Abstract

We present a simple and intuitive model based on the impulse response of linear electrical systems for describing the propagation of optical pulses through a dynamic Fabry–Perot resonator whose refractive index changes with time. Our model shows that the adiabatic wavelength conversion process in resonators results from a scaling of the round-trip time with index changes. For pulses longer than the cavity round-trip time, we find that more energy can be transferred to the new wavelength when the input pulses are slightly detuned from the cavity resonance and the refractive index does not change too rapidly. In fact, the optimum duration of index changes scales with the photon lifetime of the resonator. We describe the evolution of the shape and spectrum of picosecond pulses inside a resonator under a variety of input conditions and with the magnitude and duration of index variations. We also apply our general theory to the case of pulses whose widths are shorter than the round-trip time and derive an analytical expression for the output field under quite general conditions. This analysis reveals a shifting of the spectral comb as well as compression of the temporal pulse train that depends on the both the magnitude and sign of the index change. Our results should find applications in the area of optical signal processing using resonant photonic structures.

© 2011 Optical Society of America

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  1. M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
    [CrossRef] [PubMed]
  2. M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
    [CrossRef] [PubMed]
  3. M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. A 73, 051803(2006).
    [CrossRef]
  4. S. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photon. 1, 293–296 (2007).
    [CrossRef]
  5. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907(2009).
    [CrossRef] [PubMed]
  6. Z. Gaburro, M. Ghulinyan, F. Riboli, and L. Pavesi, “Photon energy lifter,” Opt. Express 14, 7270–7278 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
  8. M. J. Lawrence, B. Willke, M. E. Husman, E. K. Gustafson, and R. L. Byer, “Dynamic response of a Fabry–Perot interferometer,” J. Opt. Soc. Am. B 16, 523–532 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  12. M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry–Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
    [CrossRef]
  13. T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
    [CrossRef]
  14. Y. Xiao, G. P. Agrawal, and D. N. Maywar, “Spectral and temporal changes of optical pulses propagating through time-varying linear media,” Opt. Lett. 36, 505–507 (2011).
    [CrossRef] [PubMed]
  15. G. P. Agrawal, Lightwave Technology: Components and Devices (Wiley, 2004).
  16. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

2011 (1)

2010 (1)

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

2009 (1)

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907(2009).
[CrossRef] [PubMed]

2007 (1)

S. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photon. 1, 293–296 (2007).
[CrossRef]

2006 (2)

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. A 73, 051803(2006).
[CrossRef]

Z. Gaburro, M. Ghulinyan, F. Riboli, and L. Pavesi, “Photon energy lifter,” Opt. Express 14, 7270–7278 (2006).
[CrossRef] [PubMed]

2004 (2)

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
[CrossRef] [PubMed]

2002 (3)

2001 (1)

1999 (1)

1965 (1)

S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. 1, 245–262 (1965).
[CrossRef]

Agrawal, G. P.

Y. Xiao, G. P. Agrawal, and D. N. Maywar, “Spectral and temporal changes of optical pulses propagating through time-varying linear media,” Opt. Lett. 36, 505–507 (2011).
[CrossRef] [PubMed]

G. P. Agrawal, Lightwave Technology: Components and Devices (Wiley, 2004).

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Beggs, D. M.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Blatt, R.

Byer, R. L.

Eschner, J.

Fan, S.

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
[CrossRef] [PubMed]

Gaburro, Z.

Ghulinyan, M.

Gustafson, E. K.

Harris, S. E.

S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. 1, 245–262 (1965).
[CrossRef]

Husman, M. E.

Kampfrath, T.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Krauss, T. F.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Kuipers, L.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Kuramochi, E.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907(2009).
[CrossRef] [PubMed]

Lawrence, M. J.

Lipson, M.

S. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photon. 1, 293–296 (2007).
[CrossRef]

Maywar, D. N.

McDuff, O. P.

S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. 1, 245–262 (1965).
[CrossRef]

Melloni, A.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Mitsugi, S.

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. A 73, 051803(2006).
[CrossRef]

Notomi, M.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907(2009).
[CrossRef] [PubMed]

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. A 73, 051803(2006).
[CrossRef]

Pavesi, L.

Preble, S.

S. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photon. 1, 293–296 (2007).
[CrossRef]

Rakhmanov, M.

M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry–Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
[CrossRef]

M. Rakhmanov, “Doppler-induced dynamics of fields in Fabry–Perot cavities with suspended mirrors,” Appl. Opt. 40, 1942–1949 (2001).
[CrossRef]

Redding, D.

Regehr, M.

Reitze, D. H.

M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry–Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
[CrossRef]

Riboli, F.

Rohde, H.

Savage, R. L.

M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry–Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
[CrossRef]

Schmidt-Kaler, F.

Sievers, L.

Tanabe, T.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907(2009).
[CrossRef] [PubMed]

Taniyama, H.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907(2009).
[CrossRef] [PubMed]

Tanner, D. B.

M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry–Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
[CrossRef]

White, T. P.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Willke, B.

Xiao, Y.

Xu, Q.

S. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photon. 1, 293–296 (2007).
[CrossRef]

Yanik, M. F.

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. 1, 245–262 (1965).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nat. Photon. (1)

S. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photon. 1, 293–296 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (1)

M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry–Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
[CrossRef]

Phys. Rev. A (2)

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. A 73, 051803(2006).
[CrossRef]

Phys. Rev. Lett. (3)

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907(2009).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
[CrossRef] [PubMed]

Other (2)

G. P. Agrawal, Lightwave Technology: Components and Devices (Wiley, 2004).

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

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Figures (11)

Fig. 1
Fig. 1

Schematic illustration of pulse propagation through a dynamic optical medium with refractive index n ( t ) . Each temporal slice of the input pulse is delayed by a time-dependent transit time T r ( t ) .

Fig. 2
Fig. 2

Schematic illustration of multiple round trips within an FP resonator.

Fig. 3
Fig. 3

Short-pulse ( 60 fs ) propagation for an instantaneous change in the refractive index of 5 % (dashed blue curve) and + 5 % (solid red curve) at time T c = 250 fs . The round-trip time is 233 fs . Detuning Δ ν is defined as ν ν 0 , where ν 0 is the input frequency. (a) The index change alters the width, amplitude, and delay of temporal pulses after T c . (b) The index change shifts the comblike input spectrum (green dotted curve) to higher ( + 5 % ) and lower ( 5 % ) frequencies at the FP output.

Fig. 4
Fig. 4

Spectrogram for the short-pulse case shown in Fig. 3 for Δ n = 5 % at time T c = 250 fs . A shift of about 10 THz in the carrier frequency of the pulse is clearly seen after the index change at time T c .

Fig. 5
Fig. 5

Long-pulse ( 10 ps ) propagation for a linear change in refractive index of 0.1 % (solid red curve) between T i = 3 ps and T f = 6 ps (marked by arrows). The round-trip time of the resonator is 0.23 ps . (a) The index change advances the output peak and produces a long tail with a kneelike feature. (b) The output spectrum shows two spectral peaks corresponding to the original and shifted cavity modes, respectively.

Fig. 6
Fig. 6

Spectrogram for the long-pulse case shown in Fig. 5. The pulse spectrum appears to follow the cavity resonance indicated by the dotted line.

Fig. 7
Fig. 7

Impact of index-change duration Δ T = T f T i on propagation of 10 ps Gaussian pulses in the case of a 0.1 % linear change in the refractive index beginning at T i = 3 ps . (a) Output pulse narrows down and exhibits oscillations as Δ T decreases. (b) Amplitude of the AWC peak also increases as Δ T decreases.

Fig. 8
Fig. 8

AWC efficiency as a function of index-change duration Δ T for three values of photon lifetimes τ ph in the case of Δ n = 0.1 % . The efficiency is maximum for Δ T near zero when τ ph is relatively small, but it peaks at a value of Δ T > 0 that increases with τ ph .

Fig. 9
Fig. 9

Impact of the magnitude of Δ n for a fixed linear index-change duration of 3 ps , input pulse width of 10 ps , and a round-trip time of 0.23 ps . (a) The oscillation frequency in the pulse tail increases for a larger index change because of (b) a larger AWC-induced shift in the mode frequency.

Fig. 10
Fig. 10

Impact of detuning Δ ν c of the input pulse from a cavity resonance ( Δ ν c = ν 0 ν c , where ν 0 and ν c are the input carrier frequency and cavity-resonance frequency, respectively) for a 0.1 % linear change in the refractive index between T i = 3 ps and T f = 6 ps (the round-trip time is 0.23 ps ). (a) Output pulse shapes show an asymmetry with respect to the sign of Δ ν c . (b) The amplitude of the AWC peak is larger for a + 25 GHz detuning than for the 25 GHz detuning (in this negative index-change case).

Fig. 11
Fig. 11

Fraction of pulse energy transferred to the AWC peak as a function of detuning of input pulse from a cavity resonance under the same long-pulse conditions as in Fig. 10 but for an index change of 0.1 % (blue solid curve) and + 0.1 % (red dotted curve).

Equations (13)

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E out ( t ) = h ( t , t ) E in ( t ) d t ,
h ( t , t ) = δ [ t t T r ( t ) ] ,
t t + T r ( t ) [ c / n ( τ ) ] d τ = L .
h ( t , t ) = ( 1 R ) m = 0 R m δ [ t t T m ( t ) ] ,
h ( t t ) = ( 1 R ) m = 0 R m δ [ t t ( 2 m + 1 ) T 0 / 2 ] ,
H ( ω ) = ( 1 R ) e i ω T 0 / 2 / ( 1 R e i ω T 0 ) .
n ( t ) = { n 1 ( t < T c ) n 2 ( t T c ) .
t T c c n 1 d τ + T c t + T m ( t ) c n 2 d τ = ( 2 m + 1 ) L .
T m ( t ) = ( 1 s ) t / s + T em ,
T em = ( 2 m + 1 ) n 2 L / c + ( 1 1 / s ) T c .
E out ( t ) = ( 1 R ) m = 0 R m [ s E in ( s t s T em ) ] .
E in ( t ) = E 0 exp [ t 2 / ( 2 T 0 2 ) i ω 1 t ] ,
S ( ω , τ ) = | W ( t , τ ) E out ( t ) e i ω t d t | 2 .

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