Abstract

We present a one-dimensional iterative predictor-corrector finite-difference time-domain method for modeling of broadband optical pulse propagation and interaction with inhomogeneously broadened materials. The simulator is used to demonstrate two- and three-pulse photon echoes resulting from bandwidth limited pulse and matched chirp interactions with a material modeled with hundreds of equally spaced, discrete spectral lines of detuning. The results are illustrated as Bloch-sphere evolution movies.

© 2005 Optical Society of America

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  1. T.  Chang, M.  Tian, Z.W.  Barber, W. R.  Babbitt, “Numerical modeling of optical coherent transient processes with complex configurations - II. Angled beams with arbitrary phase modulations,” J. Luminescence 107, 138–145 (2004).
    [CrossRef]
  2. R.  Ziolkowski, J.  Arnold, D.  Gogny, “Ultrafast pulse interaction with two-level atoms,” Phys. Rev. 52, 3082–3094 (1995).
    [CrossRef]
  3. M.  Mitsunaga, R. G.  Brewer, “Generalized perturbation theory of coherent optical emission,” Phys. Rev. A 32, 1605–1613 (1985).
    [CrossRef] [PubMed]
  4. K.  Merkel, R. K.  Mohan, Z.  Cole, T.  Chang, A.  Olson, W.  Babbitt, “Multi-Gigahertz radar range processing of baseband and RF carrier modulated signals in Tm:YAG,” J. Luminescence 107, 62–74 (2004).
    [CrossRef]
  5. V.  Lavielle, F. D.  Seze, I.  Lorgere, J.-L. L.  Gouet, “Wideband radio frequency spectrum analyzer: Improved design and experimental results,” J. of Luminescence 107, 75–89 (2004).
    [CrossRef]
  6. Y. S.  Bai, W. R.  Babbitt, N. W.  Carlson, T. W.  Mossberg, “Real-Time Optical Waveform Convolver Cross Correlator,” Appl. Phys. Lett. 45, 714–716 (1984).
    [CrossRef]
  7. T. W.  Mossberg, “Time-Domain frequency selective optical data storage,” Opt. Lett. 7, 77–79 (1982).
    [CrossRef] [PubMed]
  8. S.  Nakamura, Applied Numerical Methods in C, 1st ed., Prentice-Hall Inc, Englewood Cliffs, New Jersey (1993).
  9. L.  Allen, J.  Eberly, Optical Resonance and Two-Level Atoms, 2nd ed., General Publishing Ltd., Don Mills, Toronto (1987).
  10. K. D.  Merkel, W. R.  Babbitt, “Coherent transient optical signal processing without brief pulses,” Appl. Opt. 35, 278–285 (1996).
    [CrossRef] [PubMed]
  11. F.  Schlottau, K. H.  Wagner, “Demonstration of a continuous scanner and time-integrating correlator using spatial-spectral holography,” J. Luminescence 107, 90–102 (2004).
    [CrossRef]

2004

T.  Chang, M.  Tian, Z.W.  Barber, W. R.  Babbitt, “Numerical modeling of optical coherent transient processes with complex configurations - II. Angled beams with arbitrary phase modulations,” J. Luminescence 107, 138–145 (2004).
[CrossRef]

K.  Merkel, R. K.  Mohan, Z.  Cole, T.  Chang, A.  Olson, W.  Babbitt, “Multi-Gigahertz radar range processing of baseband and RF carrier modulated signals in Tm:YAG,” J. Luminescence 107, 62–74 (2004).
[CrossRef]

V.  Lavielle, F. D.  Seze, I.  Lorgere, J.-L. L.  Gouet, “Wideband radio frequency spectrum analyzer: Improved design and experimental results,” J. of Luminescence 107, 75–89 (2004).
[CrossRef]

F.  Schlottau, K. H.  Wagner, “Demonstration of a continuous scanner and time-integrating correlator using spatial-spectral holography,” J. Luminescence 107, 90–102 (2004).
[CrossRef]

1996

1995

R.  Ziolkowski, J.  Arnold, D.  Gogny, “Ultrafast pulse interaction with two-level atoms,” Phys. Rev. 52, 3082–3094 (1995).
[CrossRef]

1985

M.  Mitsunaga, R. G.  Brewer, “Generalized perturbation theory of coherent optical emission,” Phys. Rev. A 32, 1605–1613 (1985).
[CrossRef] [PubMed]

1984

Y. S.  Bai, W. R.  Babbitt, N. W.  Carlson, T. W.  Mossberg, “Real-Time Optical Waveform Convolver Cross Correlator,” Appl. Phys. Lett. 45, 714–716 (1984).
[CrossRef]

1982

Allen, L.

L.  Allen, J.  Eberly, Optical Resonance and Two-Level Atoms, 2nd ed., General Publishing Ltd., Don Mills, Toronto (1987).

Arnold, J.

R.  Ziolkowski, J.  Arnold, D.  Gogny, “Ultrafast pulse interaction with two-level atoms,” Phys. Rev. 52, 3082–3094 (1995).
[CrossRef]

Babbitt, W.

K.  Merkel, R. K.  Mohan, Z.  Cole, T.  Chang, A.  Olson, W.  Babbitt, “Multi-Gigahertz radar range processing of baseband and RF carrier modulated signals in Tm:YAG,” J. Luminescence 107, 62–74 (2004).
[CrossRef]

Babbitt, W. R.

T.  Chang, M.  Tian, Z.W.  Barber, W. R.  Babbitt, “Numerical modeling of optical coherent transient processes with complex configurations - II. Angled beams with arbitrary phase modulations,” J. Luminescence 107, 138–145 (2004).
[CrossRef]

K. D.  Merkel, W. R.  Babbitt, “Coherent transient optical signal processing without brief pulses,” Appl. Opt. 35, 278–285 (1996).
[CrossRef] [PubMed]

Y. S.  Bai, W. R.  Babbitt, N. W.  Carlson, T. W.  Mossberg, “Real-Time Optical Waveform Convolver Cross Correlator,” Appl. Phys. Lett. 45, 714–716 (1984).
[CrossRef]

Bai, Y. S.

Y. S.  Bai, W. R.  Babbitt, N. W.  Carlson, T. W.  Mossberg, “Real-Time Optical Waveform Convolver Cross Correlator,” Appl. Phys. Lett. 45, 714–716 (1984).
[CrossRef]

Barber, Z.W.

T.  Chang, M.  Tian, Z.W.  Barber, W. R.  Babbitt, “Numerical modeling of optical coherent transient processes with complex configurations - II. Angled beams with arbitrary phase modulations,” J. Luminescence 107, 138–145 (2004).
[CrossRef]

Brewer, R. G.

M.  Mitsunaga, R. G.  Brewer, “Generalized perturbation theory of coherent optical emission,” Phys. Rev. A 32, 1605–1613 (1985).
[CrossRef] [PubMed]

Carlson, N. W.

Y. S.  Bai, W. R.  Babbitt, N. W.  Carlson, T. W.  Mossberg, “Real-Time Optical Waveform Convolver Cross Correlator,” Appl. Phys. Lett. 45, 714–716 (1984).
[CrossRef]

Chang, T.

K.  Merkel, R. K.  Mohan, Z.  Cole, T.  Chang, A.  Olson, W.  Babbitt, “Multi-Gigahertz radar range processing of baseband and RF carrier modulated signals in Tm:YAG,” J. Luminescence 107, 62–74 (2004).
[CrossRef]

T.  Chang, M.  Tian, Z.W.  Barber, W. R.  Babbitt, “Numerical modeling of optical coherent transient processes with complex configurations - II. Angled beams with arbitrary phase modulations,” J. Luminescence 107, 138–145 (2004).
[CrossRef]

Cole, Z.

K.  Merkel, R. K.  Mohan, Z.  Cole, T.  Chang, A.  Olson, W.  Babbitt, “Multi-Gigahertz radar range processing of baseband and RF carrier modulated signals in Tm:YAG,” J. Luminescence 107, 62–74 (2004).
[CrossRef]

Eberly, J.

L.  Allen, J.  Eberly, Optical Resonance and Two-Level Atoms, 2nd ed., General Publishing Ltd., Don Mills, Toronto (1987).

Gogny, D.

R.  Ziolkowski, J.  Arnold, D.  Gogny, “Ultrafast pulse interaction with two-level atoms,” Phys. Rev. 52, 3082–3094 (1995).
[CrossRef]

Gouet, J.-L. L.

V.  Lavielle, F. D.  Seze, I.  Lorgere, J.-L. L.  Gouet, “Wideband radio frequency spectrum analyzer: Improved design and experimental results,” J. of Luminescence 107, 75–89 (2004).
[CrossRef]

Lavielle, V.

V.  Lavielle, F. D.  Seze, I.  Lorgere, J.-L. L.  Gouet, “Wideband radio frequency spectrum analyzer: Improved design and experimental results,” J. of Luminescence 107, 75–89 (2004).
[CrossRef]

Lorgere, I.

V.  Lavielle, F. D.  Seze, I.  Lorgere, J.-L. L.  Gouet, “Wideband radio frequency spectrum analyzer: Improved design and experimental results,” J. of Luminescence 107, 75–89 (2004).
[CrossRef]

Merkel, K.

K.  Merkel, R. K.  Mohan, Z.  Cole, T.  Chang, A.  Olson, W.  Babbitt, “Multi-Gigahertz radar range processing of baseband and RF carrier modulated signals in Tm:YAG,” J. Luminescence 107, 62–74 (2004).
[CrossRef]

Merkel, K. D.

Mitsunaga, M.

M.  Mitsunaga, R. G.  Brewer, “Generalized perturbation theory of coherent optical emission,” Phys. Rev. A 32, 1605–1613 (1985).
[CrossRef] [PubMed]

Mohan, R. K.

K.  Merkel, R. K.  Mohan, Z.  Cole, T.  Chang, A.  Olson, W.  Babbitt, “Multi-Gigahertz radar range processing of baseband and RF carrier modulated signals in Tm:YAG,” J. Luminescence 107, 62–74 (2004).
[CrossRef]

Mossberg, T. W.

Y. S.  Bai, W. R.  Babbitt, N. W.  Carlson, T. W.  Mossberg, “Real-Time Optical Waveform Convolver Cross Correlator,” Appl. Phys. Lett. 45, 714–716 (1984).
[CrossRef]

T. W.  Mossberg, “Time-Domain frequency selective optical data storage,” Opt. Lett. 7, 77–79 (1982).
[CrossRef] [PubMed]

Nakamura, S.

S.  Nakamura, Applied Numerical Methods in C, 1st ed., Prentice-Hall Inc, Englewood Cliffs, New Jersey (1993).

Olson, A.

K.  Merkel, R. K.  Mohan, Z.  Cole, T.  Chang, A.  Olson, W.  Babbitt, “Multi-Gigahertz radar range processing of baseband and RF carrier modulated signals in Tm:YAG,” J. Luminescence 107, 62–74 (2004).
[CrossRef]

Schlottau, F.

F.  Schlottau, K. H.  Wagner, “Demonstration of a continuous scanner and time-integrating correlator using spatial-spectral holography,” J. Luminescence 107, 90–102 (2004).
[CrossRef]

Seze, F. D.

V.  Lavielle, F. D.  Seze, I.  Lorgere, J.-L. L.  Gouet, “Wideband radio frequency spectrum analyzer: Improved design and experimental results,” J. of Luminescence 107, 75–89 (2004).
[CrossRef]

Tian, M.

T.  Chang, M.  Tian, Z.W.  Barber, W. R.  Babbitt, “Numerical modeling of optical coherent transient processes with complex configurations - II. Angled beams with arbitrary phase modulations,” J. Luminescence 107, 138–145 (2004).
[CrossRef]

Wagner, K. H.

F.  Schlottau, K. H.  Wagner, “Demonstration of a continuous scanner and time-integrating correlator using spatial-spectral holography,” J. Luminescence 107, 90–102 (2004).
[CrossRef]

Ziolkowski, R.

R.  Ziolkowski, J.  Arnold, D.  Gogny, “Ultrafast pulse interaction with two-level atoms,” Phys. Rev. 52, 3082–3094 (1995).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

Y. S.  Bai, W. R.  Babbitt, N. W.  Carlson, T. W.  Mossberg, “Real-Time Optical Waveform Convolver Cross Correlator,” Appl. Phys. Lett. 45, 714–716 (1984).
[CrossRef]

J. Luminescence

F.  Schlottau, K. H.  Wagner, “Demonstration of a continuous scanner and time-integrating correlator using spatial-spectral holography,” J. Luminescence 107, 90–102 (2004).
[CrossRef]

T.  Chang, M.  Tian, Z.W.  Barber, W. R.  Babbitt, “Numerical modeling of optical coherent transient processes with complex configurations - II. Angled beams with arbitrary phase modulations,” J. Luminescence 107, 138–145 (2004).
[CrossRef]

K.  Merkel, R. K.  Mohan, Z.  Cole, T.  Chang, A.  Olson, W.  Babbitt, “Multi-Gigahertz radar range processing of baseband and RF carrier modulated signals in Tm:YAG,” J. Luminescence 107, 62–74 (2004).
[CrossRef]

J. of Luminescence

V.  Lavielle, F. D.  Seze, I.  Lorgere, J.-L. L.  Gouet, “Wideband radio frequency spectrum analyzer: Improved design and experimental results,” J. of Luminescence 107, 75–89 (2004).
[CrossRef]

Opt. Lett.

Phys. Rev.

R.  Ziolkowski, J.  Arnold, D.  Gogny, “Ultrafast pulse interaction with two-level atoms,” Phys. Rev. 52, 3082–3094 (1995).
[CrossRef]

Phys. Rev. A

M.  Mitsunaga, R. G.  Brewer, “Generalized perturbation theory of coherent optical emission,” Phys. Rev. A 32, 1605–1613 (1985).
[CrossRef] [PubMed]

Other

S.  Nakamura, Applied Numerical Methods in C, 1st ed., Prentice-Hall Inc, Englewood Cliffs, New Jersey (1993).

L.  Allen, J.  Eberly, Optical Resonance and Two-Level Atoms, 2nd ed., General Publishing Ltd., Don Mills, Toronto (1987).

Supplementary Material (4)

» Media 1: MPG (6427 KB)     
» Media 2: MPG (2359 KB)     
» Media 3: MPG (14635 KB)     
» Media 4: MPG (1719 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Typical inhomogeneously broadened absorption spectrum consisting of various detuned homogeneous absorption lines centered around ω 0. ΔΓI and ΔΓ H are measures of inhomogeneous and homogeneous broadening, respectively

Fig. 2.
Fig. 2.

Short-pulse photon echo simulation: The electric field due to a femtosecond pulse train (a) excites the inhomogeneously broadened absorber. The delay between pulse 1 and 2 cause the sinusoidal modulation of the inversion (b), whose periodicity is inversely proportional to the pulse seperation time. The detuned lines of the IBA dephase at their particular rate, as shown by the rotating coordinate frame, in-phase component of the Bloch-vector (c). Two and three-pulse echoes are visible as re-phasings of the different detuned lines, seen as hyperbolic Moiré patterns in (c). Finally, (d) shows the electric field detected at the output of the material. The programming pulses are visible, as well as the 2P and 3P photon echoes. The vertical scale in (d) is amplified by ten to make the echoes more apparent.

Fig. 3.
Fig. 3.

A single frame from the rotating coordinate frame Bloch-vector evolution movie. On the left is a perspective view of the Bloch-sphere, in which the color-coding of the vectors indicates the amount of detuning (blue and red colors indicate frequency up and down shifts, respectively) for a particular homogeneous line. The right view shows a projection of the coherence, or ρ 1,ρ 2 plane. The plot at the bottom indicates the rotating coordinate frame electric-field as well as an artificial decoherence function, which is applied in order to shorten the simulation time (as described in the text). [Small movie file 1.7MB] [Large movie file 6.3MB]

Fig. 4.
Fig. 4.

Chirped-pulse photon echo simulation: The electric field of a set of chirped pulses (a) (with chirp-rates b and 2b covering the same bandwidth) sequentially excites the different homogeneous lines of the inhomogeneously broadened absorber with linear frequency sweeps. The inversion that is caused due to chirps one and two is a chirped spectral grating (b). The spectrally detuned lines dephase at their particular rate, as shown by the rotating coordinate frame, in-phase component of the Bloch-vectors (c). Two and three-pulse echoes are visible in plot (c) as hyperbolic Moiré patterns indicating the re-phasing of the detuned lines. The detected output electric field (×10)(d), shows pulse compressed 2P and 3P echoes, as well as an E-field ringing after the chirps known as chirp free induction decay.

Fig. 5.
Fig. 5.

A single frame from the rotating coordinate frame Bloch-vector evolution movie. Again, the color-coding of the vectors indicates the amount of detuning (blue and red colors indicate frequency up and down shifts, respectively). The plot at the bottom indicates the rotating coordinate frame electric-field consisting of three chirps with the same power spectral density, aswell as an artificial decoherence function. The moment of re-coherence for the three-pulse echo is shown. [Small movie file 2.3MB] [Large movie file 14MB]

Equations (21)

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t H y = 1 μ 0 z E x
t E x = 1 ε 0 z H y 1 ε 0 t P x
= 1 ε 0 z H y j N j γ ε 0 T 2 ρ 1 j ( z ) + N j γ ω j ε 0 ρ 2 j ( z )
t ρ 1 , j = 1 T 2 ρ 1 , j + ω j ρ 2 , j
t ρ 2 , j = ω j ρ 1 , j 1 T 2 ρ 2 , j + 2 γ h ̅ E x ρ 3 , j
t ρ 3, j = 2 γ h ̅ E x ρ 2 , j 1 T 1 ( ρ 3 , j ρ 30 )
H y m + 1 2 n + 1 2 = H y m + 1 2 n 1 2 Δ t μ 0 Δ z [ E x m + 1 n E x m n ]
E x m n + 1 = E x m n Δ t [ 1 ε 0 Δ z [ H y m + 1 2 n + 1 2 H y m 1 2 n + 1 2 ] A 1 2 [ u 1 m n + 1 + u 1 m n ] + B 1 2 [ u 2 m n + 1 + u 2 m n ] ]
u 1 m n + 1 = u 1 m n + Δ t [ ω 0 1 2 [ u 2 m n + 1 + u 2 m n ] ]
u 2 m n + 1 = u 2 m n + Δ t [ ω 0 1 2 [ u 1 m n + 1 + u 1 m n ] + 1 2 [ E x m n + 1 + E x m n ] × { 1 2 C + [ u 3 m n + 1 + u 3 m n ] + D } ]
u 3 m n + 1 = u 3 m n Δ t [ C { 1 2 [ E x m n + 1 + E x m n ] × 1 2 [ u 2 m n + 1 + u 2 m n ] } ]
A = N j γ ε 0 T 2 exp [ t T 2 ] ,
B = N j γ ω 0 ε 0 exp [ t T 2 ] ,
C + = 2 γ h ̅ exp [ t ( 1 T 1 1 T 2 ) ] ,
C = 2 γ h ̅ exp [ t ( 1 T 2 1 T 1 ) ] ,
D = 2 γ ρ 30 h ̅ exp [ t T 2 ] .
X new = X old + Δ t f ( X old , X predicted )
E x m n + 1 = E x m n Δ t [ 1 ε 0 Δ z [ H y m + 1 2 n + 1 2 H y m 1 2 n + 1 2 ] + j = 0 N lines { A j 1 2 [ u 1 , j m n + 1 + u 1 , j m n ]
+ B j 1 2 [ u 2 , j m n + 1 + u 2 , j m n ] } ]
E o ( t ) = S 1 * ( ω ) S 2 ( ω ) S 3 ( ω ) e i ω t d ω
= e i 2 π b 1 ω 2 e i [ 2 π b 2 ω 2 + ω τ 12 ] e i [ 2 π b 3 ω 2 + ω τ 3 ] e i ω t d ω δ ( t τ 12 ) ,

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