Abstract

We present universal formulas for the spectral and temporal output optical fields from a linear traveling-wave medium whose refractive index changes during its propagation within the medium. These formulas agree with known changes in central wavelength and energy that are associated with adiabatic wavelength conversion (AWC). Moreover, they reveal new changes to the optical pulses that have not been noticed, such as pulse compression and spectral broadening. Most significantly, we find that AWC alters the pulse power, pulse chirp, and pulse delay. All of these effects depend on whether the central wavelength is blueshifted or redshifted, the first sign of asymmetry to be reported for AWC. These findings impact the applications of AWC to optical signal processing in microphotonic and nanophotonic structures as well as in lightwave systems.

© 2011 Optical Society of America

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  1. M. F. Yanik and S. Fan, Phys. Rev. Lett. 92, 083901 (2004).
    [CrossRef] [PubMed]
  2. M. F. Yanik and S. Fan, Phys. Rev. Lett. 93, 173903 (2004).
    [CrossRef] [PubMed]
  3. M. Notomi and S. Mitsugi, Phys. Rev. A 73, 051803(R)(2006).
    [CrossRef]
  4. S. Preble, Q. Xu, and M. Lipson, Nat. Photon. 1, 293 (2007).
    [CrossRef]
  5. P. Dong, S. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, Phys. Rev. Lett. 100, 033904 (2008).
    [CrossRef] [PubMed]
  6. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, Phys. Rev. Lett. 102, 043907 (2009).
    [CrossRef] [PubMed]
  7. T. Kampfrath, D. M. Beggs, T. F. Krauss, and L. Kuipers, Opt. Lett. 34, 3418 (2009).
    [CrossRef] [PubMed]
  8. T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, Phys. Rev. A 81, 043837 (2010).
    [CrossRef]
  9. Z. Gaburro, M. Ghulinyan, F. Riboli, and L. Pavesi, Opt. Express 14, 7270 (2006).
    [CrossRef] [PubMed]
  10. H. Goldstein, Classical Mechanics (Addison-Wesley, 1980).

2010 (1)

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, Phys. Rev. A 81, 043837 (2010).
[CrossRef]

2009 (2)

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

T. Kampfrath, D. M. Beggs, T. F. Krauss, and L. Kuipers, Opt. Lett. 34, 3418 (2009).
[CrossRef] [PubMed]

2008 (1)

P. Dong, S. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, Phys. Rev. Lett. 100, 033904 (2008).
[CrossRef] [PubMed]

2007 (1)

S. Preble, Q. Xu, and M. Lipson, Nat. Photon. 1, 293 (2007).
[CrossRef]

2006 (2)

2004 (2)

M. F. Yanik and S. Fan, Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, Phys. Rev. Lett. 93, 173903 (2004).
[CrossRef] [PubMed]

Beggs, D. M.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, Phys. Rev. A 81, 043837 (2010).
[CrossRef]

T. Kampfrath, D. M. Beggs, T. F. Krauss, and L. Kuipers, Opt. Lett. 34, 3418 (2009).
[CrossRef] [PubMed]

Dong, P.

P. Dong, S. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, Phys. Rev. Lett. 100, 033904 (2008).
[CrossRef] [PubMed]

Fan, S.

M. F. Yanik and S. Fan, Phys. Rev. Lett. 93, 173903 (2004).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

Gaburro, Z.

Ghulinyan, M.

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, 1980).

Kampfrath, T.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, Phys. Rev. A 81, 043837 (2010).
[CrossRef]

T. Kampfrath, D. M. Beggs, T. F. Krauss, and L. Kuipers, Opt. Lett. 34, 3418 (2009).
[CrossRef] [PubMed]

Krauss, T. F.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, Phys. Rev. A 81, 043837 (2010).
[CrossRef]

T. Kampfrath, D. M. Beggs, T. F. Krauss, and L. Kuipers, Opt. Lett. 34, 3418 (2009).
[CrossRef] [PubMed]

Kuipers, L.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, Phys. Rev. A 81, 043837 (2010).
[CrossRef]

T. Kampfrath, D. M. Beggs, T. F. Krauss, and L. Kuipers, Opt. Lett. 34, 3418 (2009).
[CrossRef] [PubMed]

Kuramochi, E.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

Lipson, M.

P. Dong, S. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, Phys. Rev. Lett. 100, 033904 (2008).
[CrossRef] [PubMed]

S. Preble, Q. Xu, and M. Lipson, Nat. Photon. 1, 293 (2007).
[CrossRef]

Manipatruni, S.

P. Dong, S. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, Phys. Rev. Lett. 100, 033904 (2008).
[CrossRef] [PubMed]

Melloni, A.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Mitsugi, S.

M. Notomi and S. Mitsugi, Phys. Rev. A 73, 051803(R)(2006).
[CrossRef]

Notomi, M.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

M. Notomi and S. Mitsugi, Phys. Rev. A 73, 051803(R)(2006).
[CrossRef]

Pavesi, L.

Preble, S.

P. Dong, S. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, Phys. Rev. Lett. 100, 033904 (2008).
[CrossRef] [PubMed]

S. Preble, Q. Xu, and M. Lipson, Nat. Photon. 1, 293 (2007).
[CrossRef]

Riboli, F.

Robinson, J. T.

P. Dong, S. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, Phys. Rev. Lett. 100, 033904 (2008).
[CrossRef] [PubMed]

Tanabe, T.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

Taniyama, H.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

White, T. P.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Xu, Q.

S. Preble, Q. Xu, and M. Lipson, Nat. Photon. 1, 293 (2007).
[CrossRef]

Yanik, M. F.

M. F. Yanik and S. Fan, Phys. Rev. Lett. 93, 173903 (2004).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

Nat. Photon. (1)

S. Preble, Q. Xu, and M. Lipson, Nat. Photon. 1, 293 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (2)

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, Phys. Rev. A 81, 043837 (2010).
[CrossRef]

M. Notomi and S. Mitsugi, Phys. Rev. A 73, 051803(R)(2006).
[CrossRef]

Phys. Rev. Lett. (4)

P. Dong, S. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, Phys. Rev. Lett. 100, 033904 (2008).
[CrossRef] [PubMed]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, Phys. Rev. Lett. 93, 173903 (2004).
[CrossRef] [PubMed]

Other (1)

H. Goldstein, Classical Mechanics (Addison-Wesley, 1980).

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Figures (3)

Fig. 1
Fig. 1

Input and output pulses of linear systems: (a) time-invariant and (b) time-variant medium. Each temporal slice of input pulse is delayed by a transit time T r , which is constant for the time-invariant case, but time dependent for the time-variant case.

Fig. 2
Fig. 2

Input (left column) and output (right column) traces showing the power and phase profiles of Gaussian pulses in the temporal and spectral domains. The dashed (blue), solid (green), and dotted (red) curves correspond to stretching factors of s = 3 / 2 , 1, and 2 / 3 , respectively. We choose a linear variation of n ( τ ) for s 1 , while preserving the interval T f T i = 0.5 n 1 L / c . .

Fig. 3
Fig. 3

Four dynamic refractive index profiles and the corresponding output pulses. In all cases, the refractive index is initially n 1 and becomes n 2 after the change. The output pulses are identical in temporal shape and spectral power, but have different time delays.

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 ,
t T i ( c / n 1 ) d τ + T i T f [ c / n ( τ ) ] d τ + T f t + T r ( t ) ( c / n 2 ) d τ = L .
T r ( t ) = ( 1 s ) t / s + T r 0 ,
T r 0 = n 2 L / c + T f T i / s n 2 T i T f [ 1 / n ( τ ) ] d τ .
E out ( t ) = s E in ( s t s T ro ) ,
E ˜ out ( ω ) = E ˜ in ( ω / s ) exp ( i ω T ro ) .
E in ( t ) = E 0 e t 2 / 2 T 0 2 i ω 1 t ,
E ˜ in ( ω ) = E 0 T 0 2 π e T 0 2 ( ω ω 1 ) 2 / 2 ,
E out ( t ) = s E 0 e s 2 ( t T r 0 ) 2 / 2 T 0 2 i s ω 1 ( t T r 0 ) ,
E ˜ out ( ω ) = E 0 T 0 2 π e T 0 2 ( ω s ω 1 ) 2 / ( 2 s 2 ) i ω 1 T r 0 .
J out = | s E in ( s t s T r 0 ) | 2 d t s ω = | E in ( t ) | 2 d t ω = J in .

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