Abstract

The ability to control the spectrum of light is of fundamental significance in many applications of light. We consider a dynamically modulated ring resonator that supports a set of resonant modes equally spaced in their resonant frequencies, and is modulated at a frequency that is slightly detuned from the modal frequency spacing. We find that such a system can be mapped into a tight-binding model of photons under a constant effective force, and, as a result, the system exhibits Bloch oscillation along the frequency axis. The sign of the detuning is mapped to the sign of the effective force and hence controls the direction of the Bloch oscillation. We also show that a periodic switching of the detuning can lead to unidirectional transport of light along the frequency axis. Our work points to a new capability for manipulating the frequencies of light using dynamic microcavity structures.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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2016 (2)

T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, “Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics,” Phys. Rev. A 93, 043827 (2016).
[Crossref]

L. Yuan, Y. Shi, and S. Fan, “Photonic gauge potential in a system with a synthetic frequency dimension,” Opt. Lett. 41, 741–744 (2016).
[Crossref]

2015 (4)

P. Lodahl, S. Mahmoodian, and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures,” Rev. Mod. Phys. 87, 347–400 (2015).
[Crossref]

L. Yuan and S. Fan, “Three-dimensional dynamic localization of light from a time-dependent effective gauge field for photons,” Phys. Rev. Lett. 114, 243901 (2015).
[Crossref]

S.-W. Huang, H. Zhou, J. Yang, J. F. McMillan, A. Matsko, M. Yu, D.-L. Kwong, L. Maleki, and C. W. Wong, “Mode-locked ultrashort pulse generation from on-chip normal dispersion microresonators,” Phys. Rev. Lett. 114, 053901 (2015).
[Crossref]

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref]

2014 (4)

2013 (1)

T. Hu, P. Yu, C. Qiu, H. Qiu, F. Wang, M. Yang, X. Jiang, H. Yu, and J. Yang, “Tunable Fano resonances based on two-beam interference in microring resonator,” Appl. Phys. Lett. 102, 011112 (2013).
[Crossref]

2012 (3)

2011 (1)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

2010 (4)

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nat. Photonics 4, 786–791 (2010).
[Crossref]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref]

M. Levy and P. Kumar, “Nonreciprocal Bloch oscillations in magneto-optic waveguide arrays,” Opt. Lett. 35, 3147–3149 (2010).
[Crossref]

2009 (4)

C. Bersch, G. Onishchukov, and U. Peschel, “Experimental observation of spectral Bloch oscillations,” Opt. Lett. 34, 2372–2374 (2009).
[Crossref]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Collective emission and absorption in a linear resonator chain,” Opt. Express 17, 15210–15215 (2009).
[Crossref]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

2008 (3)

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904 (2008).
[Crossref]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett. 101, 093902 (2008).
[Crossref]

U. Peschel, C. Bersch, and G. Onishchukov, “Discreteness in time,” Open Phys. 6, 619–627 (2008).
[Crossref]

2006 (3)

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

S. Longhi, “Optical Zener-Bloch oscillations in binary waveguide arrays,” Europhys. Lett. 76, 416–421 (2006).
[Crossref]

2005 (1)

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref]

2000 (1)

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[Crossref]

1999 (2)

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[Crossref]

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[Crossref]

1986 (1)

D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged particle moving under the influence of an electric field,” Phys. Rev. B 34, 3625–3633 (1986).
[Crossref]

1929 (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[Crossref]

Aitchison, J. S.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).

Beausoleil, R. G.

Beha, K.

Bender, C. M.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Bersch, C.

Bloch, F.

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[Crossref]

Cardenas, J.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref]

Carusotto, I.

T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, “Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics,” Phys. Rev. A 93, 043827 (2016).
[Crossref]

Danzl, J. G.

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

de Sterke, C. M.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[Crossref]

Del’haye, P.

Desyatnikov, A. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

Diddams, S. A.

S. B. Papp, K. Beha, P. Del’haye, F. Quinlan, H. Lee, K. J. Vahala, and S. A. Diddams, “Microresonator frequency comb optical clock,” Optica 1, 10–14 (2014).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

Dignam, M. M.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

Dong, P.

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904 (2008).
[Crossref]

Dreisow, F.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

Dunlap, D. H.

D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged particle moving under the influence of an electric field,” Phys. Rev. B 34, 3625–3633 (1986).
[Crossref]

Fain, R.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref]

Fan, S.

L. Yuan, Y. Shi, and S. Fan, “Photonic gauge potential in a system with a synthetic frequency dimension,” Opt. Lett. 41, 741–744 (2016).
[Crossref]

L. Yuan and S. Fan, “Three-dimensional dynamic localization of light from a time-dependent effective gauge field for photons,” Phys. Rev. Lett. 114, 243901 (2015).
[Crossref]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref]

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[Crossref]

Foster, M. A.

Gaeta, A. L.

Gianfreda, M.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Goldman, N.

T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, “Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics,” Phys. Rev. A 93, 043827 (2016).
[Crossref]

Griffith, A. G.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref]

Haller, E.

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

Hart, R.

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

Haus, H. A.

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[Crossref]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Heinrich, M.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

Hu, T.

T. Hu, P. Yu, C. Qiu, H. Qiu, F. Wang, M. Yang, X. Jiang, H. Yu, and J. Yang, “Tunable Fano resonances based on two-beam interference in microring resonator,” Appl. Phys. Lett. 102, 011112 (2013).
[Crossref]

Huang, S.-W.

S.-W. Huang, H. Zhou, J. Yang, J. F. McMillan, A. Matsko, M. Yu, D.-L. Kwong, L. Maleki, and C. W. Wong, “Mode-locked ultrashort pulse generation from on-chip normal dispersion microresonators,” Phys. Rev. Lett. 114, 053901 (2015).
[Crossref]

Ilchenko, V. S.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Collective emission and absorption in a linear resonator chain,” Opt. Express 17, 15210–15215 (2009).
[Crossref]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett. 101, 093902 (2008).
[Crossref]

Ippen, E. P.

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[Crossref]

Iyer, R.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

Jiang, X.

T. Hu, P. Yu, C. Qiu, H. Qiu, F. Wang, M. Yang, X. Jiang, H. Yu, and J. Yang, “Tunable Fano resonances based on two-beam interference in microring resonator,” Appl. Phys. Lett. 102, 011112 (2013).
[Crossref]

Joannopoulos, J. D.

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[Crossref]

Johnson, A. R.

Joushaghani, A.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

Kenkre, V. M.

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Figures (5)

Fig. 1.
Fig. 1. (a) Ring resonator modulated by an electro-optic modulator (EOM). (b) In the absence of group velocity dispersion, the ring in (a) supports a set of resonant modes (blue dashed lines) with their resonant frequencies equally spaced by Ω R . The modulation frequency Ω is chosen to be slightly different from Ω R . For the modes that are located at ω 0 , the modulation generates sidebands that are detuned from the other resonant modes. (c) The system as described in (b) can be mapped onto a tight-binding model of a photon under an effective force [Eq. (4)].
Fig. 2.
Fig. 2. Evolution in time of the total energy for each resonant mode ( I m ( t ) ring | E m ( t , z ) | 2 d z ): (a)  Δ = 0.1 c / n 0 L , and (b)  Δ = 0.5 c / n 0 L , when only the m = 0 mode is excited.
Fig. 3.
Fig. 3. Evolution in time of the total energy for each resonant mode ( I m ( t ) ring | E m ( t , z ) | 2 d z ): (a)  Δ = 0.055 c / n 0 L , and (b)  Δ = 0.055 c / n 0 L , when several modes are initially excited.
Fig. 4.
Fig. 4. (a) The evolution in time of the total energy for each resonant mode ( I m ( t ) ) , as the modulation frequency is periodically switched between Δ = ± 0.055 c / n 0 L , as described in (b).
Fig. 5.
Fig. 5. (a) Ring resonator under dynamic modulation by the electro-optic modulator (EOM), and in addition coupled to two external waveguides. (b) and (c) Detected signals in external waveguide 2: the top panels are the input spectra injected through external waveguide 1. The ring is modulated for a period from 0 to t f . The output spectra, after the modulation is turned off, are plotted in the remaining panels. In (b), the modulation frequency periodically switched between Δ = ± 0.055 c / n 0 L as described in Fig. 4(b). In (c), the modulation frequency is maintained constant at Δ = 0.055 c / n 0 L .

Equations (16)

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ω m = ω 0 + m Ω R ,
H = m ω m a m a m + m 2 g cos ( Ω t ) ( a m a m + 1 + a m + 1 a m ) ,
H RWA = g m ( e i Δ t c m c m + 1 + e i Δ t c m + 1 c m ) ,
H ˜ RWA = g m ( c m c m + 1 + c m + 1 c m ) + m m Δ c m c m ,
| Ψ = m v m c m | 0 | Ψ ˜ = m v ˜ m c m | 0 = m v m e i m Δ t c m | 0 ,
E ( t , r , z ) = m E m ( t , z ) E m ( r ) e i ω m t ,
( z + i β ( ω m ) ) E m ( t , z ) n g ( ω m ) c t E m ( t , z ) = 0 ,
E ( t + , r , z 0 ) = E ( t , r , z 0 ) e i α cos ( Ω t ) ,
E m ( t + , z 0 ) = J 0 ( α ) E m ( t , z 0 ) + q i q J q ( α ) [ E m + q ( t , z 0 ) e i q Δ t + E m q ( t , z 0 ) e i q Δ t ] ,
Δ E m ( t , z 0 ) = i α 2 [ E m + 1 ( t , z 0 ) e i Δ t + E m 1 ( t , z 0 ) e i Δ t ] .
T R E m T = i α 2 ( E m + 1 e i Δ T + E m 1 e i Δ T ) .
E m ( t + , z s ) = E m ( t , z s ) + S m ( t ) .
S m ( t ) = f m e 2 ln 2 ( t 20 ) 2 / 100 ,
f m = exp [ ( ω m ω c ) 2 / ( 5 Ω R ) 2 ] ,
E m ( t + , z 1,2 ) = 1 γ 1,2 2 ( t ) E m ( t , z 1,2 ) i γ 1,2 ( t ) E m ( 1,2 ) ( t , z 1,2 ) ,
E m ( 1,2 ) ( t + , z 1,2 ) = 1 γ 1,2 2 ( t ) E m ( 1,2 ) ( t , z 1,2 ) i γ 1,2 ( t ) E m ( t , z 1,2 ) ,

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