##
*χ*
^{(2)} and *χ*
^{(3)} harmonic generation at a critical power in inhomogeneous doubly resonant cavities

Optics Express, Vol. 15, Issue 12, pp. 7303-7318 (2007)

http://dx.doi.org/10.1364/OE.15.007303

Acrobat PDF (268 KB)

### 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

*χ*

^{(2)}nonlinearities in large Fabry-Perot etalons or two-dimensional ring resonators where a scalar approximation applied [1–9

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]

*χ*

^{(2)}and

*χ*

^{(3)}nonlinearities and handle the inhomogenous fully vectorial case. We then develop several results from this generalization: whereas it is well known that 100% harmonic conversion is possible, at least in

*χ*

^{(2)}media, in this work we further explore and identify the conditions under which this can be achieved. First, we demonstrate the existence of a critical input power at which harmonic generation is maximized, in contrast to previous work that focused largely on the low-power limit in which generation efficiency increased monotonically with input power [3

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]

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

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

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]

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

*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.

*χ*

^{(2)}materials where 3

*ω*is generated by cascading two nonlinear processes (harmonic generation and frequency summing) [14

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]

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

*χ*

^{(3)}materials so that only a single resonant process need be designed and a different set of materials becomes available. (

*χ*

^{(3)}third-harmonic generation in a bulk periodic structure, with no cavity, was considered in Ref. 43.) In a

*χ*

^{(3)}medium, there are also self/cross-phase modulation phenomena (nonlinear frequency shifts) that, unchecked, will prevent 100% conversion by making the frequency ratio ≠ 3. To address this mismatch, we describe how one can use two materials with opposite-sign

*χ*

^{(3)}to cancel the frequency-shifting effect; it may also be possible to pre-shift the cavity resonant frequency to correct for the nonlinear shift. On the other hand, a

*χ*

^{(2)}medium has no self-phase modulation, and so in this case it is sufficient to increase the input power until 100% frequency conversion is reached. (An “effective” self-phase modulation occurs in

*χ*

^{(2)}media due to cascaded up- and down-conversion processes [44

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]

*χ*

^{(2)}and

*χ*

^{(3)}nonlinearities, and show that the latter can be made negligible.) If the critical field were too intense, then material breakdown might also be an obstacle, but we show that it is sufficient to use modes with a large lifetime

*Q*and small volume

*V*so that a slow conversion due to a weak nonlinear effect has enough time to occur.

*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]: light with frequency

*ω*is coupled to light with frequency ℓ

*ω*. Therefore, if we design the cavity so that it supports two modes, one at

*ω*and one at ℓ

*ω*, then input power at

*ω*can be converted, at least partially, to output power at ℓ

*ω*.

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]

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

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

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]

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]

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]

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]

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]

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]

*χ*

^{(2)}and

*χ*

^{(3)}media. (The optimization of these coupling coefficients is then the generalization of the phase-matching criteria used in one-dimensional geometries [5

**14**, 1351 (1997). [CrossRef]

## 2. Temporal coupled-mode theory

*s*

_{+}) and out (

*s*

_{-}) of the cavity. A schematic illustration of the system is shown in Fig. 1. Specifically, we follow the formalism described in Ref. 50, adapted to handle nonlinearly coupled modes with frequencies

*ω*. Although similar equations for the case of

_{k}*χ*

^{(2)}media were derived in the past [3

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]

*χ*

^{(3)}harmonic generation in cavities. Moreover, a derivation via the temporal coupled-mode formalism of Ref. 50 is arguably more general than earlier developments based on a particular scalar nonlinear wave equation, because this formalism (for a given-order nonlinearity) depends only on general considerations such as weak coupling and energy conservation (the resulting equations hold for vector or scalar waves in electromagnetism, acoustics, or any other weakly-coupled problem with a few simple properties). In the next section, we will then specialize the equations to electromagnetism by deriving explicit equations for the coupling coefficients from Maxwell’s equations.

*a*denote the time-dependent complex amplitude of the

_{k}*k*th mode, normalized so that |

*a*|

_{k}^{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 |

*s*̃

_{±}(

*ω*)|

^{2}is the power at

*ω*. Later, we will let

*s*

_{k±}denote the input/output power at

*ω*.) [In 1d, the units of |

_{k}*a*|

_{k}^{2}and |

*s*

_{±}|

^{2}are those of energy and power per unit area, respectively. More generally, in

*d*dimensions, the units of |

*a*|

_{k}^{2}and |

*s*

_{±}|

^{2}are those of energy and power per length

^{3-d}.] By itself, a linear cavity mode decaying with a lifetime

*τ*would be described by

_{k}*da*/

_{k}*dt*= (

*iω*-1/

_{k}*τ*)

_{k}*a*. [Technically, such a decaying mode is not a true eigenmode, but is rather a “leaky mode” [52], corresponding to a “quasi-bound state” in the Breit-Wigner scattering theory [53].] The decay rate 1/

_{k}*τ*can be decomposed into 1/

_{k}*τ*=1/

_{k}*τ*+ 1/

_{e,k}*τ*where 1/

_{s,k}*τ*is the “external” loss rate (absorption etc.) and 1/

_{e,k}*τ*is the decay rate into

_{s,k}*s*

_{-}. When the weak coupling (

*ω*≫ 1) to

_{k}τ_{k}*s*

_{±}is included, energy conservation and similar fundamental constraints lead to equations of the form [54]:

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]

*ω*and the decay rates 1/

_{k}*τ*, which can be determined by any numerical method to solve for the cavity modes (e.g. FDTD, below). Instead of

_{k}*τ*, one commonly uses the quality factor

_{k}*Q*=

_{k}*ω*/2.

_{k}τ_{k}*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

*a*that have frequency near

_{k}*ω*. In particular, we suppose that

_{k}*ω*≈

_{k}*kω*

_{1}, so that

*ω*is the

_{k}*k*th 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:

*χ*

^{(3)}nonlinearity with two modes

*ω*

_{1}and its third harmonic

*ω*

_{3}, the coupled-mode equations must take the form:

*frequency-shifting*terms, with coefficients

*α*, dependent on one of the field amplitudes. For

_{ij}*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

*β*, corresponding to four-wave mixing for

_{i}*χ*

^{(3)}. Furthermore, we can constrain the coupling terms

*β*, by energy conservation:

_{i}*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.)

*a*+

_{k}*a*

_{k}^{*})/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

*a*term is proportional to

_{k}*e*multiplied by a slowly varying envelope, and we discard any product term whose total frequency differs from

^{kiω}*kω*for the

*da*/

_{k}*dt*equation. Thus, a term like

*a*

_{1}

^{*}

*a*

_{3}

*a*

_{3}would be proportional to

*e*, and would only appear in a

^{5iω}*da*

_{5}/

*dt*equation. (We focus on the simpler case of doubly resonant cavities in this paper.)

*α*and

_{ij}*β*(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.

_{i}## 3. Perturbation theory and coupling coefficients

*δε*, a well-known result of perturbation theory states that the corresponding change

*δω*in an eigenfrequency

*ω*is, to first order [45]:

**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.

**E**present from all of the modes. Once the frequency perturbations

*δω*are known, we can re-introduce these into the coupled-mode theory by simply setting

_{k}*ω*→

_{k}*ω*+

_{k}*δω*in Eq. 1. By comparison with Eqs. 3–6, we can then identify the

_{k}*α*and

*β*coefficients.

*χ*

^{(2)}nonlinearity, with the nonlinear polarization

*δ*

**P**given by

*δP*= ∑

_{i}

_{ijk}εχ_{ijk}^{(2)}

*E*, in a cavity with two modes

_{j}E_{k}**E**

_{1}and

**E**

_{2}. As before, we require that the modes oscillate with frequency

*ω*

_{1}and

*ω*

_{2}≈ 2

*ω*

_{1}, respectively. Taking

**E**= Re[

**E**

_{1}

*e*

^{iω1t}+

**E**

_{2}

*e*

^{iω2t}] and using the rotating-wave approximation, we can separate the contribution of

*δ*

**P**to each

*δω*, to obtain the following frequency perturbations:

_{k}*χ*

^{(3)}medium,

*δ*

**P**is given by

*δ*

**P**=

*εχ*

^{(3)}|

**E**|

^{2}

**E**, with

**E**= Re[

**E**

_{1}

*e*

^{iω1t}+

**E**

_{3}

*e*

^{iω3t}]. We obtain the following frequency perturbations:

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]

*ω*→

_{k}*ω*+

_{k}*δω*in Eq. 1. However, the frequency perturbations

_{k}*δω*are time-independent quantities, and we need to connect them to the time-dependent

_{k}*a*amplitudes. Therefore, to re-introduce the time dependence, one can use the slowly varying envelope approximation: a slowly varying, time-dependent amplitude

_{k}*a*(

_{k}*t*) is introduced into the unperturbed fields

**E**

_{k}→

**E**

*(*

_{k}a_{k}*t*). The eigenmode must be normalized so that |

*a*|

_{k}^{2}is the energy, as assumed for the coupled-mode theory. Thus, we divide each

**E**

_{k}by

*χ*

^{(2)}medium. Carrying out the above substitutions in Eq. 1 and grouping terms proportional

*a*yields Eqs. 3–4 with

_{k}*α*and

_{ij}*β*given by:

_{i}*χ*

^{(3)}coupled-mode equations with coefficients given by:

*ω*

_{1}

*β*

_{1}=

*ω*

_{2}

*β*

^{*}

_{2}and

*ω*

_{1}

*β*

_{1}=

*ω*

_{3}

*β*

^{*}

_{3}, previously derived from conservation of energy—for

*χ*

^{(2)}, this requires that one apply the symmetries of the

*χ*

^{(2)}

_{ijk}tensor, which is invariant under permutations of

*ijk*for a frequency-independent

*χ*

^{(2)}[46]. Furthermore, we can relate the coefficients

*α*and

*β*to an effective modal volume

*V*, similar to Ref. 55. In particular, the strongest possible nonlinear coupling will occur if the eigenfields are a constant in the nonlinear material and zero elsewhere. In this case, any integral over the fields will simply yield the geometric volume

*V*of the nonlinear material. Thus, for the

*χ*

^{(2)}effect we would obtain

*β*~

_{i}*χ*

^{(2)}/√

*Vε*; similarly, for the

*χ*

^{(3)}effect we would obtain

*α*,

_{ij}*β*~

_{i}*χ*

^{(3)}/

*Vε*. This proportionality to 1/√

*V*and 1/

*V*carries over to more realistic field profiles (and in fact could be used to

*define*a modal volume for these effects).

## 4. Numerical validation

*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 2

*πc*/

*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}= 3

*ω*

_{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.

*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]

*ω*

_{1}and

*ω*

_{3}= 3

*ω*

_{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}= 3

*χ*

^{(3)}/4

*cε*. 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 Δ

*n*/

*n*≡

*χ*

^{(3)}max|

*E*|

^{2}/2

*ε*> 1%. This is why we also plot the maximum index shift Δ

*n*/

*n*in the same figure.

*ω*

_{1}/

*ω*

_{1}= Re[

*δω*

_{1}/

*ω*

_{1}]. As expected, when Δ

*ω*

_{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 Δ

*n*/

*n*and Δ

*ω*

_{1}/

*ω*

_{1}should be comparable, but this is not the case because Δ

*n*/

*n*is the maximum index shift while Δ

*ω*

_{1}/

*ω*

_{1}is due to an average index shift.)

*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 (2

*π*c/a),

*ω*

_{2}= 0.95454 (2

*π*c/a) and quality factors

*Q*

_{1}= 1286 and

*Q*

_{3}= 3726, respectively. Given these field patterns, we computed the

*α*and

_{ij}*β*, coefficients. We obtained the following coupling coefficients, in units of

_{i}*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 - 0.7244

*i*) × 10

^{-5}.

## 5. Complete frequency conversion

*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.

*χ*

^{(2)}medium is that there are no frequency-shifting terms (

*α*= 0), so the resonance condition

*ω*

_{2}= 2

*ω*

_{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 (

*τ*≫

_{e,k}*τ*). In this case, 100% conversion corresponds to setting

_{s,k}*s*

_{1-}= 0 in the steady-state. Using this fact, Eqs. 3-4 for an input source

*s*

^{+}(

*t*) =

*s*

^{1+}exp(

*iω*

_{1}

*t*) yields the following condition on the input power for 100% conversion:

*Q*

^{2}

_{1}

*Q*

_{2}was previously observed [5

**14**, 1351 (1997). [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]

*Q*

_{1}~

*Q*

_{2}, then this critical power scales as

*V*/

*Q*

^{3}where

*V*is the modal volume (recall that

*β*~ 1/√

*V*).

*χ*

^{(2)}medium, we plot the solution to the coupled-mode equations as a function of input power in Fig. 3. The 100% conversion at the predicted critical power is clearly visible. For this calculation, we chose modal parameters similar to the ones from the FDTD computation before:

*ω*

_{1}= 0.3,

*ω*

_{2}= 0.6,

*Q*

_{1}= 10

^{4},

*Q*

_{2}= 2 × 10

^{4}, with dimensionless

*β*

_{1}= (4.55985 - 0.7244) × 10

^{-5}.

*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

*α*= 0, which can be done in two ways. First, one could employ

_{ij}*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

*α*will vanish while the

_{ij}*β*integrals will not. (In practice, a

*α*≪

*β*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}+ Δ

*ω*

_{3}= 3(

*ω*

_{1}+ Δ

*ω*

_{1}). Equivalently, design the device for

*α*= 0 and then adjust the linear cavity frequencies

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

*α*is thereby forced to be zero, and we can also neglect external losses (absorption, etc.) as above, then 100% third-harmonic conversion (

_{ij}*s*

_{1-}= 0) is obtained when:

*Q*

_{1}~

*Q*

_{3}, then this critical power scales as

*V*/

*Q*

^{2}where

*V*is the modal volume (recall that

*β*~ 1/

*V*). This is precisely the scaling that was predicted for the power to obtain nonlinear bistability in a single-mode cavity [59]. Similarly, one finds that the energy density in the cavity (|

*a*|

_{i}^{2}) decreases proportional to

*V*/√

*Q*

_{1}

*Q*

_{3}.

*α*= 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

_{ij}*α*= 0. In this case, comparing with Fig. 2, we observe that complete frequency conversion occurs at a power corresponding to Δ

*n*/

*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*.

*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, 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]

*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.)

*χ*

^{(2)}medium will generally also have

*χ*

^{(3)}≠ 0, so if the power is large enough this could conceivably cause a frequency shift that would spoil the second-harmonic resonance even in the

*χ*

^{(2)}device. Here, we perform a simple scaling analysis to determine when this will occur. (Although the frequency shifting could potentially be compensated for as described above, one prefers that it be negligible to begin with.) In order to preserve the resonance condition, any fractional frequency shift Δ

*ω*/

*ω*must be much smaller than the bandwidth 1/

*Q*, or equivalently we must have

*Q*Δ

*ω*/

*ω*≪ 1. From above, Δ

*ω*~

*ωα*|

*a*|

^{2}, and |

*a*|

^{2}~ |

*s*

_{1+}|

^{2}

*Q*/

*ω*. Suppose that we are operating at the critical input power

*P*

^{(2)}for second-harmonic conversion, from Eq. 19. It then follows that we desire

*Q*Δ

*ω*/

*ω*~

*Q*

^{2}

*α*/

*ωP*

^{(2)}≪ 1. It is convenient to re-express this relationship in terms of

*P*

^{(3)}~

*ω*/

*βQ*

^{2}, the third-harmonic critical power from Eq. 19, by assuming

*α*~

*β*as discussed in the previous section. We therefore find that

*χ*

^{(3)}self-phase modulation can be ignored for

*χ*

^{(2)}second-harmonic generation as long as

*P*

^{(2)}/

*P*

^{(3)}≪ 1. As discussed in the concluding remarks, this is indeed the case for common materials such as gallium arsenide, where

*P*

^{(2)}/

*P*

^{(3)}≈ 1/30 for

*Q*~ 1000 and for typical values of the cavity lifetime and volume. Moreover, since

*P*

^{(2)}/

*P*

^{(3)}~ 1/

*Q*, one can make the ratio arbitrarily smaller if necessary (at the expense of bandwidth) by increasing

*Q*.

## 6. The Effect of Losses

*Q*. We also solve for the critical input power to achieve maximal conversion efficiency in the presence of losses.

*χ*

^{(2)}medium with a linear loss rate 1/

*τ*, we solve Eqs 3–4 for |

_{e,k}*s*

_{2-}|

^{2}and enforce the condition for maximal conversion efficiency:

*τ*=

_{k}*τ*, Eq. 22 gives 100% conversion and Eq. 21 reduces to Eq. 19. For small external losses

_{s,k}*τ*≪

_{s,k}*τ*, the optimal efficiency is reduced by the ratio of the loss rates, to first order:

_{e,k}*χ*

^{(3)}yields the following critical input power and optimal efficiency:

*τ*

_{2}replaced by

*τ*

_{3}.

*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)

*τ*of the following form, to lowest order in the loss:

_{e,k}*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:

## 7. Conclusion

*χ*

^{(3)}cavity. The motivation for this work was the hope that a doubly resonant cavity would lead to 100% conversion at very low input powers, and so we conclude our paper by estimating the critical power for reasonable material and geometry assumptions.

*χ*

^{(2)}≈ 145 pm/V and

*n*

_{2}= 1.5 × 10

^{-13}cm

^{2}/Wat 1.5

*μ*m. (Al doping is usually employed to decrease nonlinear losses near resonance [61

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]

*χ*

^{(2)}and

*χ*

^{(3)}effects, we can selectively enhance one or the other by choosing the cavity to have resonances at either the second or third harmonic. Many well confined optical cavity geometries are available at these wavelengths and have been used for nonlinear devices, such as ring resonators [62

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]

*Q*

_{1}= 1000,

*Q*

_{2}= 2000,

*Q*

_{3}= 3000, and a modal volume of 10 cubic half-wavelengths (

*V*≈ 10(

*λ*/2

*n*)

^{3}) with roughly constant field amplitude in the nonlinear material (worse than a realistic case of strongly peaked fields). In this case, the critical input power, from Eqs. 19–20, becomes approximately 70

*μ*W for second-harmonic generation and 2 mW for third-harmonic generation (with a moderate peak index shift Δ

*n*/

*n*≈ 10

^{-3}, justifying our first-order approximation)

*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

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**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

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