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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7526–7543
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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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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.

© 2012 OSA

1. Introduction

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).

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

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 Qrad were obtained by Harminv, which is a free program to solve the problem of harmonic inversion [54

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

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/Qtot can be written as the sum of two decay rates: 1/Qtot = 1/Qw + 1/Qrad, where 1/Qw and 1/Qrad are the waveguide and radiative decay rates, respectively. We obtain Qtot from a filter-diagonalization analysis of the field decay in FDTD [53

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 Qw and Qrad 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

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)=PoutPin=ω02(1/Qw1/Qrad)2+4(ωω0)2ω02(1/Qw+1/Qrad)2+4(ωω0)2,
(1)
one can solve for both Qrad and Qw given T (ω0) (the minimum T) and Qtot.

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

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.

3. 2D Design

In Ref. [2

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]

], 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

    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

    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/Q12Q2.
    • – 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

    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

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

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 Q2Q1. 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 χzzz(2)41×1012m/V [42

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

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 χzzz(2) [56

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

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 ∼ eimϕ, determined by the conserved angular momentum “quantum number” m ∈ 𝕑. By simple inspection of Eq. (2) for χzzz(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.

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 Qw,1 and Qw,2 at ω1 and ω2. In order to obtain small radiation losses, these coupling lifetimes should be much smaller than the radiative lifetimes Qrad,1 and Qrad,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 Qw ∼ 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 Qw 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

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 Qw 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 Qw,1 and Qrad,1 in Fig. 3(a), which illustrates two effects. First, the coupling Qw 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 Qrad 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 Qrad and Qw from a combination of mode decay and transmission simulations, and several of these transmission spectra are shown in Fig. 3(b). When QradQw, the linear ring–waveguide system approaches an all-pass filter with 100% transmission (but a resonant delay), while as Qw approaches Qrad for large d the radiation loss increases and one observes a resonant dip in the transmission.

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

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

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]

]: 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 Qw,1 = 2000 and Qw,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 Ez 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: Ez 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 Qrad/Qw). 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 Qrad/Qw, 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 = PSH/Pin versus Pin, 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) Ez 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

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 Pin = 2Pc, 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

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

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

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

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.

The < 100% efficiency was due to the finite Qrad/Qw and in particular the limiting factor is the loss at ω1 (Qrad,1/Qw,1 = 9847/1998 ≈ 4.9), so we can obtain higher efficiency by making Qw,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 Qrad/Qw = 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

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

For purely off-diagonal components χxyz(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(E1xE1yE2z + E1xE1zE2y + E1yE1zE2x) in polar coordinates, using the coordinate transformations Ex = Er cos(θ) – Eθ sin(θ) and Ey = Er sin(θ) + Eθ cos(θ) we find coupling terms of the form:
E1xE1y=(E1r2E1θ2)sin(2θ)/2+E1rE1θcos(2θ)
(4)
E1xE2y+E2xE1y=2(E1rE2rE1θE2θ)sin(θ)cos(θ)+(E1rE2θ+E2rE1θ)[cos2(θ)sin2(θ)].
(5)

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

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 E1θ2E2z terms that couple two TM-like modes (the symmetry allows for a large overlap if E2z is even because E1θ 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 Qrad 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 (Er component) and (b) ω2 (Ez 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

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

60

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]

].)

Fig. 10: Band diagram of the 3d waveguides. The bottom and upper insets show the Er and Ez 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.

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 Qw 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

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

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

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

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).

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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

  1. A. Rodriguez, M. Soljačić, J. D. Joannopulos, S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15(12), 7303–7318 (2007). [CrossRef] [PubMed]
  2. H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljacic, 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, L. Maleki, “Nonlinear Optics and Crystalline Whispering Gallery Mode Cavities,” Phys. Rev. Lett. 92(4), 043,903 (2004). [CrossRef] [PubMed]
  4. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, G. Leuchs, “Naturally Phase-Matched Second-Harmonic Generation in a Whispering-Gallery-Mode Resonator,” Phys. Rev. Lett. 104(15), 153,901 (2010). [CrossRef] [PubMed]
  5. P. S. Kuo, G. S. Solomon, “On- and off-resonance second-harmonic generation in GaAs microdisks,” Opt. Express 19(18), 16,898–16,918 (2011). [CrossRef]
  6. Y. Dumeige, 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, E. C. Cheung, “Optical parametric oscillation with intracavity second-harmonic generation,” Optics Communications 113, 463 (1995). [CrossRef]
  8. M. Liscidini, L. A. Andreani, “Highly efficient second-harmonic generation in doubly resonant planar microcavities,” Appl. Phys. Lett. 85, 1883 (2004).
  9. L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, W. Stolz, “Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling,” Appl. Phys. Lett 88, 117–251,119 (2006). [CrossRef]
  10. P. Scotto, P. Colet, M. San Miguel, “All-optical image processing with cavity type II second-harmonic generation,” Opt. Lett. 28, 1695 (2003). [CrossRef] [PubMed]
  11. M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25–32 (1994). [CrossRef]
  12. G. McConnell, A. I. Ferguson, N. Langford, “Cavity-augmented frequency tripling of a continuous wave mode-locked laser,” J. Phys. D: Appl.Phys 34, 2408 (2001). [CrossRef]
  13. R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron. 6, 215–223 (1970). [CrossRef]
  14. A. I. Gerguson, M. H. Dunn, “Intracavity second harmonic generation in continuous-wave dye lasers,” IEEE J. Quantum Electron. 13, 751–756 (1977). [CrossRef]
  15. M. Brieger, H. Busener, A. Hese, F. V. Moers, A. Renn, “Enhancement of single frequency SHG in a passive ring resonator,” Opt. Commun. 38, 423–426 (1981). [CrossRef]
  16. S. Pearl, H. Lotem, Y. Shimony, “Optimization of laser intracavity second-harmonic generation by a linear dispersion element,” J. Opt. Soc. Am. B 16, 1705 (1999). [CrossRef]
  17. A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band edge under the quasiphase matching condition,” Phys. Rev. E 63, 046,609 (2001). [CrossRef]
  18. G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, “Generalized coupled-mode theory for χ(2) interactions in finite multi-layered structures,” J. Opt. Soc. Am. B 19, 2111–2122 (2002). [CrossRef]
  19. A. H. Norton, C. M. de Sterke, “Optimal poling of nonlinear photonic crystals for frequency conversion,” Opt. Lett. 28, 188 (2002).
  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, M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016,609 (2001).
  21. A. R. Cowan, J. F. Young, “Mode matching for second-harmonic generation in photonic crystal waveguides,” Phys. Rev. E 65, 085,106 (2002).
  22. A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161,306 (2003). [CrossRef]
  23. R. Paschotta, K. Fiedler, P. Kurz, J. Mlynek, “Nonlinear mode coupling in doubly resonant frequency doublers,” Appl. Phys. Lett. 58, 117 (1994).
  24. V. Berger, “Second-harmonic generation in monolithic cavities,” J. Opt. Soc. Am. B 14, 1351 (1997). [CrossRef]
  25. I. I. Zootoverkh, K. N. V, E. G. Lariontsev, “Enhancement of the efficiency of second-harmonic generation in microlaser,” Quantum Electronics 30, 565 (2000). [CrossRef]
  26. B. Maes, P. Bienstman, R. Baets, “Modeling second-harmonic generation by use of mode expansion,” J. Opt. Soc. Am. B 22, 1378 (2005). [CrossRef]
  27. M. Liscidini, L. A. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73, 016,613 (2006). [CrossRef]
  28. J. A. Armstrong, N. loembergen, J. Ducuing, P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). [CrossRef]
  29. A. Ashkin, G. D. Boyd, J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2, 109–124 (1966). [CrossRef]
  30. J. Bravo-Abad, A. W. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, M. Soljačić, “Enhanced nonlinear optics in photonic-crystal nanocavities,” Opt. Express 15(24), 16,161–16,176 (2007). [CrossRef]
  31. Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, 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, R. Morandotti, “Optical frequency conversion in integrated devices,” J. Opt. Soc. Am. B 28(12), A67–A82 (2011). [CrossRef]
  33. S. Schiller, “Principles and Applications of Optical Monolithic Total-Internal-Reflection Resonators,” Ph.D. thesis, Stanford University, Stanford, CA (1993).
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