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Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper

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Abstract

A technique is demonstrated which efficiently transfers light between a tapered standard single-mode optical fiber and a high-Q, ultra-small mode volume, silicon photonic crystal resonant cavity. Cavity mode quality factors of 4.7×104 are measured, and a total fiber-to-cavity coupling efficiency of 44% is demonstrated. Using this efficient cavity input and output channel, the steady-state nonlinear absorption and dispersion of the photonic crystal cavity is studied. Optical bistability is observed for fiber input powers as low as 250 µW, corresponding to a dropped power of 100 µW and 3 fJ of stored cavity energy. A high-density effective free-carrier lifetime for these silicon photonic crystal resonators of ~0.5 ns is also estimated from power dependent loss and dispersion measurements.

©2005 Optical Society of America

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Figures (8)

Fig. 1.
Fig. 1. (a) Schematic of the fiber taper to PC cavity coupling scheme. The blue arrow represents the input light, some of which is coupled contradirectionally into the PCWG. The green arrow represents the light reflected by the PC cavity and recollected in the backwards propagating fiber mode. The red colored region represents the cavity mode and its radiation pattern. (b) Illustration of the fiber-PC cavity coupling process. The dashed line represents the “local” band-edge frequency of the photonic crystal along the waveguide axis. The step discontinuity in the bandedge at the PCWG - PC cavity interface is due to a jump in the longitudinal () lattice constant (see Fig. 2). The parabolic “potential” is a result of the longitudinal grade in hole radius of the PC cavity. The bandwidth of the waveguide is represented by the gray shaded area. Coupling between the cavity mode of interest (frequency ω0) and the mode matched PCWG mode (ωWG0) is represented by γ0e, coupling to radiating PCWG modes is represented by γje >0, and intrinsic cavity loss is represented by γi . (c,d) Magnetic field profile, calculated using FDTD, of the high-Q PC cavity A20 mode and the fundamental TE1 PCWG mode, respectively.
Fig. 2.
Fig. 2. SEM image of an integrated PCWG-PC cavity sample. The PC cavity and PCWG have lattice constants Λ~430 nm, Λ x ~430 nm, and Λ z ~550 nm. The surrounding silicon material has been removed to form a diagonal trench and isolated mesa structure to enable fiber taper probing.
Fig. 3.
Fig. 3. (a) Illustration of the device and fiber taper orientation for (i) efficient PCWG mediated taper probing of the cavity, and (ii) direct taper probing of the cavity. (b) Normalized depth of the transmission resonance (Δ) at λ o ~1589.7, as a function of lateral taper displacement relative to the center of the PC cavity, during direct taper probing (taper in orientation (ii)).
Fig. 4.
Fig. 4. (a) Measured reflected taper signal as a function of input wavelength (taper diameter d~1 µm, taper height g=0.80 µm). The sharp dip at λ~1589.7 nm, highlighted in panel (b), corresponds to coupling to the A20 cavity mode. (c) Maximum reflected signal (slightly detuned from the A20 resonance line), and resonance reflection contrast as a function of taper height. The dashed line at ΔR=0.6 shows the PCWG-cavity drop efficiency, which is independent of the fiber taper position for g≥0.8 µm.
Fig. 5.
Fig. 5. (a) Measured cavity response as a function of input wavelength, for varying PCWG power (taper diameter d~1 µm, taper height g=0.80 µm).
Fig. 6.
Fig. 6. (a) Power dropped (Pd ) into the cavity as a function of power in the PCWG (Pi ). The dashed line shows the expected result in absence of nonlinear cavity loss. (b) Resonance wavelength shift as a function of internal cavity energy. Solid blue lines in both Figs. show simulated results.
Fig. 7.
Fig. 7. (a) Simulated effective quality factors for the different PC cavity loss channels as a function of power dropped into the cavity. (b) Contributions from the modeled dispersive processes to the PC cavity resonance wavelength shift as a function of power dropped into the cavity. (Simulation parameters: ηlin~0.40, Γth dT/dP abs=27 K/mW, τ-1~0.0067+(1.4×10-7)N 0.94 where N has units of cm-3 and τ has units of ns.)
Fig. 8.
Fig. 8. Dependance of free-carrier lifetime on free-carrier density (red dots) as found by fitting Δλo (Pi ) and Pd (Pi ) with the constant material and modal parameter values of Table 1, and for effective PC cavity thermal resistance of Γth dT/dP abs=27 K/mW and linear absorption fraction ηlin=0.40. The solid blue line corresponds to a smooth curve fit to the point-by-point least-squared fit data given by τ-1~0.0067+(1.4×10-7)N 0.94, where N is in units of cm-3 and τ is in ns.

Tables (1)

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Table 1. Fixed parameters used in the model.

Equations (41)

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K γ 0 e γ i + j 0 γ j e ,
I γ 0 e Σ j γ j e ,
R o ( ω o ) = ( 1 K ) 2 ( 1 + K ) 2 .
Q i + P = 2 Q T 1 1 ± R o ( ω o ) = Q T ( 1 + K ) ,
η 0 = γ 0 e γ i + j γ j e = 1 1 + 1 K .
Q T Q i = 1 K ( I ( 1 + K ) ) = 1 η 0 I .
U = ( 1 R o ( ω o ) ) Q i + P ω o P i = 4 K ( 1 + K ) 2 I K ( 1 I ) I Q i ω o P i
γ i ( U ) = γ rad + γ lin + γ ̅ TPA ( U ) + γ ̅ FCA ( U ) .
U = 4 K ( U ) ( 1 + K ( U ) ) 2 Q i + p ( U ) ω o P i ,
K ( U ) = γ o e γ i ( U ) + j > 0 γ j e ,
ω o Q i + P ( U ) = γ i ( U ) + j > 0 γ j e .
γ TPA ( r ) = β ' ( r ) 1 2 ε o n 2 ( r ) E 2 ( r ) ,
γ ̅ TPA = γ TPA ( r ) n 2 ( r ) E 2 ( r ) d r n 2 ( r ) E 2 ( r ) d r = β ' ̅ U V TPA ,
β ' ̅ = β ' ( r ) n 4 ( r ) E 4 ( r ) d r n 4 ( r ) E 4 ( r ) d r
V TPA = ( n 2 ( r ) E 2 ( r ) d r ) 2 n 4 ( r ) E 4 ( r ) d r .
γ ̅ TPA = Γ TPA β ' Si U V TPA
Γ TPA = Si n 4 ( r ) E 4 ( r ) d r n 4 ( r ) E 4 ( r ) d r ,
γ FCA = σ ' ( r ) N ( r ) ,
N ( r ) = τ p TPA ( r ) 2 h ̅ ω o ,
p TPA ( r ) = 1 2 ε o n 2 ( r ) E 2 ( r ) γ TPA ( r ) .
γ ̅ FCA = τ 2 h ̅ ω o ( σ ' ( r ) 1 2 ε o n 2 ( r ) E 2 ( r ) γ TPA ( r ) ) n 2 ( r ) E 2 ( r ) d r n 2 ( r ) E 2 ( r ) d r .
γ ̅ FCA = Γ FCA ( τ σ ' Si β ' Si 2 h ̅ ω o U 2 V FCA 2 ) ,
Γ FCA = Si n 6 ( r ) E 6 ( r ) d r n 6 ( r ) E 6 ( r ) d r
V FCA 2 = ( n 2 ( r ) E 2 ( r ) d r ) 3 n 6 ( r ) E 6 ( r ) d r .
Δ ω o ( U ) ω o = Δ n ̅ ( U ) ,
Δ n ̅ ( U ) = ( Δ n ( r ) n ( r ) ) n 2 ( r ) E 2 ( r ) d r n 2 ( r ) E 2 ( r ) d r .
R o ( ω ) = 1 4 K ( U ) ( 1 + K ( U ) ) 2 ( δ ω 2 ) 2 ( ω ω o Δ ω o ( U ) ) 2 + ( δ ω ( U ) 2 ) 2 .
U = P d γ i + P = ( 1 R o ( ω ) ) Q i + P ( U ) ω o P i ,
Δ n Kerr ( r ) = n ' 2 ( r ) 1 2 ε o n 2 ( r ) E 2 ( r ) ,
Δ n ̅ Kerr ( U ) = Γ Kerr n Si ( n ' 2 , Si U V Kerr ) ,
Γ Kerr = Γ TPA
V Kerr = V TPA .
Δ n FCD ( r ) = ζ ( r ) N ( r ) ,
Δ n ̅ FCD ( U ) = Γ FCD n Si ( τ ζ Si β ' Si 2 h ̅ ω o U 2 V FCD 2 ) ,
Γ FCD = Γ FCA
V FCD = V FCA .
Δ n ̅ th = ( 1 n ( r ) d n d T ( r ) Δ T ( r ) ) n 2 ( r ) E 2 ( r ) d r n 2 ( r ) E 2 ( r ) d r .
Δ n ̅ th ( U ) = Γ th n Si ( d n Si d T d T d P abs P abs ( U ) )
P abs ( U ) = ( γ lin + γ ̅ TPA ( U ) + γ ̅ FCA ( U 2 ) ) U .
Q i + P ( P i ) = K ( Δ R o ( P i ) ) Q T ( P i = 0 ) η 0 ( P i = 0 ) .
Δ n FCD , Si = [ ζ e , Si N e + ( ζ h , Si N h ) 0.8 ] .
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