maximum power before coupled-mode/perturbation theory becomes invalid (either because of second-order effects or material breakdown), but we could easily decrease the critical power by increasing Q.

For both the χ (2) and the χ (3) effects, in Figs. 3–4, we see that the harmonic conversion efficiency goes to zero if the input power (or χ) is either too small or too large. It is not surprising that frequency conversion decreases for low powers, but the decrease in efficiency for high powers is less intuitive. It corresponds to a well-known phenomenon in coupled-mode systems: in order to get 100% transmission from an input port to an output port, the coupling rates to the two ports must be matched in order to cancel the back-reflected wave [50

50. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984). Ch. 7.

, 60

60. S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18(2), 162–165 (2001). [CrossRef]

]. In the present case, the coupling rate to the input port is ~ 1/Q 1, and the coupling rate to the output “port” (the harmonic frequency) is determined by the strength of the nonlinear coupling. If the nonlinear coupling is either too small or too large, then the rates are not matched and the light is reflected instead of converted. (On the other hand, we find that for large input powers, while the conversion efficiency as a fraction of input power goes to zero, the absolute converted power (|s 2-|2 or |s 3-|2) goes to a constant.)

6. The Effect of Losses

In practice, a real device will have some additional losses, such as linear or nonlinear absorption and radiative scattering. Such losses will lower the peak conversion efficiency below 100%. As we show in this section, their quantitative effect depends on the ratio of the loss rate to the total loss rate 1/Q. We also solve for the critical input power to achieve maximal conversion efficiency in the presence of losses.

For a χ (2) medium with a linear loss rate 1/τe,k, we solve Eqs 3–4 for |s 2-|2 and enforce the condition for maximal conversion efficiency: dds1+2(s22s1+2)=0. We thus obtain the following optimal input power and conversion efficiency:

s1+2=2τs,1ω12β12τ13τ2
(21)
s22s1+2=τ1τ2τs,1τs,2
(22)

It immediately follows that for zero external losses, i.e. τk = τs,k, Eq. 22 gives 100% conversion and Eq. 21 reduces to Eq. 19. For small external losses τs,kτe,k, the optimal efficiency is reduced by the ratio of the loss rates, to first order:

s22s1+21(τs,2τe,2+τs,1τe,1).
(23)

(A similar transmission reduction occurs in coupled-mode theory when any sort of loss is introduced into a resonant coupling process [54

54. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).

].)

The same analysis for χ (3) yields the following critical input power and optimal efficiency:

s1+2=[4τs,12ω12β12τ15τ3]12
(24)
s32s1+2=τ1τ3τs,1τs,3
(25)

where by comparison with Eq. 22, a first-order expansion for low-loss yields an expression of the same form as Eq. 23: the efficiency is reduced by the ratio of the loss rates, with τ 2 replaced by τ 3.

A χ (3) medium may also have a nonlinear “two-photon” absorption, corresponding to a complex-valued χ (3), which gives an absorption coefficient proportional to the field intensity. This enters the coupled-mode equations as a small imaginary part added to α, even if we have set the real part of α to zero. (The corresponding effect on β is just a phase shift.) That yields a nonlinear (NL) τe,k of the following form, to lowest order in the loss:

1τe,1NLω1Im[α11τs,12s1+2+α13τs,32τs,138ω32β32s1+6]
(26)
1τe,3NLω3Im[α31τs,12s1+2+α33τs,32τs,138ω32β32s1+6].
(27)

where we have simply substituted the values for the critical fields a1=2τ1s1+ and a 3 given by Eq. 6, and grouped terms that correspond to imaginary frequency shifts. These loss rates can then be substituted in the expression for the losses above, i.e. Eq. 25, in which case one obtains the following optimal efficiency of third-harmonic generation, to lowest-order in the loss, not including linear losses:

s32s1+21τs,3β1τs,3τs,1Im[α11+3α13τs,3+α13+3α33τs,1]
(28)

(The linear and nonlinear losses can be combined by simply multiplying Eq. 25 and Eq. 28.) Thus, the nonlinear loss is proportional to the ratio Im α/|β|, which is proportional to Im χ (3)/|χ (3)|.

7. Conclusion

Future work will involve designing specific doubly resonant cavity geometries and more precise power predictions. Using our expressions for α and β, optimized cavities for harmonic generation can be designed using standard methods to compute the linear eigenmodes. In practice, experimentally achieving cavity modes with “exactly” harmonic frequencies, matched to within the fractional bandwidth 1/Q, is a challenge and may require some external tuning mechanism. For example, one could use the nonlinearity itself for tuning, via external illumination of the cavity with an intense “tuning” beam at some other frequency. Also, although we can directly integrate the coupled-mode equations in time, we intend to supplement this with a linearized stability analysis at the critical power. This is particularly important for the χ (3) case, where pre-correcting the frequency to compensate the nonlinear frequency shift (self-phase modulation) may require some care to ensure a stable solution.

Acknowledgements

We would like to thank Zheng Wang and Karl Koch for useful discussions, as well as the anonymous referees for many helpful suggestions. This work was supported in part by the Materials Research Science and Engineering Center program of the National Science Foundation under award DMR-9400334, by a Department of Energy (DOE) Computational Science Fellowship under grant DE–FG02-97ER25308, and also by the Paul E. Gray Undergraduate Research Opportunities Program Fund at MIT.

References and links

1.

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation I: Semiclassical theory,” Optica Acta. 27(3), 321–335 (1980). [CrossRef]

2.

L.-A. Wu, M. Xiao, and H. J. Kimble, “Squeezed states of light from an optical parametric oscillator,” JOSA-B 4, 1465–1476 (1987). [CrossRef]

3.

Z. Y. Ou and H. J. Kimble, “Enhanced conversion efficiency for harmonic generation with double resonance,” Opt. Lett. 18, 1053–1055 (1993). [CrossRef] [PubMed]

4.

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, “Nonlinear mode coupling in doubly resonant frequency dou-blers,” Appl. Phys. Lett. 58, 117 (1994).

5.

V. Berger, “Second-harmonic generation in monolithic cavities,” J. Opt. Soc. Am. B 14, 1351 (1997). [CrossRef]

6.

I. I. Zootoverkh, K. N. V., and E. G. Lariontsev, “Enhancement of the efficiency of second-harmonic generation in microlaser,” Quantum Electron . 30, 565 (2000). [CrossRef]

7.

B. Maes, P. Bienstman, and R. Baets, “Modeling second-harmonic generation by use of mode expansion,” J. Opt. Soc. Am. B 22, 1378 (2005). [CrossRef]

8.

M. Liscidini and L. A. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73, 016,613 (2006). [CrossRef]

9.

Y. Dumeige and P. Feron, “Wispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” PRA 74, 063,804 (2006).

10.

G. T. Moore, K. Koch, and E. C. Cheung, “Optical parametric oscillation with intracavity second-harmonic generation,” Opt. Commun. 113, 463 (1995). [CrossRef]

11.

M. Liscidini and L. A. Andreani, “Highly efficient second-harmonic generation in doubly resonant planar micro-cavities,” Appl. Phys. Lett. 85, 1883 (2004). [CrossRef]

12.

L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, and W. Stolz, “Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling,” Appl. Phys. Lett. 88, 2251,117 (2006). [CrossRef]

13.

P. Scotto, P. Colet, and M. San Miguel, “All-optical image processing with cavity type II second-harmonic generation,” Opt. Lett. 28, 1695 (2003). [CrossRef] [PubMed]

14.

G. McConnell, A. I. Ferguson, and N. Langford, “Cavity-augmented frequency tripling of a continuous wave mode-locked laser,” J. Phys. D: Appl.Phys 34, 2408 (2001). [CrossRef]

15.

G. S. Dutta and J. Jolly, “Third harmonic generation in layered media in presence of optical bistability of the fundamental,” Pramana J. Phys. 50, 239 (1988).

16.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Generalized coupled-mode theory for χ(2) interactions in finite multi-layered structures,” J. Opt. Soc. Am. B 19, 2111–2122 (2002). [CrossRef]

17.

A. R. Cowan and J. F. Young, “Mode matching for second-harmonic generation in photonic crystal waveguides,” Phys. Rev. E 65, 085,106 (2002).

18.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161,306 (2003). [CrossRef]

19.

S. Pearl, H. Lotem, and Y. Shimony, “Optimization of laser intracavity second-harmonic generation by a linear dispersion element,” J. Opt. Soc. Am. B 16, 1705 (1999). [CrossRef]

20.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band edge under the quasiphase matching condition,” Phys. Rev. E 63, 046,609 (2001). [CrossRef]

21.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016,609 (2001).

22.

A. H. Norton and C. M. de Sterke, “Optimal poling of nonlinear photonic crystals for frequency conversion,” Opt. Lett. 28, 188 (2002). [CrossRef]

23.

J. A. Armstrong, N. loembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). [CrossRef]

24.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2, 109–124 (1966). [CrossRef]

25.

R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron. 6, 215–223 (1970). [CrossRef]

26.

A. I. Gerguson and M. H. Dunn, “Intracavity second harmonic generation in continuous-wave dye lasers,” IEEE J. Quantum Electron. 13, 751–756 (1977). [CrossRef]

27.

M. Brieger, H. Busener, A. Hese, F. V. Moers, and A. Renn, “Enhancement of single frequency SHG in a passive ring resonator,” Opt. Commun. 38, 423–426 (1981). [CrossRef]

28.

J. C. Berquist, H. Hemmati, and W. M. Itano, “High power second harmonic generation of 257 nm radiation in an external ring cavity,” Opt. Commun. 43, 437–442 (1982). [CrossRef]

29.

W. J. Kozlovsky, W. P. Risk, W. Lenth, B. G. Kim, G. L. Bona, H. Jaeckel, and D. J. Webb, “Blue light generation by resonator-enhanced frequency doubling of an extended-cavity diode laser,” Appl. Phys. Lett. 65, 525–527 (1994). [CrossRef]

30.

G. J. Dixon, C. E. Tanner, and C. E. Wieman, “432-nm source based on efficient second-harmonic generation of GaAlAs diode-laser radiation in a self-locking external resonant cavity,” Opt. Lett. 14, 731–733 (1989). [CrossRef] [PubMed]

31.

M. J. Collet and R. B. Levien, “Two-photon loss model of intracavity second-harmonic generation,” PRA 43(9), 5068–5073 (1990).

32.

M. A. Persaud, J. M. Tolchard, and A. I. Ferguson, “Efficient generation of picosecond pulses at 243 nm,” IEEE J. Quantum Electron. 26, 1253–1258 (1990). [CrossRef]

33.

K. Schneider, S. Schiller, and J. Mlynek, “1.1-W single-frequency 532-nm radiation by second-harmonic generation of a miniature Nd:YAG ring laser,” Opt. Lett. 21, 1999–2001 (1996). [CrossRef] [PubMed]

34.

X. Mu, Y. J. Ding, H. Yang, and G. J. Salamo, “Cavity-enhanced and quasiphase-matched mutli-order reflection-second-harmonic generation from GaAs/AlAs and GaAs/AlGaAs multilayers,” Appl. Phys. Lett. 79, 569 (2001). [CrossRef]

35.

J. Hald, “Second harmonic generation in an external ring cavity with a Brewster-cut nonlinear cystal: theoretical considerations,” Opt. Commun. 197, 169 (2001). [CrossRef]

36.

T. V. Dolgova, A. I. Maidykovski, M. G. Martemyanov, A. A. Fedyanin, O. A. Aktsipetrov, G. Marowsky, V. A. Yakovlev, G. Mattei, N. Ohta, and S. Nakabayashi, “Giant optical second-harmonic generation in single and coupled microcavities formed from one-dimensional photonic crystals,” J. Opt. Soc. Am. B 19, 2129 (2002). [CrossRef]

37.

T.-M. Liu, C.-T. Yu, and C.-K. Sun, “2 Ghz repetition-rate femtosecond blue sources by second-harmonic generation in a resonantly enhanced cavity,” Appl. Phys. Lett. 86, 061,112 (2005). [CrossRef]

38.

L. Scaccabarozzi, M. M. Fejer, Y. Huo, S. Fan, X. Yu, and J. S. Harris, “Enchanced second-harmonic generation in AlGaAs/AlxOy tightly confining waveguides and resonant cavities,” OL 31(24), 3626–3630 (2006). [CrossRef]

39.

A. Di Falco, C. Conti, and G. Assanto, “Impedance matching in photonic crystal microcavities for second-harmonic generation,” Opt. Lett. 31, 250 (2006). [CrossRef] [PubMed]

40.

H. Schnitzler, U. Fröhlich, T. K. W. Boley, A. E. M. Clemen, J. Mlynek, A. Peters, and S. Schiller, “All-solid-state tunable continuous-wave ultraviolet source with high spectral purity and frequency stability,” Appl. Opt. 41, 7000–7005 (2002). [CrossRef] [PubMed]

41.

F.-F. Ren, R. Li, C. Cheng, and H.-T. Wang, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” PRB 70, 245,109 (2004). [CrossRef]

42.

K. Koch and G. T. Moore, “Singly resonant cavity-enhanced frequency tripling,” J. Opt. Soc. Am. B 16, 448 (1999). [CrossRef]

43.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, N. N. Lepeshkin, and R. W. Boyd, “Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals,” Phys. Rev. Lett. 92(083903) (2004). [CrossRef] [PubMed]

44.

G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. pp. 13–15 (1993). [CrossRef] [PubMed]

45.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

46.

R. W. Boyd, Nonlinear Optics (Academic Press, California, 1992).

47.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1998).

48.

D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques,” J. Opt. Soc. Am. B 6, 910–916 (1989). [CrossRef]

49.

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]

50.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984). Ch. 7.

51.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004). [CrossRef]

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A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

53.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Butterworth-Heinemann, Oxford, 1977).

54.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).

55.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Rapid Commun. 66, 055,601(R) (2002).

56.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, Norwood, MA, 2000).

57.

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. Burr, J. D. Joannopoulos, and S. G. Johnson, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972–2974 (2006). [CrossRef] [PubMed]

58.

D. D. Smith, G. Fischer, R. W. Boyd, and D. A. Gregory, “Cancellation of photoinduced absorption in metal nanoparticle composites through a counterintuitive consequence of local field effects,” J. Opt. Soc. Am. B 14, 1625 (1997). [CrossRef]

59.

M. F. Yanik, S. Fan, M. Soljačić, J. D. Joannopoulos, and Yanik, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 68, 2506 (2004).

60.

S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18(2), 162–165 (2001). [CrossRef]

61.

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. Lin, and H. Lin, “Nonlinear refractive-index and two-photon absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62, 2465–2467 (1993). [CrossRef]

62.

Q. Xu and M. Lipson, “Carrier-induced optical bistability in Silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2005). [CrossRef]

63.

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005). [CrossRef] [PubMed]

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(230.4320) Optical devices : Nonlinear optical devices

ToC Category:
Nonlinear Optics

History
Original Manuscript: February 23, 2007
Revised Manuscript: May 22, 2007
Manuscript Accepted: May 23, 2007
Published: May 31, 2007

Citation
Alejandro Rodriguez, Marin Soljacic, J. D. Joannopoulos, and Steven G. Johnson, "Χ(2) and Χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities," Opt. Express 15, 7303-7318 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7303


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References

  1. P. D. Drummond, K. J. McNeil, and D. F. Walls, "Non-equilibrium transitions in sub/second harmonic generation I: Semiclassical theory," Opt. Acta. 27, 321-335 (1980). [CrossRef]
  2. L.-A. Wu, M. Xiao, and H. J. Kimble, "Squeezed states of light from an optical parametric oscillator," J. Opt. Soc. Am. B 4, 1465-1476 (1987). [CrossRef]
  3. Z. Y. Ou and H. J. Kimble, "Enhanced conversion efficiency for harmonic generation with double resonance," Opt. Lett. 18, 1053-1055 (1993). [CrossRef] [PubMed]
  4. R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, "Nonlinear mode coupling in doubly resonant frequency doublers," Appl. Phys. Lett. 58, 117 (1994).
  5. V. Berger, "Second-harmonic generation in monolithic cavities," J. Opt. Soc. Am. B 14, 1351 (1997). [CrossRef]
  6. I. I. Zootoverkh, K. N. V., and E. G. Lariontsev, "Enhancement of the efficiency of second-harmonic generation in microlaser," Quantum Electron. 30, 565 (2000). [CrossRef]
  7. B. Maes, P. Bienstman, and R. Baets, "Modeling second-harmonic generation by use of mode expansion," J. Opt. Soc. Am. B 22, 1378 (2005). [CrossRef]
  8. M. Liscidini and L. A. Andreani, "Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors," Phys. Rev. E 73, 016,613 (2006). [CrossRef]
  9. Y. Dumeige and P. Feron, "Wispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation," PRA 74, 063,804 (2006).
  10. G. T. Moore, K. Koch, and E. C. Cheung, "Optical parametric oscillation with intracavity second-harmonic generation," Opt. Commun. 113, 463 (1995). [CrossRef]
  11. M. Liscidini and L. A. Andreani, "Highly efficient second-harmonic generation in doubly resonant planar microcavities," Appl. Phys. Lett. 85, 1883 (2004). [CrossRef]
  12. L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, and W. Stolz, "Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling," Appl. Phys. Lett. 88, 2251,117 (2006). [CrossRef]
  13. P. Scotto, P. Colet, and M. San Miguel, "All-optical image processing with cavity type II second-harmonic generation," Opt. Lett. 28, 1695 (2003). [CrossRef] [PubMed]
  14. G. McConnell, A. I. Ferguson, and N. Langford, "Cavity-augmented frequency tripling of a continuous wave mode-locked laser," J. Phys. D: Appl. Phys 34, 2408 (2001). [CrossRef]
  15. G. S. Dutta and J. Jolly, "Third harmonic generation in layered media in presence of optical bistability of the fundamental," Pramana J. Phys. 50, 239 (1988).
  16. G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for |(2) interactions in finite multi-layered structures," J. Opt. Soc. Am. B 19, 2111-2122 (2002). [CrossRef]
  17. A. R. Cowan and J. F. Young, "Mode matching for second-harmonic generation in photonic crystal waveguides," Phys. Rev. E 65, 085,106 (2002).
  18. A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003). [CrossRef]
  19. S. Pearl, H. Lotem, and Y. Shimony, "Optimization of laser intracavity second-harmonic generation by a linear dispersion element," J. Opt. Soc. Am. B 16, 1705 (1999). [CrossRef]
  20. A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, "Enhancement of sum frequency generation near the photonic band edge under the quasiphase matching condition," Phys. Rev. E 63, 046,609 (2001). [CrossRef]
  21. G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).
  22. A. H. Norton and C. M. de Sterke, "Optimal poling of nonlinear photonic crystals for frequency conversion," Opt. Lett. 28, 188 (2002). [CrossRef]
  23. J. A. Armstrong, N. loembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962). [CrossRef]
  24. A. Ashkin, G. D. Boyd, and J. M. Dziedzic, "Resonant optical second harmonic generation and mixing," IEEE J. Quantum Electron. 2, 109-124 (1966). [CrossRef]
  25. R. G. Smith, "Theory of intracavity optical second-harmonic generation," IEEE J. Quantum Electron. 6, 215-223 (1970). [CrossRef]
  26. A. I. Gerguson and M. H. Dunn, "Intracavity second harmonic generation in continuous-wave dye lasers," IEEE J. Quantum Electron. 13, 751-756 (1977). [CrossRef]
  27. M. Brieger, H. Busener, A. Hese, F. V. Moers, and A. Renn, "Enhancement of single frequency SHG in a passive ring resonator," Opt. Commun. 38, 423-426 (1981). [CrossRef]
  28. J. C. Berquist, H. Hemmati, and W. M. Itano, "High power second harmonic generation of 257 nm radiation in an external ring cavity," Opt. Commun. 43, 437-442 (1982). [CrossRef]
  29. W. J. Kozlovsky, W. P. Risk,W. Lenth, B. G. Kim, G. L. Bona, H. Jaeckel, and D. J. Webb, "Blue light generation by resonator-enhanced frequency doubling of an extended-cavity diode laser," Appl. Phys. Lett. 65, 525-527 (1994). [CrossRef]
  30. G. J. Dixon, C. E. Tanner, and C. E. Wieman, "432-nm source based on efficient second-harmonic generation of GaAlAs diode-laser radiation in a self-locking external resonant cavity," Opt. Lett. 14, 731-733 (1989). [CrossRef] [PubMed]
  31. M. J. Collet and R. B. Levien, "Two-photon loss model of intracavity second-harmonic generation," PRA 43(9), 5068-5073 (1990).
  32. M. A. Persaud, J. M. Tolchard, and A. I. Ferguson, "Efficient generation of picosecond pulses at 243 nm," IEEE J. Quantum Electron. 26, 1253-1258 (1990). [CrossRef]
  33. K. Schneider, S. Schiller, and J. Mlynek, "1.1-W single-frequency 532-nm radiation by second-harmonic generation of a miniature Nd:YAG ring laser," Opt. Lett. 21, 1999-2001 (1996). [CrossRef] [PubMed]
  34. X. Mu, Y. J. Ding, H. Yang, and G. J. Salamo, "Cavity-enhanced and quasiphase-matched mutli-order reflectionsecond-harmonic generation from GaAs/AlAs and GaAs/AlGaAs multilayers," Appl. Phys. Lett. 79, 569 (2001). [CrossRef]
  35. J. Hald, "Second harmonic generation in an external ring cavity with a Brewster-cut nonlinear cystal: theoretical considerations," Opt. Commun. 197, 169 (2001). [CrossRef]
  36. T. V. Dolgova, A. I. Maidykovski, M. G. Martemyanov, A. A. Fedyanin, O. A. Aktsipetrov, G. Marowsky, V. A. Yakovlev, G. Mattei, N. Ohta, and S. Nakabayashi, "Giant optical second-harmonic generation in single and coupled microcavities formed from one-dimensional photonic crystals," J. Opt. Soc. Am. B 19, 2129 (2002). [CrossRef]
  37. T.-M. Liu, C.-T. Yu, and C.-K. Sun, "2 Ghz repetition-rate femtosecond blue sources by second-harmonic generation in a resonantly enhanced cavity," Appl. Phys. Lett. 86, 061,112 (2005). [CrossRef]
  38. L. Scaccabarozzi, M. M. Fejer, Y. Huo, S. Fan, X. Yu, and J. S. Harris, "Enchanced second-harmonic generation in AlGaAs/AlxOy tightly confining waveguides and resonant cavities," Opt. Lett. 31, 3626-3630 (2006). [CrossRef]
  39. A. Di Falco, C. Conti, and G. Assanto, "Impedance matching in photonic crystal microcavities for secondharmonic generation," Opt. Lett. 31, 250 (2006). [CrossRef] [PubMed]
  40. H. Schnitzler, U. Fröhlich, T. K. W. Boley, A. E. M. Clemen, J. Mlynek, A. Peters, and S. Schiller, "All-solidstate tunable continuous-wave ultraviolet source with high spectral purity and frequency stability," Appl. Opt. 41, 7000-7005 (2002). [CrossRef] [PubMed]
  41. F.-F. Ren, R. Li, C. Cheng, and H.-T. Wang, "Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes," PRB 70, 245,109 (2004). [CrossRef]
  42. K. Koch and G. T. Moore, "Singly resonant cavity-enhanced frequency tripling," J. Opt. Soc. Am. B 16, 448 (1999). [CrossRef]
  43. P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, N. N. Lepeshkin, and R. W. Boyd, "Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals," Phys. Rev. Lett. 92, 083903 (2004). [CrossRef] [PubMed]
  44. G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, "Large nonlinear phase shifts in second-order nonlinear-optical processes," Opt. Lett. 18, 13-15 (1993). [CrossRef] [PubMed]
  45. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).
  46. R. W. Boyd, Nonlinear Optics (Academic Press, California, 1992).
  47. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1998).
  48. D. S. Bethune, "Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques," J. Opt. Soc. Am. B 6, 910-916 (1989). [CrossRef]
  49. 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]
  50. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984). Chap. 7.
  51. W. Suh, Z. Wang, and S. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004). [CrossRef]
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