High-efficiency second-harmonic generation in doubly-resonant χ<sup>(2)</sup> microring resonators

doi:10.1364/OE.20.007526

High-efficiency second-harmonic generation in doubly-resonant χ⁽²⁾ microring resonators

Zhuan-Fang Bi, Alejandro W. Rodriguez, Hila Hashemi, David Duchesne, Marko Loncar, Ke-Ming Wang, and Steven G. Johnson »View Author Affiliations

Optics Express, Vol. 20, Issue 7, pp. 7526-7543 (2012)
http://dx.doi.org/10.1364/OE.20.007526

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▸ Article Outline
– Abstract
1. Introduction
2. Computational methods
3. 2D Design
4. 3D Design
5. Concluding Remarks
– References and links

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Abstract

By directly simulating Maxwell’s equations via the finite-difference time-domain (FDTD) method, we numerically demonstrate the possibility of achieving high-efficiency second harmonic generation (SHG) in a structure consisting of a microscale doubly-resonant ring resonator side-coupled to two adjacent waveguides. We find that ≳ 94% conversion efficiency can be attained at telecom wavelengths, for incident powers in the milliwatts, and for reasonably large bandwidths (Q ∼ 1000s). We demonstrate that in this high efficiency regime, the system also exhibits limit-cycle or bistable behavior for light incident above a threshold power. Our numerical results agree to within a few percent with the predictions of a simple but rigorous coupled-mode theory framework.

1. Introduction

In this paper, we develop concrete designs for ring-resonator devices (depicted schematically in Fig. 1) that achieve compact, high-efficiency second-harmonic generation (SHG: ω₁  ω₂ = 2ω₁) by second-order (χ⁽²⁾) optical nonlinearities. By employing a combination of abstract coupled-mode theory (CMT) models and full nonlinear Maxwell finite-difference time-domain (FDTD) simulations, we identify key design criteria for achieving low-power high efficiency operation. Our results extend earlier work which established the theoretical parameters of high-efficiency intra-cavity SHG independent of any particular geometry choice [2

H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant Kerr cavities,” Phys. Rev. A 79(1), 013,812 (2009). [CrossRef]

, 3

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear Optics and Crystalline Whispering Gallery Mode Cavities,” Phys. Rev. Lett. 92(4), 043,903 (2004). [CrossRef] [PubMed]

], and demonstrates the predictive accuracy of CMT when confronted with the complexities of realistic systems. Intra-cavity SHG in ring-resonator geometries is attractive because of the amenability of such geometries to lithographic fabrication in a variety of materials, but we show that special care is required to satisfy both the frequency-matching condition (ω₂ = 2ω₁) and also the symmetry-induced selection rules (below). Unlike previous designs for SHG in ring resonators, which focused on either large (millimeter) resonators with ultra-small bandwidths [4

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally Phase-Matched Second-Harmonic Generation in a Whispering-Gallery-Mode Resonator,” Phys. Rev. Lett. 104(15), 153,901 (2010). [CrossRef] [PubMed]

–6

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

] or on low-power schemes that ignore down-conversion (or depletion) [7

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

], our design focuses on micron (wavelength-scale) resonators with relatively small bandwidths and large (≫ 1%) conversion efficiencies, and fully incorporates competing effects such as bending losses, down-conversion, and instabilities, which become relevant in this regime. Key to our design is the presence of input/output coupling ports based on waveguides with intentional cutoff frequencies, that allow independent control of the input/output coupling lifetimes, and which play a crucial role in establishing the sensitivity, bandwidth, and critical power of the device. To illustrate these design principles, we begin with a 2d example, and then consider a full 3d design involving a AlGaAs resonator on oxide designed for input light at 1.55 μm, in which we predict ≈ 94% conversion efficiency with a Q of only 1000 (0.1% bandwidth), for an input power of 30 mW. (We expect that much lower powers can be readily achieved with further optimization, or at the expense of bandwidth and/or efficiency.) Although 2d calculations are quicker, we show that some aspects of the design actually simplify in 3d, as more complicated ring and waveguide structures turn out to be required in 2d in order to achieve the required frequency-matching. In 2d, we are able to validate the semi-analytical CMT against nonlinear FDTD simulations that capture the full complexities of the system. Such low-power high conversion efficiencies could find applications in high-frequency sources [8

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

–10

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]

], imaging [11

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25–32 (1994). [CrossRef]

], optical storage [12

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]

], and other spectroscopic devices [13

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

] that suffer from a lack of coherent lasers at various wavelengths.

Fig. 1: Schematic ring-resonator waveguide-cavity system: input light from a waveguide supporting a propagating mode of frequency ω₁ (input power P1, in²) is coupled to a ring-resonator cavity mode of frequency ω₁, converted to a cavity mode of twice the frequency ω₂ = 2ω₁ by a nonlinear χ⁽²⁾ process, and coupled out by another waveguide supporting a propagating mode of frequency ω₂ (the waveguide does not support a propagating ω₁ mode).

SHG has been studied in many different contexts [14

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]

–20

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).

], most commonly in waveguides [21

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

–23

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

], in the so-called undepleted pump limit in which down-conversion effects are negligible [7

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

, 24

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

–28

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]

]. More recently, as devices have become smaller and as these nonlinear processes have become more important for integrated and ultracompact applications, there has been increased interest in designing techniques and/or devices that lower the power requirements and dimensions, as well as increase the efficiency of these systems [29

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]

–33

S. Schiller, “Principles and Applications of Optical Monolithic Total-Internal-Reflection Resonators,” Ph.D. thesis, Stanford University, Stanford, CA (1993).

]. Microcavities serve as a promising avenue for such applications: when light is confined in a cavity, restricted to small modal volumes V and confined for long times Q, nonlinear effects are enhanced due to the increased field strength and temporal confinement of the interacting modes, leading to significantly lower power requirements [2

, 31

Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett. 32(7), 826–828 (2007). [CrossRef] [PubMed]

, 32

L. Caspani, D. Duchesne, K. Dolgaleva, S. J. Wagner, M. Ferrera, L. Razzari, A. Pasquazi, M. Peccianti, D. J. Moss, J. S. Aitchison, and R. Morandotti, “Optical frequency conversion in integrated devices,” J. Opt. Soc. Am. B 28(12), A67–A82 (2011). [CrossRef]

, 34

Y. Dumeige and P. Feron, “Stability and time-domain analysis of the dispersive tristability in microresonators under modal coupling,” Phys. Rev. A 84(4), 043,847 (2011). [CrossRef]

] in addition to novel dynamical effects [3

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear Optics and Crystalline Whispering Gallery Mode Cavities,” Phys. Rev. Lett. 92(4), 043,903 (2004). [CrossRef] [PubMed]

, 35

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

]. In doubly-resonant cavities, such benefits are obtained both at the pump ω₁ and harmonic ω₂ frequencies, leading to orders of magnitude improvements in efficiency and power requirements [2

,24

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

,36

M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction (Wiley, New York, 1989).

]. Specifically, SHG in doubly-resonant cavities bears a number of unique features [2

]: First, there exists a critical power P_crit, proportional to V/Q³, at which 100% efficiency is achieved. For input powers below and above this critical power, the SHG efficiency is significantly reduced. Second, for large powers, the stability of the conversion process is compromised due to bistable behaviors and the system exhibits limit cycles (mathematically known as Hopf bifurcations [37

J. S. Levy, M. A. Foster, A. L. Gaeta, and M. Lipson, “Harmonic generation in silicon nitride ring resonators,” Opt. Express 19(12), 11,415–11,421 (2011). [CrossRef]

]). Thus far, and recently, experiments have demonstrated significantly higher conversion efficiencies ≲ 10% in these doubly-resonant cavities [5

P. S. Kuo and G. S. Solomon, “On- and off-resonance second-harmonic generation in GaAs microdisks,” Opt. Express 19(18), 16,898–16,918 (2011). [CrossRef]

], albeit in large-etalon cavities with narrow-bandwidths (Q ≳ 10⁸) for which it is only feasible to probe the undepleted (≪ 100%) conversion regime.

Recently, ring resonator geometries have become an attractive venue for realizing efficient and device-integrated SHG [4

–7

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

, 31

–33

S. Schiller, “Principles and Applications of Optical Monolithic Total-Internal-Reflection Resonators,” Ph.D. thesis, Stanford University, Stanford, CA (1993).

, 38

C. Xiong, W. Pernice, K. K. Ryu, C. Schuck, K. Y. Fong, T. Palacios, and H. X. Tang, “Integrated GaN photonic circuits on silicon (100) for second harmonic generation,” Opt. Express 19(11), 10,462–10,470 (2011). [CrossRef]

–40

G. Nielson, D. Seneviratne, F. Lopez-Royo, P. Rakich, Y. Avrahami, M. Watts, H. Haus, H. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” Photonics Technology Letters, IEEE 17(6), 1190 –1192 (2005). [CrossRef]

]. Ring resonators, like waveguides, offer an advantage over other geometries in that they readily yield modes satisfying the frequency-matching requirement ω₂ = 2ω₁. To date, however, most works have focused on large (millimeter) ring resonators with long modal lifetimes (Q ≳ 10⁸), in which the long lifetimes compensate for the relatively large modal volumes [4

, 5

P. S. Kuo and G. S. Solomon, “On- and off-resonance second-harmonic generation in GaAs microdisks,” Opt. Express 19(18), 16,898–16,918 (2011). [CrossRef]

, 32

, 38

–40

], or on smaller-scale (micron) resonators with more moderate bandwidths (Q = 10⁴) that operate at low efficiencies and therefore neglect down conversion effects [6

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

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

], a drawback for applications involving ultracompact integrated devices requiring broad-bandwidth operations, e.g. integrated optical switches [41

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

]. In this paper and unlike previous works, we focus on small (micron-scale) doubly-resonant χ⁽²⁾ resonators supporting modes at frequencies ω₁ and ω₂ = 2ω₁, with moderate lifetimes Q ∼ 1000s. Here, the significantly smaller Qs are offset by the smaller V, and we find that nearly 100% conversion efficiency is feasible for input powers in the mW range. Our geometry is advantageous in several ways: First, the smaller V translates into more compact operation as well as lower power requirements. Second, in addition to providing a larger bandwidth of conversion, a lower Q also makes the device less sensitive to manufacturing errors—a key consideration since the ring parameters must be carefully chosen to match the frequencies of the fundamental and second-harmonic modes. Third, as noted below, and unlike large-etalon or waveguide setups, the conventional quasi-phase matching requirement is replaced by a selection rule on the symmetry of the modes of the resonator that proves significantly simpler to satisfy [42

M. Fejer, G. Magel, D. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Elec. 28(11), 2631–2654 (1992). [CrossRef]

, 43

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Letters to Nature 391, 463–466 (1997).

], both experimentally and theoretically. The lack of a quasi-phase matching condition is attractive in that it opens up the possibility of studying SHG in materials (e.g. Gallium Arsenide) that are highly nonlinear and practical, but for which phase-matching is difficult because of material dispersion and other issues [44

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q ring resonators in thin silicon-on-insulator,” Appl. Phys. Lett. 85(16), 3346–3347 (2004). [CrossRef]

]. A challenging aspect of designing low-power SHG in small ring resonators is the well-known tradeoff between modal volume V and radiation or bending losses (lifetime Q_rad) [45

I. Tomita, M. Asobe, H. Suzuki, J. Yumoto, and Y. Yoshikuni, “Broadband quasi-phase-matched second-harmonic generation in a nonlinear photonic crystal,” J. of Appl. Phys. 100(2), 023,120 (2006). [CrossRef]

]. (Such a tradeoff is absent in complete-gap photonic crystals [2

] where SHG has also been considered [46

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light , 2nd ed. (Princeton University Press, 2008). URL http://ab-initio.mit.edu/book.

], but in these systems it seems challenging to procure a design that satisfies the frequency matching condition because of the difficulty of obtaining simultaneous complete gaps at harmonic frequencies in more than one dimension [47

K. Rivoire, S. Buckley, and J. Vuckovic, “Multiply resonant high quality photonic crystal nanocavities,” Appl. Phys. Lett. 99(1), 013,114 (2011). [CrossRef]

]. Recent designs involving PhC nanobeam cavities [48

K. Rivoire, S. Buckley, and J. Vuckovic, “Multiply resonant photonic crystal nanocavities for nonlinear frequency conversion,” Opt. Express 19(22), 22,198–22,207 (2011). [CrossRef]

, 49

K. Rivoire, Z. Lin, F. Hatami, and J. Vuckovic, “Sum-frequency generation in doubly resonant GaP photonic crystal nanocavities,” Appl. Phys. Lett. 97(4), 043,103 (2010). [CrossRef]

] are promising for both sum-frequency generation [50

K. Rivoire, Z. Lin, F. Hatami, W. T. Masselink, and J. Vuckovic, “Second harmonic generation in gallium phosphide photonic crystal nanocavities with ultralow continuous wave pump power,” Opt. Express 17(25), 22,609–22,615 (2009). [CrossRef]

] and SHG [51

K. Rivoire, S. Buckley, F. Hatami, and J. Vuckovic, “Second harmonic generation in GaP photonic crystal waveguides,” Appl. Phys. Lett. 98(26), 263,113 (2011). [CrossRef]

,52

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Steven, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm. 181, 687–702 (2010). [CrossRef]

].) More importantly, it is well known that side-coupling a waveguide to a small resonator yields additional (and ultimately prohibitive) radiation losses [45

], and so we design our geometry to overcome these and other limitations. To aid our design, we employ a CMT formalism that allows us to predict the critical power at which the highest efficiency occurs, and the largest such efficiency, in terms of the lifetimes, modal volumes, and coupling coefficients of the linear modes of the resonator [2

]. We supplement this with a full Maxwell simulation via the FDTD method, which incorporates all the complexities of the full system that are neglected by CMT, and we find excellent agreement with CMT.

2. Computational methods

In order to develop the harmonic-generation design, we needed to compute microcavity modes, frequencies, and lifetimes (Q), as well as waveguide dispersion relations. The final design was evaluated both semi-analytically with coupled-mode theory (CMT) and with a full nonlinear Maxwell simulation. We began by studying a two-dimensional (2d) model system, and continued to full 3d calculations. The computational methods for these calculations are described here.

The basic cavity design is that of a ring resonator coupled with one or two adjacent waveguides, as depicted in Fig. 1. To begin with, we studied the isolated cavities, uncoupled to any waveguide. Since the isolated waveguide is axisymmetric, it can be modeled in cylindrical coordinates. We did so using a free finite-difference time-domain (FDTD) software package (Meep) [53

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107(17), 6756–6769 (1997). See erratum. [CrossRef]

]. The simulation cell is surrounded by a perfectly matched layer (PML) absorbing boundary region. The use of cylindrical coordinates in this simulation reduces the 2d problems to a 1d problem, thereby reducing simulation times significantly. Another advantage is that, the angular dependence of the fields in systems with continuous rotational symmetry can be given by the angular momentum parameter (input variable) m, which is easy to control. To begin with, we inserted a broad Gaussian pulse in the structure in order to excite all of the (TM polarized) modes within a chosen bandwidth and with a fixed m; we then re-ran the simulation with a narrow-band source around each mode and outputted the corresponding fields at the end. The resonance frequency and lifetime Q_rad were obtained by Harminv, which is a free program to solve the problem of harmonic inversion [54

V. A. Mandelshtam and H. S. Taylor, “Erratum: “Harmonic inversion of time signals and its applications”,” J. Chem. Phys. 109, 4128 (1998). [CrossRef]

]. The waveguide modes were computed using an iterative eigenmode solver in a planewave basis, using a freely available software package (MPB) [55

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001).

The combined waveguide-cavity system, with waveguides adjacent to the ring resonator, is not axisymmetric and requires a full 2d FDTD calculation. Bringing in the waveguides, there are two decay mechanisms for the modes in this cavity: the mode can decay into the adjacent waveguides, and it can radiate into the surrounding air. The total dimensionless decay rate 1/Q_tot can be written as the sum of two decay rates: 1/Q_tot = 1/Q_w + 1/Q_rad, where 1/Q_w and 1/Q_rad are the waveguide and radiative decay rates, respectively. We obtain Q_tot from a filter-diagonalization analysis of the field decay in FDTD [53

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107(17), 6756–6769 (1997). See erratum. [CrossRef]

]. However, we also need to know Q_w and Q_rad individually, both of which are modified for different ring–waveguide separations. Therefore, for each separation, we computed the linear transmission spectrum in FDTD. Then, comparing with the transmission equation obtained from coupled mode theory [47

K. Rivoire, S. Buckley, and J. Vuckovic, “Multiply resonant high quality photonic crystal nanocavities,” Appl. Phys. Lett. 99(1), 013,114 (2011). [CrossRef]

T (ω) = \frac{P_{out}}{P_{in}} = \frac{ω_{0}^{2} {(1 / Q_{w} - 1 / Q_{rad})}^{2} + 4 {(ω - ω_{0})}^{2}}{ω_{0}^{2} {(1 / Q_{w} + 1 / Q_{rad})}^{2} + 4 {(ω - ω_{0})}^{2}},

(1)

one can solve for both Q_rad and Q_w given T (ω₀) (the minimum T) and Q_tot.

Given the waveguide modes, frequencies, and Q values, the resulting nonlinear system can be modelled semi-analytically using the coupled-mode equations of Ref. [2

]. A key parameter of this model is the overlap integral β₁ between the fundamental (ω₁) and harmonic (ω₂ = 2ω₁) modes, which is computed from the following integral:

β_{1} = \frac{1}{4} \frac{\int d^{3} x \sum_{i j k} ε_{0} χ_{i j k}^{(2)} [E_{1 i}^{*} (E_{2 j} E_{1 k}^{*} + E_{1 j}^{*} E_{2 k})]}{(\int d^{3} x ε {| E_{1} |}^{2}) {({\int d^{3} x ε | E_{2} |}^{2})}^{1 / 2}},

(2)

This coupling coefficient can be obtained by applying perturbation theory to Maxwell’s equations in the presence of a χ⁽²⁾ nonlinearity, as explained in Ref. [2

]. Within the CMT, the β₁ coefficient is related to the rate of energy down-conversion. Similarly, there is a corresponding coefficient β₂ that relates to the rate of frequency up-conversion [2

]. As shown in [2

], these two coefficients are related to one another via conservation of energy:

ω_{1} β_{1} = ω_{2} β_{2}^{*}

CMT makes several approximations: it assumes that cavity-waveguide and cavity-radiation coupling is weak (high Q), it neglects nonlinear coupling to modes not at 2ω₁ (the rotating-wave approximation), it assumes that the input waveguide couples only to a single direction mode of propagation around the ring (clockwise or counterclockwise), and correspondingly that each ring mode couples out to only a single direction of propagation in the waveguides (despite the fact that we use waveguides that are not identical to the ring structure for reasons described below). To ensure that the full complexities of the nonlinear Maxwell equations in this geometry were accurately captured by the coupled-mode equations, we also performed fully nonlinear FDTD simulations (where the nonlinear constitutive equations are solved by Padé approximants [53

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107(17), 6756–6769 (1997). See erratum. [CrossRef]

]). In particular, we excited the input waveguide with a continuous plane wave of frequency ω₁, and computed the SHG power escaping through the output waveguide at ω₂. The nonlinear FDTD calculations were then checked against the CMT predictions, using the frequencies, decay rates, and β₁ coefficient computed from a set of linear FDTD simulations.

In the absence of losses or reflections, 100% conversion is achieved in the steady state. Including losses [2

] (Q_rad finite), the critical power can be related to the various model parameters,

{| P_{crit} |}^{2} = \frac{ω_{1} Q_{1}}{2 {| β_{1} |}^{2} Q_{w, 2} Q_{w, 1}^{3}},

(3)

and ends up scaling as V/Q³ [2

]. Hence, one can in principle obtain very low-power efficient harmonic conversion by increasing Q and/or decreasing V (i.e., maximize β) [2

]. The efficiency decreases if the power is either too low (in the linear regime) or too high (dominated by down-conversion) compared to |P_crit|².

3. 2D Design

In Ref. [2

], we showed, conceptually, that 100% second-harmonic conversion efficiency, in the absence of loss, can be obtained by coupling a single input channel to a single output channel via a cavity with two resonant frequencies, ω₁ and ω₂ = 2ω₁, that are coupled by a second-order nonlinearity. However, a number of difficulties must be addressed in order to obtain a realistic cavity design that achieves the desired characteristics.

First, the cavity should be compact and support modes of the requisite frequencies. Generally, in order for them to couple nonlinearly, one must also ensure that the modes of the cavity satisfy certain selection rules arising from the presence of any cavity symmetries (exact or approximate), a generalization of the “phase-matching” requirement of earlier works [43
A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Letters to Nature 391, 463–466 (1997).
, 44
T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q ring resonators in thin silicon-on-insulator,” Appl. Phys. Lett. 85(16), 3346–3347 (2004). [CrossRef]
].
Second, the cavity quality factors Q should be carefully controlled, as they affect several tradeoffs:
- – The (fractional) bandwidth of conversion is 1/max(Q₁, Q₂).
- – The critical power is proportional to $1 / Q_{1}^{2} Q_{2}$ .
- – The sensitivity to perturbations in the structure is determined by min(Q₁, Q₂).
Therefore, it is desirable to have a design in which one can choose Q₁ and Q₂ independently (e.g. to obtain Q₁ = Q₂ to minimize power for a given bandwidth).
Third, the design of the input/output waveguide(s) is critical to ensure that the system supports only a single incoming and a single outgoing wave at both ω₁ and ω₂—additional channels will lower the efficiency (unless they have much larger coupling Q) [47
K. Rivoire, S. Buckley, and J. Vuckovic, “Multiply resonant high quality photonic crystal nanocavities,” Appl. Phys. Lett. 99(1), 013,114 (2011). [CrossRef]
].

Before delving into the details of our design, we briefly summarize our findings. To obtain strongly confined modes at both fundamental and harmonic frequencies, the most attractive candidate cavities seem to be microring cavities and related geometries (whereas mechanisms like photonic bandgaps typically cannot confine light at both ω and 2ω in more than one dimension). However, we find (below) that the basic ring-resonator design must be somewhat modified to obtain a strong coupling (β₁) between the ω and 2ω modes. We chose to operate at total Q’s of only a few thousand, which allows our design to cover significant bandwidths (e.g. a 10 Gbit/s telecom channel) and this proved computationally convenient because it required simulations of only a few thousand optical periods. It also means that fabrication errors of up to about 0.1% frequency-mismatch can be tolerated (although some mismatch can be compensated by post-fabrication tuning [6

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

, 43

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Letters to Nature 391, 463–466 (1997).

]). As described below, we found that obtaining much smaller Qs by decreasing the ring-waveguide separation is also possible (at the cost of higher critical power), although this is ultimately limited by the increasing radiation losses. For the same ring-waveguide separation, it is well known that higher-frequency modes will have higher Q, so coupling the ring to a single waveguide would yield Q₂ ≫ Q₁. Thus our design goals above favored two waveguides coupled to the ring: one waveguide for the ω₁ input and another for the ω₂ output. The third requirement of a single in/out channels led to an unusual requirement in the waveguide design: the output (ω₂) waveguide was designed to have a low-frequency cutoff > ω₁ in order to eliminate its ω₁ guided mode.

As a 2d proof of concept, we considered LiNbO₃ (permittivity ε = 4.84) as the nonlinear material. When it is poled in the vertical (z) direction (in the direction of the axis of symmetry), LiNbO₃ has a nonlinear susceptibility

χ_{z z z}^{(2)} \approx 41 \times 10^{- 12} m / V

[42

M. Fejer, G. Magel, D. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Elec. 28(11), 2631–2654 (1992). [CrossRef]

] coupling the E_z field to itself, which means that we can work with purely TM-polarized (E || ẑ) waves. (The use of a diagonal χ⁽²⁾ component also simplifies FDTD calculations [53

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107(17), 6756–6769 (1997). See erratum. [CrossRef]

].) Our design is such that no additional quasi-phase-matching is necessary, e.g. we need not resort to schemes like alternatingly poling the LiNbO₃ in the ±z directions to have an oscillating

χ_{zzz}^{(2)}

[56

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. O. H. Suche, R. Nouroozi, and Y. Min, “Integrated Optical Devices in Lithium Niobate,” Opt. Photon. News 19(1), 24–31 (2008). [CrossRef]

] (which requires complicated electrical contacting for the oscillating poling field).

3.1. Ring-resonator design

Previously, achieving efficient SHG in waveguides or Fabry–Perot etalons required techniques to obtain “phase-matching” of the fundamental and harmonic modes [44

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q ring resonators in thin silicon-on-insulator,” Appl. Phys. Lett. 85(16), 3346–3347 (2004). [CrossRef]

]. Phase matching is a selection rule arising from the approximate translational symmetry for propagation over long uniform regions, according to which the fundamental and harmonic modes must have the same phase velocities in order to couple efficiently. In microcavities where the fields are confined to within a few wavelengths, such a constraint is instead replaced by selection rules resulting from symmetry considerations, which determine whether the overlap integral in Eq. (2) is nonzero. In our geometry, involving cylindrically symmetric cavities, the fields can be chosen to have azimuthal dependence ∼ e^imϕ, determined by the conserved angular momentum “quantum number” m ∈ 𝕑. By simple inspection of Eq. (2) for

χ_{zzz}^{(2)}

coupling, this leads to the requirement m₂ = 2m₁, where m₁ and m₂ are the corresponding quantum numbers of the fundamental and harmonic modes, respectively. Because m is constrained to integer values, perturbing the cavity parameters does not alter the m of a given mode, so the m₂ = 2m₁ condition is easy to satisfy and robust. On the other hand, perturbing the cavity parameters does change the frequencies of the modes at given m values, so the key difficulty is to find modes at a given pair of m’s that satisfy ω₂ = 2ω₁.

For any given geometry and given m, we compute the resonant modes of the linear system by FDTD in cylindrical coordinates, which reduces to a 1d problem in r for a given m. We then compute the modes at m₁ and 2m₁ for various choices of m₁, and vary the geometry until we find a pair of modes with ω₂ = 2ω₁. (More precisely, the finite bandwidth means that there is some tolerance ∼ 1/Q on the frequency mismatch Δω = ω₂ – 2ω₁ ≲ ω₂/Q.) We began exploring the simplest possible space of designs: we considered a fixed ring width a and varied the inner radius R, as depicted in the right inset of Fig. 2. However finding modes that satisfy both phase- and frequency-matching requirements for this single-ring design, at chosen index contrast, turned out to be unfeasible. In particular, Fig. 2 shows the frequency mismatch Δω (units of 2πc/a) corresponding to two phase-matched modes with m₁ = 15 and m₂ = 30, as a function of the ring radius R. As observed, Δω is always positive and large ≳ 0.04 (2πc/a) relative to the mode bandwidths (Q ∼ 1000s) considered here, and we obtained similar results for other values of m₁ and other pairs of modes. We therefore abandoned the single-ring design and instead considered a double-ring structure, depicted schematically in the left inset of Fig. 2. (Note that Ref. [7

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

] considered a set of different double-ring structures for SHG, consisting of either multi-layered stacks in the z direction or adjacent rings, in larger resonator geometries. Similar multilayering designs have been studied in the context of phase-matching in waveguides as well [44

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q ring resonators in thin silicon-on-insulator,” Appl. Phys. Lett. 85(16), 3346–3347 (2004). [CrossRef]

].) The reason to consider such a structure is based on the fact that, for the single ring resonator, the ω₂ = 2ω₁ condition was most nearly matched by a pair of modes where the ω₁ mode has approximately even symmetry through the middle of the waveguide and the ω₂ mode was approximately odd-symmetrical. We therefore expect that an air groove introduced within the resonator will affect the ω₁ mode more than the ω₂ mode (which has a node), thereby allowing us to shift ω₁ into ω₂/2. Furthermore, by introducing the air groove off-center in the resonator, we break the approximate mirror symmetry (that is only slightly broken by the small curvature), and thereby allow the ω₁ mode and ω₂ modes to have a much larger overlap integral. (Exactly even and odd modes would not couple at all, a fact we return to in 3d.) Thus, this structure yields two additional degrees of freedom, the widths of the inner and outer rings, w₁ and w₂ respectively, with which it is possible to shift ω₁ relative to ω₂. After a relatively small search through the design space, we settled on a structure having w₁ = 0.45a and w₂ = 0.35a, where a denotes the total width of the double-ring structure including the air groove of width a – w₁ – w₂ = 0.2a (labeled in the left inset of Fig. 2). For this configuration, we find that, for the same azimuthal modes m₁ = 15 and m₂ = 30 as above, Δω switches sign as R increases and is zero at R ≈ 4.51a, as shown by the red line of Fig. 2. Using the field patterns of these modes, we also computed the coupling coefficient β₁, which we found to be β₁ ≈ 3.178×10⁻⁵ (in units of χ⁽²⁾/a^3/2).

Fig. 2: Plot of the frequency difference Δω = ω₂ – 2ω₁ (units of 2πc/a) of two LiNbO₃ ring-resonator modes of frequencies ω₁ and ω₂, and azimuthal momentum m₁ = 15 and m₂ = 30, respectively, corresponding to two different ring-resonator geometries (insets), as a function of inner radius R. The blue and red lines correspond to the single-ring (right inset) and double-ring resonators.

As described below, the final ring parameters are slightly perturbed by the presence of the input/output waveguides, so our design procedure was to design the isolated ring, design the coupling waveguides as described below, and then tweak the ring design to restore the ω₂ = 2ω₁ condition. We obtained a final ring radius of R = 4.585a for frequencies ω₁ = 0.277172 · 2πc/a (vacuum wavelength 3.6a) and ω₂ = 0.554344 · 2πc/a (vacuum wavelength 1.8a).

3.2. Input/output coupling waveguides

Given a ring resonator with appropriate modes as described above, we must then design the coupling to adjacent waveguides so that they have the desired coupling lifetimes Q_w,₁ and Q_w,₂ at ω₁ and ω₂. In order to obtain small radiation losses, these coupling lifetimes should be much smaller than the radiative lifetimes Q_rad,₁ and Q_rad,₂ (approximately 10⁵ and 10⁷ for the isolated ring), the so-called “overcoupled” regime. For this reason, and also to obtain a reasonable bandwidth of conversion, and to limit computation times, we chose to work with Q_w ∼ 10⁴. Furthermore, we don’t want to have to bring the waveguide too close to the ring, which would require a high computational resolution and might also induce additional radiative scattering.

In order to obtain Q_w values that are not too large for moderate ring–waveguide separations, we phase-match the waveguide mode to the ring-resonator mode. Conceptually one designs the waveguide to have a phase velocity equal to m/rω, the phase velocity of the ring mode, but this condition is ambiguous because the “phase velocity” of the ring mode varies with radius r. For a large R, the difference between R and R+a is negligible and so one can simply use a waveguide of the same width as the ring [57

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

]. In our case, however, R/a is too small for this to be a good approximation and an identical waveguide is not optimal. Instead, we varied the waveguide width to minimize Q_w for a given ring–waveguide separation d. Furthermore, this allowed us to have good coupling between the double-ring structure and a simple dielectric waveguide with no air groove, as well as with the cutoff waveguide described below. For example, for the ω₁ mode we found that this procedure corresponded to an optimal phase velocity m/rω with r = R + 1.3a.

Fixing the waveguide width and varying the ring–waveguide gap d, we obtain the plot of the Q_w,₁ and Q_rad,₁ in Fig. 3(a), which illustrates two effects. First, the coupling Q_w decreases exponentially with d, thanks to the exponentially increasing overlap of the evanescent tails of the waveguide and cavity modes. Second, although for large d the losses Q_rad asymptotes to a constant given by the radiation loss of the isolated ring, for sufficiently small d the radiation losses increase due to scattering of the cavity mode from the waveguide. As explained in Sec. 2, we obtain Q_rad and Q_w from a combination of mode decay and transmission simulations, and several of these transmission spectra are shown in Fig. 3(b). When Q_rad ≫ Q_w, the linear ring–waveguide system approaches an all-pass filter with 100% transmission (but a resonant delay), while as Q_w approaches Q_rad for large d the radiation loss increases and one observes a resonant dip in the transmission.

Fig. 3: (a) Semilog plot of the radiative (Q_rad), waveguide-coupling (Q_w), and total (Q_tot) lifetimes of the ω₁ mode of Fig. 5, as a function of the ring-waveguide separation d₁. (b) Corresponding transmission spectrum at various separations.

In principle, we could use a single waveguide to couple ω₁ into the ring resonator and ω₂ out after SHG. However, for the same ring–waveguide separation d the ω₂ mode will normally have Q_w,₂ ≫ Q_w,₁ because the evanescent tails of a waveguide mode decay more rapidly at higher frequencies. (Furthermore, if the waveguide is optimized to couple ω₁, the coupling will not be optimized at ω₂.) For example, if we designed the ring to couple to a single dielectric waveguide at ω₁ with a gap d₁ = 0.5a, we would obtain Q_w,₁ = 191 and Q₂_,w = 155, 000, and the difference increases for larger d₁. Instead, therefore, we couple ω₂ out with a second waveguide that is optimized to couple at that frequency, on the other side of the ring. This introduces a new problem, however: we must prevent the second waveguide from coupling ω₁, since CMT predicts that the introduction of multiple ω₁ channels will lower the attainable conversion efficiency significantly. We solve this problem by designing the second waveguide to have a low-frequency cutoff > ω₁, so that ω₁ cannot couple out via that channel. In 3d below, a low-frequency cutoff is introduced simply by the waveguide substrate [58

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, 2002).

–60

K. K. Y. Lee, Y. Avniel, and S. G. Johnson, “Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides,” Opt. Express 16, 9261–9275 (2008). [CrossRef] [PubMed]

]. In our 2d model system here we obtain a cutoff by the simple expedient of placing the ω₂ waveguide next to a perfect electric conductor (PEC). [A low-index dielectric “substrate” (wall) could have been used in 2d to introduce a cutoff similar to 3d, but PEC is computationally convenient and yields qualitatively the same results—the only purpose of the 2d simulations is to qualitatively explore the design space and validate the theory in preparation for the more realistic 3d design of Sec. 4.]

By this procedure, we obtain an input waveguide of width 0.5a and an output waveguide of width 0.35a (adjacent to PEC), with dispersion relations shown in Fig. 4. Note the cutoff in the ω₂ mode, as desired. In Fig. 5, we show the field distribution of the two modes in the ring resonator coupled with these two waveguides. Note that the ω₂ mode has negligible leakage into the upper waveguide because the coupling Q in that direction is so much larger. Also, note that each ring mode (propagating counter-clockwise) couples primarily to waveguide modes traveling in the same direction. This is critical in order to mimic the theoretically optimal situation as described in Ref. [2

]: ω₁ must enter the resonator from a single channel and exit in a single channel for ω₁ and in a single channel for ω₂. For the nonlinear simulations below, we use larger values of d₁ and d₂ in order to obtain a lower critical power. In particular, we use d₁ = 1.05a and d₂ = 0.7a, obtaining Q_w,₁ = 2000 and Q_w,₂ = 8992.

Fig. 4: Band diagram or frequency ω (units of 2πc/a) as a function of wave-vector k (units of 2π/a), corresponding to the fundamental (red line) and second-order (blue line) modes of two different LiNbO₃ waveguides of thickness w₁ = 0.5a and w₂ = 0.35a, respectively. Here, a denotes the thickness of the double-ring resonator of Fig. 2. The bottom and upper insets show the E_z field profile (blue/white/red denote positive/zero/negative amplitude) of two different modes, with frequencies ω₁ = 0.277(2πc/a) and ω₂ = 2ω₁, and corresponding wave-vectors k₁ = 0.39(2π/a) and k₂ = 2k₁, respectively.

Fig. 5: E_z field snapshot of two double-ring (Fig. 2 inset) resonator modes propagating counter-clockwise, with frequencies ω₁ = 0.277(2πc/a) (left) and ω₂ = 2ω₁ (right) and azimuthal momentum m₁ = 15 and m₂ = 30 (effective k₁ = 0.39(2π/a) and k₂ = 2k₁). The ring resonator is side-coupled to two adjacent waveguides, separated by a distance d₁ = d₂ = 0.5a, supporting phase-matched propagating modes at ω₁ (top waveguide) and ω₂ (bottom waveguide).

3.3. Nonlinear characterization and SHG efficiency

Given these parameters of the linear resonator system, CMT can predict the behavior of the nonlinear system when a χ⁽²⁾ is introduced. In particular, it predicts that 82% SHG efficiency should be obtained at a certain critical power (< 100% because of radiations from the finite Q_rad/Q_w). However, as noted in Sec. 2, CMT makes many approximations with respect to the full Maxwell equations, and while each of these approximations seems justified in the present case, it is desirable to validate the CMT predictions against a full nonlinear simulation as described in Sec. 2.

In Fig. 6(a), we plot the SHG conversion efficiency (output to the lower waveguide) versus the input power at ω₁, incident from the upper-left port, as computed by both CMT and by nonlinear FDTD (run long enough to reach steady state from zero initial fields). The nonlinear FDTD results agree well with the CMT: the FDTD efficiency peaks at the predicted critical power with a maximum efficiency of 78%, which is reasonable agreement especially considering that it is difficult to determine the resonator Q value from the transmission fits with more than a few percent accuracy. A snapshot of the nonlinear FDTD simulation is shown in Fig. 6(c), in which both the ω₁ input and the ω₂ output are visible (with a complicated superposition of the two modes in the cavity). Most of the 22% of unconverted power is lost to radiation (visible in the plot) due to the finite Q_rad/Q_w, but the imperfections represented by these losses also give rise to a few percent of the ω₁ power escaping into the upper-right port and ≈ 2% of the ω₂ power escaping to the lower-right port.

Fig. 6: (a) Plot of SHG efficiency η = P_SH/P_in versus P_in, for the double-ring resonator system of Fig. 5 with waveguide-separations d₁ = 1.05a and d₂ = 0.7a, obtained both via FDTD simulations (red circles) and CMT (blue line). The gray region denotes the presence of instabilities that lead to limit-cycle behavior. (b) An example of a limit cycle at point A. (c) E_z temporal snapshot of the nonlinear conversion process at point B (the efficiency peak).

Another intriguing prediction of CMT for intra-cavity SHG is that, once the input power is significantly larger than the critical power, the steady-state solution is replaced by a “self-pulsing” solution in which a constant input power at ω₁ produces an oscillating output power at ω₂ [61

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]

]. These “limit cycles” occur in the shaded region of Fig. 6(a), and because the system never reaches a steady state simply taking the efficiency at the end of the simulation gives a somewhat noisy value as seen in this plot from the CMT data. A plot of efficiency vs. time, from the CMT for P_in = 2P_c, is shown in Fig. 6(b), and the limit cycles are clearly visible (after an initial transient). Note that similar limit cycle (self-pulsing) behaviors in doubly-resonant systems have been theoretically predicted, see e.g. [3

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear Optics and Crystalline Whispering Gallery Mode Cavities,” Phys. Rev. Lett. 92(4), 043,903 (2004). [CrossRef] [PubMed]

, 35

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

, 61

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]

, 62

K. Grygiel and P. Szlatchetka, “Chaos in second-harmonic generation of light. The case of a strain of pulses.” Opt. Comm. 91, 241–246 (1992). [CrossRef]

]; such limit cycles form a kind of optical oscillator (clock) with a period in the hundreds of GHz or THz (possibly lower depending on the cavity parameters). We have also examined the FDTD behavior in this regime and verified that the FDTD simulations also exhibit limit cycles of the expected period and amplitude.

When it comes to realistic numbers, we are most interested in the 3d results of the next section, but it is still interesting to estimate the critical power for the 2d system in experimental units, assuming a vacuum wavelength λ₁ = 1.55μm for ω₁ (and hence a = 0.277 × 1.55 = 0.43μm). To obtain a power in W from a 2d calculation, we must assume a certain finite thickness in the z direction, and in this case we assume 200 nm (good vertical confinement), and obtain a critical input power of 800 mW for LiNbO₃ at these Q values.

The < 100% efficiency was due to the finite Q_rad/Q_w and in particular the limiting factor is the loss at ω₁ (Q_rad,₁/Q_w,₁ = 9847/1998 ≈ 4.9), so we can obtain higher efficiency by making Q_w,₁ smaller. This is accomplished by bringing the input waveguide closer to the ring resonator. Substituting the Q values from Fig. 3 into CMT, we predict an increase in efficiency with decreasing d₁ as shown by the black curve in Fig. 7. The inset of Fig. 7 shows the results in both CMT and FDTD for a separation d₁ = 0.9a, for which Q_rad/Q_w = 9616/1043 ≈ 9.2, and obtain 90% and 85% efficiency from CMT and FDTD, respectively.

Fig. 7: Maximum efficiency vs. separation distance between input waveguide and ring resonator. (inset: Conversion efficiency from CMT and FDTD in the case d₁ = 0.9a)

4. 3D Design

We now apply the same basic principles that we validated in the 2d example to a more realistic 3d design. In 3d, we will consider SHG from λ₁ = 1.55μm in a AlGaAs film (aluminum composition x = 70%) bonded to an SiO₂ substrate. Note that there are many other possible material choices, e.g. GaAs, GaP and ZnSe. The main criterion for choosing a material is not the usual phase-matching requirement that has driven previous theoretical and experimental works to consider AlGaAs [33

S. Schiller, “Principles and Applications of Optical Monolithic Total-Internal-Reflection Resonators,” Ph.D. thesis, Stanford University, Stanford, CA (1993).

], but rather the requirement that the material exhibit relatively small absorption at these wavelengths. Note also that the choice of 70% Aluminum content is not special; for example, x = 30% also exhibits relatively low absorption, i.e. Q_abs ≫ 10⁴, and yields even larger index contrasts, making the design problem even easier.) As in 2d, we will design a ring resonator to have resonances at ω₁ and ω₂ = 2ω₁, which is side-coupled to two waveguides for input and output, as shown schematically in Fig. 8. Also as in 2d, we will design the output waveguide to have a low-frequency cutoff > ω₁ so that we can independently control the coupling Q_w for the two frequencies via the respective ring–waveguide gaps. However, there are also several key differences from 2d. First, we include realistic material dispersion of the AlGaAs: its index is 3.023 for ω₁ and 3.178 for ω₂ [63

E. D. Palik, ed., Handbook of optical constants of solids II (Academic Press, 1991).

], and these large indices relative to our 2d structure allow a smaller-radius ring for the same Q. AlGaAs has a large off-diagonal χ⁽²⁾ coefficient ≈ 100 pm/V [6

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

, 7

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

, 42

M. Fejer, G. Magel, D. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Elec. 28(11), 2631–2654 (1992). [CrossRef]

, 64

M. Ohashi, T. Kondo, and R. Ito, “Determination of quadratic nonlinear optical coefficient of Al_xGa₁₋_xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74(1), 596–601 (1993). [CrossRef]

Fig. 8: Schematic diagram of 3d ring-resonator waveguide-cavity system.

For purely off-diagonal components

χ_{x y z}^{(2)}

, the expression for β₁ in Eq. (2) is slightly more complicated than those involving diagonal χ⁽²⁾ components (coupling modes with the same polarization). In this off-diagonal case, it is instructive to explicitly write down the coupling coefficients. Moreover, since our geometry has cylindrical symmetry, it is useful to write down Eq. (2) in cylindrical coordinates. Specifically, if we write the overlap integral in the numerator ∼4(E₁_xE₁_yE₂_z + E₁_xE₁_zE₂_y + E₁_yE₁_zE₂_x) in polar coordinates, using the coordinate transformations E_x = E_r cos(θ) – E_θ sin(θ) and E_y = E_r sin(θ) + E_θ cos(θ) we find coupling terms of the form:

E_{1 x} E_{1 y} = (E_{1 r}^{2} - E_{1 θ}^{2}) sin (2 θ) / 2 + E_{1 r} E_{1 θ} cos (2 θ)

(4)

\begin{array}{l} E_{1 x} E_{2 y} + E_{2 x} E_{1 y} & = 2 (E_{1 r} E_{2 r} - E_{1 θ} E_{2 θ}) sin (θ) cos (θ) \\ + (E_{1 r} E_{2 θ} + E_{2 r} E_{1 θ}) [{cos}^{2} (θ) - {sin}^{2} (θ)] . \end{array}

(5)

If we also assume that the fields that are rotationally symmetric functions multiplied by e^im_1,2θ (from the rotational symmetry of the ring), and we write cos(θ) = (eⁱ^θ + e⁻ⁱ^θ)/2 and sin(θ) = (eⁱ^θ – e⁻ⁱ^θ)/2i, we obtain an overlap function whose integral in θ is zero unless m₂ = 2m₁ ± 2; this modification to the previous selection rule for a diagonal χ⁽²⁾ tensor in section 3, also derived in Ref. [7

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

,32

], results from the additional e^±ⁱ²^θ terms in the integral. Having integrated over θ, we find the following simplified cylindrical overlap integral:

\begin{array}{l} β_{1} = \frac{1}{2} \frac{\int r d r d z ε_{0} χ_{x y z}^{(2)} [E_{1 r} E_{1 θ} E_{2 z} + (E_{1 r} E_{2 θ} + E_{2 r} E_{1 θ}) E_{1 z}]}{(\int d^{3} x ε {| E_{1} |}^{2}) {(\int d^{3} x ε {| E_{2} |}^{2})}^{1 / 2}} \\ \pm \frac{i}{4} \frac{\int r d r d z ε_{0} χ_{x y z}^{(2)} [2 (E_{1 r} E_{2 r} - E_{1 θ} E_{2 θ}) E_{1 z} + (E_{1 r}^{2} - E_{1 θ}^{2}) E_{2 z}]}{(d^{3} x ε {| E_{1} |}^{2}) {(\int d^{3} x ε {| E_{2} |}^{2})}^{1 / 2}}, \end{array}

(6)

While the modes in 3d are not purely polarized, because of the near mirror symmetries (both laterally and vertically) for strongly confined modes, in the center of the waveguide they are mostly TE-like (E in-plane) or TM-like (E out-of-plane), and therefore it can be convenient to describe the modes as TE-like or TM-like. (Even for purely symmetric waveguides the modes are only purely polarized in the mid-planes, outside of which they have other components [47

K. Rivoire, S. Buckley, and J. Vuckovic, “Multiply resonant high quality photonic crystal nanocavities,” Appl. Phys. Lett. 99(1), 013,114 (2011). [CrossRef]

]). On the other hand, this terminology can be misleading, because, for example, TM-like modes often have significant in-plane components and hence there can be significant coupling between two TM-like modes. More explicitly, the overlap integral in the numerator of Eq. (6) has nonzero

\sim E_{1 θ}^{2} E_{2 z}

terms that couple two TM-like modes (the symmetry allows for a large overlap if E₂_z is even because E₁_θ is squared). (As mentioned in the concluding remarks, preliminary work suggests that the overlap can be significantly improved by optimizing over a wider parameter space to consider additional modes, and at the same time one should obtain modes with higher radiative Q_rad to reduce losses.) The ability to couple modes having the same polarization is a dramatic departure from 2d and belies some of the conventional wisdom on this subject.

As a consequence of all of these changes, it turns out that a single ring (no air groove) is sufficient to obtain the desired two modes with both matched frequencies and excellent mode overlap (large β₁). Here, we chose a conventional TE-like to TM-like design that we found by searching through a small space of parameters. We chose a square a×1.35a cross-section of the ring waveguide, with an inner radius R and thickness h. The key factor in the ring design is the modified selection rule from the overlap integral of Eq. (6). For several choices of m₁, we varied the ring radius R and looked for ω₂ = 2ω₁ pairs of modes at each of the two possible m₂ values. We found a suitable pair of modes for R = 4.6a, m₁ = 16, and m₂ = 34, for which ω₁ = 0.244 · 2πc/a, corresponding to a = 0.244×1.55μm = 378 nm. The resulting field patterns are shown in Fig. 9. As mentioned above, in 3d we use the substrate itself to induce the required low-frequency cutoff, rather than an unphysical perfect metal. We now describe these differences and the resulting 3d design in more detail.

Fig. 9: Field distribution (left) and corresponding lateral cross-section (right) for the (a) ω₁ (E_r component) and (b) ω₂ (E_z component) modes.

The fields in Fig. 9 are especially attractive because they satisfy a second, approximate, selection rule. Because the waveguide width is small compared to the ring radius, the modes closely resemble those of a straight waveguide with the same cross-section, and this cross-section has a mirror symmetry plane bisecting the waveguide perpendicular to the substrate. In a straight waveguide, therefore, all modes would have fields patterns that are either symmetric or anti-symmetric (even or odd) with respect to this mirror. In a ring, the curvature breaks the symmetry but the modes are still nearly even or odd with respect to this midplane. In the coupling integral, even or odd ω₁ modes are squared to become even, and hence can only couple to even-symmetry ω₂ modes. Therefore, we should consider only ω₂ modes that have nearly even symmetry such as the one in Fig. 9, as nearly odd ω₂ modes would have nearly zero β₁. As a figure of merit, we can compare the overlap integral β₁ to a “perfect” overlap integral β₀ in which we assume that the fields = 1 inside the ring and = 0 outside, and we find that |β₁/β₀| ≈ 0.09 for our modes, indicating reasonably good overlap. (Better overlaps should be easily achieved.)

For input coupling, we use an a × 1.35a waveguide identical to the ring cross-section. For output coupling, we reduce the width to 0.14a = 53 nm so that the waveguide has a low-frequency cutoff > ω₁. The corresponding dispersion relations are shown in Fig. 10. (Note that the substrate causes a cutoff in both waveguides, but the cutoff is < ω₁ for the input waveguide [58

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, 2002).

–60

].)

Fig. 10: Band diagram of the 3d waveguides. The bottom and upper insets show the E_r and E_z field profile (blue/white/red denote positive/zero/negative amplitude) of two different modes, with frequencies ω₁ = 0.24(2πc/a) and ω₂ = 2ω₁, respectively.

Since we use separate input/output waveguides, it should be possible to independently control their coupling Q’s by varying the waveguide–ring separations. The radiation Q values for the isolated ring are Q_rad,₁ ≈ 1.7×10⁴ and Q_rad,₂ ≈ 10⁷. So, if we choose Q_w,₁ = 1000 and Q_w,₂ = 2000, CMT predicts a peak of 94% efficiency at a critical power of ≈ 30 mW. As mentioned above, significantly lower critical powers can be easily achieved with further optimization of the structure (larger overlap integrals), and indeed preliminary work suggests the possibility of almost an order of magnitude improvement. Note that as in the 2d example above, and unlike previous work on SHG in GaAs resonator geometries [6

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

, 7

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

] where down-conversion effects were negligible, one can readily observe limit cycle behavior at larger input powers.

5. Concluding Remarks

This work demonstrates that a simple two-port, two-mode CMT can accurately capture the complexities involved in a full nonlinear Maxwell system involving rings, losses, and multiple output ports. We have also presented proof-of-concept designs for ring-resonator intra-cavity SHG at high efficiencies and low powers, illustrating the care that is required to obtain appropriate modes and symmetries and in the design of the input/output coupling. However, these designs could be altered in many ways depending upon the needs of a particular experiment.

For example, lower powers could be achieved by going to larger Q_w values, at the expense of bandwidth and sensitivity, while increasing the ring radius R in order to prevent radiation loss from increasing. Conversion from 10.6 μm to 5.3 μm is especially attractive, both because of the paucity of sources at 5 μm and also because a 10× increase in lengthscales should simplify fabrication. As can be seen from the comparison of the 2d and 3d designs, the specific parameters of the ring design depend very strongly on the materials (refractive indices and dispersion), the form of the χ⁽²⁾ susceptibility, and on the details of the vertical confinement. So, the specific parameter choices here are far from universal, but the basic design principles, especially the selection rules, the role of the different Q values, and the advantage of cutoffs for separate input/output coupling, will remain.

A key practical concern in any intra-cavity frequency-conversion design such as this one is the sensitivity to fabrication imperfections, which will slightly shift both ω₁ and ω₂. Any overall shift in the frequencies can be compensated by a tunable laser source for the input, in order to match ω₁. However, another tuning parameter is required if imperfections spoil the ω₂ = 2ω₁ condition. Fortunately, we found in our designs that varying a single parameter, in our case the radius R, was sufficient to bring the frequencies into alignment. Although R cannot easily be changed post-fabrication, other dynamically tunable parameters should play a similar role. For example, strain-induced deformation of the cavity [65

C. W. Wong, P. T. Rakich, S. G. Johnson, M. Qi, H. I. Smith, E. P. Ippen, L. C. Kimerling, Y. Jeon, G. Barbastathis, and S.-G. Kim, “Strain-tunable silicon photonic band gap microcavities in optical waveguides,” Appl. Phys. Lett. 84, 1242–1245 (2004). [CrossRef]

] or strain-induced birefringence should affect the ω₁ and ω₂ modes differently and hence be capable of correcting small errors in ω₂ – 2ω₁. Alternatively, postfabrication wet etching [66

D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, “Postfabrication fine-tuning of photonic crystal microcavities in InAs/InP quantum dot membranes,” Appl. Phys. Lett. 87(15), 151,107 (2005). [CrossRef]

] or ion-beam milling [67

H. Lohmeyer, J. Kalden, K. Sebald, C. Kruse, D. Hommel, and J. Gutowski, “Fine tuning of quantum-dot pillar microcavities by focused ion beam milling,” Appl. Phys. Lett. 92(1), 011,116 (2008). [CrossRef]

] can gradually change the geometry for the same purpose. Laser-induced thermal gradients have also been used for postfabrication frequency alignment of cavity frequencies [68

J. Pan, Y. Hio, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103,114 (2008). [CrossRef]

Acknowledgments

This work was supported in part by the MRSEC Program of the NSF under Award No. DMR-0819762, by the US ARO through the ISN under Contract No. W911NF-07-D-0004, DARPA Contract No. N66001-09-1-2070-DOD, and The work is supported by the National Natural Science Foundation of China (Grant No. 10735070 and No. 10925524).

References and links

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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: January 11, 2012
Revised Manuscript: March 5, 2012
Manuscript Accepted: March 5, 2012
Published: March 19, 2012

Citation
Zhuan-Fang Bi, Alejandro W. Rodriguez, Hila Hashemi, David Duchesne, Marko Loncar, Ke-Ming Wang, and Steven G. Johnson, "High-efficiency second-harmonic generation in doubly-resonant χ⁽²⁾ microring resonators," Opt. Express 20, 7526-7543 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7526

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References

A. Rodriguez, M. Soljačić, J. D. Joannopulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express15(12), 7303–7318 (2007). [CrossRef] [PubMed]
H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant Kerr cavities,” Phys. Rev. A79(1), 013,812 (2009). [CrossRef]
V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear Optics and Crystalline Whispering Gallery Mode Cavities,” Phys. Rev. Lett.92(4), 043,903 (2004). [CrossRef] [PubMed]
J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally Phase-Matched Second-Harmonic Generation in a Whispering-Gallery-Mode Resonator,” Phys. Rev. Lett.104(15), 153,901 (2010). [CrossRef] [PubMed]
P. S. Kuo and G. S. Solomon, “On- and off-resonance second-harmonic generation in GaAs microdisks,” Opt. Express19(18), 16,898–16,918 (2011). [CrossRef]
Y. Dumeige and P. Feron, “Wispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A74, 063,804 (2006). [CrossRef]
G. T. Moore, K. Koch, and E. C. Cheung, “Optical parametric oscillation with intracavity second-harmonic generation,” Optics Communications113, 463 (1995). [CrossRef]
M. Liscidini and L. A. Andreani, “Highly efficient second-harmonic generation in doubly resonant planar microcavities,” Appl. Phys. Lett.85, 1883 (2004).
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. Lett88, 117–251,119 (2006). [CrossRef]
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]
M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today47, 25–32 (1994). [CrossRef]
G. McConnell, A. I. Ferguson, and N. Langford, “Cavity-augmented frequency tripling of a continuous wave mode-locked laser,” J. Phys. D: Appl.Phys34, 2408 (2001). [CrossRef]
R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron.6, 215–223 (1970). [CrossRef]
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]
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]
S. Pearl, H. Lotem, and Y. Shimony, “Optimization of laser intracavity second-harmonic generation by a linear dispersion element,” J. Opt. Soc. Am. B16, 1705 (1999). [CrossRef]
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. E63, 046,609 (2001). [CrossRef]
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. B19, 2111–2122 (2002). [CrossRef]
A. H. Norton and C. M. de Sterke, “Optimal poling of nonlinear photonic crystals for frequency conversion,” Opt. Lett.28, 188 (2002).
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. E64, 016,609 (2001).
A. R. Cowan and J. F. Young, “Mode matching for second-harmonic generation in photonic crystal waveguides,” Phys. Rev. E65, 085,106 (2002).
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. B68, 161,306 (2003). [CrossRef]
R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, “Nonlinear mode coupling in doubly resonant frequency doublers,” Appl. Phys. Lett.58, 117 (1994).
V. Berger, “Second-harmonic generation in monolithic cavities,” J. Opt. Soc. Am. B14, 1351 (1997). [CrossRef]
I. I. Zootoverkh, K. N. V, and E. G. Lariontsev, “Enhancement of the efficiency of second-harmonic generation in microlaser,” Quantum Electronics30, 565 (2000). [CrossRef]
B. Maes, P. Bienstman, and R. Baets, “Modeling second-harmonic generation by use of mode expansion,” J. Opt. Soc. Am. B22, 1378 (2005). [CrossRef]
M. Liscidini and L. A. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E73, 016,613 (2006). [CrossRef]
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]
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]
J. Bravo-Abad, A. W. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Enhanced nonlinear optics in photonic-crystal nanocavities,” Opt. Express15(24), 16,161–16,176 (2007). [CrossRef]
Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett.32(7), 826–828 (2007). [CrossRef] [PubMed]
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J. S. Levy, M. A. Foster, A. L. Gaeta, and M. Lipson, “Harmonic generation in silicon nitride ring resonators,” Opt. Express19(12), 11,415–11,421 (2011). [CrossRef]
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M. Gandomkar and V. Ahmadi, “Design and analysis of enhanced second harmonic generation in AlGaAs/AlOx microring waveguide,” Opt. Express19(10), 9408–9418 (2011). [CrossRef] [PubMed]
G. Nielson, D. Seneviratne, F. Lopez-Royo, P. Rakich, Y. Avrahami, M. Watts, H. Haus, H. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” Photonics Technology Letters, IEEE17(6), 1190 –1192 (2005). [CrossRef]
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M. Fejer, G. Magel, D. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Elec.28(11), 2631–2654 (1992). [CrossRef]
A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Letters to Nature391, 463–466 (1997).
T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q ring resonators in thin silicon-on-insulator,” Appl. Phys. Lett.85(16), 3346–3347 (2004). [CrossRef]
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K. Rivoire, S. Buckley, and J. Vuckovic, “Multiply resonant photonic crystal nanocavities for nonlinear frequency conversion,” Opt. Express19(22), 22,198–22,207 (2011). [CrossRef]
K. Rivoire, Z. Lin, F. Hatami, and J. Vuckovic, “Sum-frequency generation in doubly resonant GaP photonic crystal nanocavities,” Appl. Phys. Lett.97(4), 043,103 (2010). [CrossRef]
K. Rivoire, Z. Lin, F. Hatami, W. T. Masselink, and J. Vuckovic, “Second harmonic generation in gallium phosphide photonic crystal nanocavities with ultralow continuous wave pump power,” Opt. Express17(25), 22,609–22,615 (2009). [CrossRef]
K. Rivoire, S. Buckley, F. Hatami, and J. Vuckovic, “Second harmonic generation in GaP photonic crystal waveguides,” Appl. Phys. Lett.98(26), 263,113 (2011). [CrossRef]
A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Steven, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm.181, 687–702 (2010). [CrossRef]
V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys.107(17), 6756–6769 (1997). See erratum. [CrossRef]
V. A. Mandelshtam and H. S. Taylor, “Erratum: “Harmonic inversion of time signals and its applications”,” J. Chem. Phys.109, 4128 (1998). [CrossRef]
S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express8(3), 173–190 (2001).
W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. O. H. Suche, R. Nouroozi, and Y. Min, “Integrated Optical Devices in Lithium Niobate,” Opt. Photon. News19(1), 24–31 (2008). [CrossRef]
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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]
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D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, “Postfabrication fine-tuning of photonic crystal microcavities in InAs/InP quantum dot membranes,” Appl. Phys. Lett.87(15), 151,107 (2005). [CrossRef]
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J. Pan, Y. Hio, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett.92(10), 103,114 (2008). [CrossRef]

Aers, G. C.
Aguanno, G. D.
Ahmadi, V.
Aitchison, J. S.
Andersen, U. L.
Andreani, L. A.
Andreani, L. C.
Armstrong, J. A.
Ashkin, A.
Asobe, M.
Avniel, Y.
Avrahami, Y.
Baehr-Jones, T.
Baets, R.
Balakin, A. V.
Barbastathis, G.
Bedford, R.
Berger, V.
Bermel, P.
Bertolotti, M.
Bienstman, P.
Bloemer, M. J.
Bowden, C. M.
Boyd, G. D.
Boyd, R. W.
Bravetti, P.
Bravo-Abad, J.
Brieger, M.
Bristow, A. D.
Büchter, D.
Buckley, S.
Busener, H.
Bushuev, V. A.
Businaro, L.
Byer, R.
Caspani, L.
Centini, M.
Chak, P.
Chen, C. L.
Cheung, E. C.
Colet, P.
Cowan, A. R.
Dalacu, D.
de Sterke, C. M.
Dolgaleva, K.
Drummond, P. D.
Duchesne, D.
Ducuing, J.
Dumeige, Y.
Dunn, M. H.
Dziedzic, J. M.
Elser, D.
Fabrizio, E. D.
Fallahi, M.
Fan, L.
Fan, S.
Fejer, M.
Fejer, M. M.
Ferguson, A. I.
Feron, P.
Ferrera, M.
Fiedler, K.
Fiore, A.
Fong, K. Y.
Foster, M. A.
Frederick, S.
Fürst, J. U.
Gaeta, A. L.
Gandomkar, M.
Gerguson, A. I.
Grundkötter, W.
Grygiel, K.
Guizzeti, G.
Gutowski, J.
Hader, J.
Harris, J. S.
Hashemi, H.
Hatami, F.
Haus, H.
Haus, J. W.
Herrmann, H.
Hese, A.
Hio, Y.
Hochberg, M.
Hommel, D.
Hu, H.
Hunsperger, R. G.
Ibanescu, M.
Ilchenko, V. S.
Ippen, E. P.
Ito, R.
Iyer, R.
Jeon, Y.
Joannopoulos, J. D.
Joannopulos, J. D.
Johnson, S. G.
Jundt, D.
Kalden, J.
Kaneda, Y.
Kim, S.-G.
Kimble, H. J.
Kimerling, L. C.
Koch, K.
Koch, S. W.
Kondo, T.
Kruse, C.
Kuo, P. S.
Kurz, P.
Langford, N.
Lariontsev, E. G.
Lassen, M.
Lee, K. K. Y.
Leuchs, G.
Levenson, J. A.
Levy, J. S.
Lin, Z.
Lipson, M.
Liscidini, M.
loembergen, N.
Lohmeyer, H.
Lopez-Royo, F.
Lotem, H.
Maes, B.
Magel, G.
Maleki, L.
Malvezzi, A. M.
Mandelshtam, V. A.
Mantsyzov, B. I.
Marquardt, C.
Masselink, W. T.
Matsko, A. B.
McConnell, G.
McNeil, K. J.
Min, Y.
Mlynek, J.
Moers, F. V.
Moloney, J.
Moore, G. T.
Morandotti, R.
Moss, D. J.
Murray, J. T.
Nagle, J.
Nielson, G.
Norton, A. H.
Nouroozi, R.
Ohashi, M.
Oskooi, A. F.
Ou, Z. Y.
Ozheredov, I. A.
Palacios, T.
Pan, J.
Paschotta, R.
Pasquazi, A.
Passaseo, A.
Patrini, M.
Pearl, S.
Peccianti, M.
Pernice, W.
Pershan, P. S.
Petrov, E. V.
Poole, P. J.
Povinelli, M. L.
Qi, M.
Quiring, V.
Rakich, P.
Rakich, P. T.
Razzari, L.
Renn, A.
Reza, S.
Ricken, R.
Rivoire, K.
Rodriguez, A.
Rodriguez, A. W.
Romanato, F.
Rosencher, E.
Roundy, D.
Ryu, K. K.
San Miguel, M.
Sandhu, S.
Savchenkov, A. A.
Scaccabarozzi, L.
Scalora, M.
Scherer, A.
Schiller, S.
Schuck, C.
Scotto, P.
Sebald, K.
Seneviratne, D.
Shimony, Y.
Shkurinov, A. P.
Sibilia, C.
Sipe, J. E.
Smirl, A. L.
Smith, H. I.
Smith, R. G.
Sohler, W.
Soljacic, M.
Solomon, G. S.
Steven, S. G.
Stolz, W.
Strekalov, D. V.
Suche, S. O. H.
Suzuki, H.
Szlatchetka, P.
Tabor, M.
Ta-Chen, H.
Tang, H. X.
Taylor, H. S.
Timp, R.
Tomita, I.
Tuller, H.
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van Driel, H. M.
Vannahme, C.
Vecchi, G.
Vidavovic, P.
Vittorio, M. D.
Vuckovic, J.
Wagner, S. J.
Walker, C.
Walls, D. F.
Watts, M.
Williams, R. L.
Wong, C. W.
Xiong, C.
Xu, Q.
XZakharian, A. R.
Yamanaka, K.
Yang, Z.
Yoshikuni, Y.
Young, J. F.
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Appl. Phys. Lett

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. Lett88, 117–251,119 (2006). [CrossRef]

Appl. Phys. Lett.

M. Liscidini and L. A. Andreani, “Highly efficient second-harmonic generation in doubly resonant planar microcavities,” Appl. Phys. Lett.85, 1883 (2004).
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K. Rivoire, S. Buckley, and J. Vuckovic, “Multiply resonant high quality photonic crystal nanocavities,” Appl. Phys. Lett.99(1), 013,114 (2011). [CrossRef]
K. Rivoire, Z. Lin, F. Hatami, and J. Vuckovic, “Sum-frequency generation in doubly resonant GaP photonic crystal nanocavities,” Appl. Phys. Lett.97(4), 043,103 (2010). [CrossRef]
K. Rivoire, S. Buckley, F. Hatami, and J. Vuckovic, “Second harmonic generation in GaP photonic crystal waveguides,” Appl. Phys. Lett.98(26), 263,113 (2011). [CrossRef]
C. W. Wong, P. T. Rakich, S. G. Johnson, M. Qi, H. I. Smith, E. P. Ippen, L. C. Kimerling, Y. Jeon, G. Barbastathis, and S.-G. Kim, “Strain-tunable silicon photonic band gap microcavities in optical waveguides,” Appl. Phys. Lett.84, 1242–1245 (2004). [CrossRef]
D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, “Postfabrication fine-tuning of photonic crystal microcavities in InAs/InP quantum dot membranes,” Appl. Phys. Lett.87(15), 151,107 (2005). [CrossRef]
H. Lohmeyer, J. Kalden, K. Sebald, C. Kruse, D. Hommel, and J. Gutowski, “Fine tuning of quantum-dot pillar microcavities by focused ion beam milling,” Appl. Phys. Lett.92(1), 011,116 (2008). [CrossRef]
J. Pan, Y. Hio, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett.92(10), 103,114 (2008). [CrossRef]

Comp. Phys. Comm.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Steven, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm.181, 687–702 (2010). [CrossRef]

IEEE J. Quant. Elec.

M. Fejer, G. Magel, D. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Elec.28(11), 2631–2654 (1992). [CrossRef]

IEEE J. Quantum Electron.

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]
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J. Appl. Phys.

M. Ohashi, T. Kondo, and R. Ito, “Determination of quadratic nonlinear optical coefficient of AlxGa1−xAs system by the method of reflected second harmonics,” J. Appl. Phys.74(1), 596–601 (1993). [CrossRef]

J. Chem. Phys.

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys.107(17), 6756–6769 (1997). See erratum. [CrossRef]
V. A. Mandelshtam and H. S. Taylor, “Erratum: “Harmonic inversion of time signals and its applications”,” J. Chem. Phys.109, 4128 (1998). [CrossRef]

J. of Appl. Phys.

I. Tomita, M. Asobe, H. Suzuki, J. Yumoto, and Y. Yoshikuni, “Broadband quasi-phase-matched second-harmonic generation in a nonlinear photonic crystal,” J. of Appl. Phys.100(2), 023,120 (2006). [CrossRef]

J. Opt. Soc. Am. B

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. B19, 2111–2122 (2002). [CrossRef]
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J. Phys. D: Appl.Phys

G. McConnell, A. I. Ferguson, and N. Langford, “Cavity-augmented frequency tripling of a continuous wave mode-locked laser,” J. Phys. D: Appl.Phys34, 2408 (2001). [CrossRef]

Letters to Nature

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Letters to Nature391, 463–466 (1997).

Opt. Comm.

K. Grygiel and P. Szlatchetka, “Chaos in second-harmonic generation of light. The case of a strain of pulses.”Opt. Comm.91, 241–246 (1992). [CrossRef]

Opt. Commun.

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]

Opt. Express

P. S. Kuo and G. S. Solomon, “On- and off-resonance second-harmonic generation in GaAs microdisks,” Opt. Express19(18), 16,898–16,918 (2011). [CrossRef]
A. Rodriguez, M. Soljačić, J. D. Joannopulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express15(12), 7303–7318 (2007). [CrossRef] [PubMed]
J. Bravo-Abad, A. W. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Enhanced nonlinear optics in photonic-crystal nanocavities,” Opt. Express15(24), 16,161–16,176 (2007). [CrossRef]
K. K. Y. Lee, Y. Avniel, and S. G. Johnson, “Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides,” Opt. Express16, 9261–9275 (2008). [CrossRef] [PubMed]
J. S. Levy, M. A. Foster, A. L. Gaeta, and M. Lipson, “Harmonic generation in silicon nitride ring resonators,” Opt. Express19(12), 11,415–11,421 (2011). [CrossRef]
C. Xiong, W. Pernice, K. K. Ryu, C. Schuck, K. Y. Fong, T. Palacios, and H. X. Tang, “Integrated GaN photonic circuits on silicon (100) for second harmonic generation,” Opt. Express19(11), 10,462–10,470 (2011). [CrossRef]
M. Gandomkar and V. Ahmadi, “Design and analysis of enhanced second harmonic generation in AlGaAs/AlOx microring waveguide,” Opt. Express19(10), 9408–9418 (2011). [CrossRef] [PubMed]
S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express8(3), 173–190 (2001).
K. Rivoire, Z. Lin, F. Hatami, W. T. Masselink, and J. Vuckovic, “Second harmonic generation in gallium phosphide photonic crystal nanocavities with ultralow continuous wave pump power,” Opt. Express17(25), 22,609–22,615 (2009). [CrossRef]
K. Rivoire, S. Buckley, and J. Vuckovic, “Multiply resonant photonic crystal nanocavities for nonlinear frequency conversion,” Opt. Express19(22), 22,198–22,207 (2011). [CrossRef]

Opt. Lett.

Q. Xu and M. Lipson, “Carrier-induced optical bistability in Silicon ring resonators,” Opt. Lett.31(3), 341–343 (2005). [CrossRef]
Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett.32(7), 826–828 (2007). [CrossRef] [PubMed]
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Opt. Photon. News

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. O. H. Suche, R. Nouroozi, and Y. Min, “Integrated Optical Devices in Lithium Niobate,” Opt. Photon. News19(1), 24–31 (2008). [CrossRef]

Optica Acta.

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]

Optics Communications

G. T. Moore, K. Koch, and E. C. Cheung, “Optical parametric oscillation with intracavity second-harmonic generation,” Optics Communications113, 463 (1995). [CrossRef]

Photonics Technology Letters, IEEE

G. Nielson, D. Seneviratne, F. Lopez-Royo, P. Rakich, Y. Avrahami, M. Watts, H. Haus, H. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” Photonics Technology Letters, IEEE17(6), 1190 –1192 (2005). [CrossRef]

Phys. Rev.

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]

Phys. Rev. A

Y. Dumeige and P. Feron, “Stability and time-domain analysis of the dispersive tristability in microresonators under modal coupling,” Phys. Rev. A84(4), 043,847 (2011). [CrossRef]
Y. Dumeige and P. Feron, “Wispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A74, 063,804 (2006). [CrossRef]
H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant Kerr cavities,” Phys. Rev. A79(1), 013,812 (2009). [CrossRef]

Phys. Rev. B

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. B68, 161,306 (2003). [CrossRef]

Phys. Rev. E

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