Abstract

We use a pulse-compression chirp-transform algorithm to generate broadband photonic arbitrary waveforms. A phase-locked frequency agile laser provides the needed broadband frequency scans. We theoretically show and experimentally demonstrate that the residual laser frequency errors do not significantly alter the waveform generation. The experiment exhibits features such as the operation at the telecom wavelength of 1.5μm, the counterpropagating, crossed-polarized beam configuration, and the storage of the initial waveform in the active medium. The last specificity gives access to large time-bandwidth product values.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Ménager, J.-L. Le Gouët, and I. Lorgeré, “Time-to-frequency Fourier transformation with photon echoes,” Opt. Lett. 26, 1397-1399 (2001).
    [CrossRef]
  2. V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. Le Gouët, and I. Lorgeré, “High resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450-1457 (2004).
    [CrossRef]
  3. V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, and I. Lorgeré, “Wideband and high resolution coherent optical transients with a frequency agile laser oscillator,” Opt. Lett. 31, 3264-3266 (2006).
    [CrossRef] [PubMed]
  4. Z. W. Barber, M. Tian, R. R. Reibel, and W. R. Babbitt, “Optical pulse shaping using optical coherent transients,” Opt. Express 10, 1145-1150 (2002).
    [PubMed]
  5. C. J. Renner, R. R. Reibel, M. Tian, T. Chang, and W. R. Babbitt, “Broadband photonic arbitrary waveform generation based on spatial-spectral holographic materials,” J. Opt. Soc. Am. B 24, 2979-2987 (2007).
    [CrossRef]
  6. L. I. Bluestein, “A linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. AU-18, 451-455 (1970).
    [CrossRef]
  7. M. A. Jack, P. M. Grant, and J. H. Collins, “The theory design and applications of surface acoustic wave Fourier transform processors,” Proc. IEEE 68, 450-468 (1980).
    [CrossRef]
  8. V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, C.Gagnol, E. Ducloux, and I. Lorgeré, “Phase locking of a frequency agile laser oscillator,” Appl. Phys. Lett. 89, 261115 (2006).
    [CrossRef]
  9. G. Gorju, V. Crozatier, I. Lorgeré, J.-L. Le Gouët, and F. Bretenaker, “Wideband RF spectral analyzer based on spectral-spatial holography in Tm3+:YAG achieved with a highly stabilized frequency chirped laser,” J. Lumin. 127, 110-115 (2007).
    [CrossRef]
  10. T. Böttger, “Laser frequency stabilization to spectral hole burning frequency reference in erbium-doped crystals: material and device optimization,” Ph.D.dissertation (Montana State University, 2002).

2007

G. Gorju, V. Crozatier, I. Lorgeré, J.-L. Le Gouët, and F. Bretenaker, “Wideband RF spectral analyzer based on spectral-spatial holography in Tm3+:YAG achieved with a highly stabilized frequency chirped laser,” J. Lumin. 127, 110-115 (2007).
[CrossRef]

C. J. Renner, R. R. Reibel, M. Tian, T. Chang, and W. R. Babbitt, “Broadband photonic arbitrary waveform generation based on spatial-spectral holographic materials,” J. Opt. Soc. Am. B 24, 2979-2987 (2007).
[CrossRef]

2006

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, and I. Lorgeré, “Wideband and high resolution coherent optical transients with a frequency agile laser oscillator,” Opt. Lett. 31, 3264-3266 (2006).
[CrossRef] [PubMed]

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, C.Gagnol, E. Ducloux, and I. Lorgeré, “Phase locking of a frequency agile laser oscillator,” Appl. Phys. Lett. 89, 261115 (2006).
[CrossRef]

2004

V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. Le Gouët, and I. Lorgeré, “High resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450-1457 (2004).
[CrossRef]

2002

2001

1980

M. A. Jack, P. M. Grant, and J. H. Collins, “The theory design and applications of surface acoustic wave Fourier transform processors,” Proc. IEEE 68, 450-468 (1980).
[CrossRef]

1970

L. I. Bluestein, “A linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. AU-18, 451-455 (1970).
[CrossRef]

Babbitt, W. R.

Barber, Z. W.

Bluestein, L. I.

L. I. Bluestein, “A linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. AU-18, 451-455 (1970).
[CrossRef]

Böttger, T.

T. Böttger, “Laser frequency stabilization to spectral hole burning frequency reference in erbium-doped crystals: material and device optimization,” Ph.D.dissertation (Montana State University, 2002).

Bretenaker, F.

G. Gorju, V. Crozatier, I. Lorgeré, J.-L. Le Gouët, and F. Bretenaker, “Wideband RF spectral analyzer based on spectral-spatial holography in Tm3+:YAG achieved with a highly stabilized frequency chirped laser,” J. Lumin. 127, 110-115 (2007).
[CrossRef]

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, and I. Lorgeré, “Wideband and high resolution coherent optical transients with a frequency agile laser oscillator,” Opt. Lett. 31, 3264-3266 (2006).
[CrossRef] [PubMed]

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, C.Gagnol, E. Ducloux, and I. Lorgeré, “Phase locking of a frequency agile laser oscillator,” Appl. Phys. Lett. 89, 261115 (2006).
[CrossRef]

V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. Le Gouët, and I. Lorgeré, “High resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450-1457 (2004).
[CrossRef]

Chang, T.

Collins, J. H.

M. A. Jack, P. M. Grant, and J. H. Collins, “The theory design and applications of surface acoustic wave Fourier transform processors,” Proc. IEEE 68, 450-468 (1980).
[CrossRef]

Crozatier, V.

G. Gorju, V. Crozatier, I. Lorgeré, J.-L. Le Gouët, and F. Bretenaker, “Wideband RF spectral analyzer based on spectral-spatial holography in Tm3+:YAG achieved with a highly stabilized frequency chirped laser,” J. Lumin. 127, 110-115 (2007).
[CrossRef]

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, and I. Lorgeré, “Wideband and high resolution coherent optical transients with a frequency agile laser oscillator,” Opt. Lett. 31, 3264-3266 (2006).
[CrossRef] [PubMed]

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, C.Gagnol, E. Ducloux, and I. Lorgeré, “Phase locking of a frequency agile laser oscillator,” Appl. Phys. Lett. 89, 261115 (2006).
[CrossRef]

V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. Le Gouët, and I. Lorgeré, “High resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450-1457 (2004).
[CrossRef]

Ducloux, E.

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, C.Gagnol, E. Ducloux, and I. Lorgeré, “Phase locking of a frequency agile laser oscillator,” Appl. Phys. Lett. 89, 261115 (2006).
[CrossRef]

Gagnol, C.

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, C.Gagnol, E. Ducloux, and I. Lorgeré, “Phase locking of a frequency agile laser oscillator,” Appl. Phys. Lett. 89, 261115 (2006).
[CrossRef]

Gorju, G.

G. Gorju, V. Crozatier, I. Lorgeré, J.-L. Le Gouët, and F. Bretenaker, “Wideband RF spectral analyzer based on spectral-spatial holography in Tm3+:YAG achieved with a highly stabilized frequency chirped laser,” J. Lumin. 127, 110-115 (2007).
[CrossRef]

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, and I. Lorgeré, “Wideband and high resolution coherent optical transients with a frequency agile laser oscillator,” Opt. Lett. 31, 3264-3266 (2006).
[CrossRef] [PubMed]

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, C.Gagnol, E. Ducloux, and I. Lorgeré, “Phase locking of a frequency agile laser oscillator,” Appl. Phys. Lett. 89, 261115 (2006).
[CrossRef]

Grant, P. M.

M. A. Jack, P. M. Grant, and J. H. Collins, “The theory design and applications of surface acoustic wave Fourier transform processors,” Proc. IEEE 68, 450-468 (1980).
[CrossRef]

Jack, M. A.

M. A. Jack, P. M. Grant, and J. H. Collins, “The theory design and applications of surface acoustic wave Fourier transform processors,” Proc. IEEE 68, 450-468 (1980).
[CrossRef]

Lavielle, V.

V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. Le Gouët, and I. Lorgeré, “High resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450-1457 (2004).
[CrossRef]

Le Gouët, J. -L.

G. Gorju, V. Crozatier, I. Lorgeré, J.-L. Le Gouët, and F. Bretenaker, “Wideband RF spectral analyzer based on spectral-spatial holography in Tm3+:YAG achieved with a highly stabilized frequency chirped laser,” J. Lumin. 127, 110-115 (2007).
[CrossRef]

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, and I. Lorgeré, “Wideband and high resolution coherent optical transients with a frequency agile laser oscillator,” Opt. Lett. 31, 3264-3266 (2006).
[CrossRef] [PubMed]

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, C.Gagnol, E. Ducloux, and I. Lorgeré, “Phase locking of a frequency agile laser oscillator,” Appl. Phys. Lett. 89, 261115 (2006).
[CrossRef]

V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. Le Gouët, and I. Lorgeré, “High resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450-1457 (2004).
[CrossRef]

L. Ménager, J.-L. Le Gouët, and I. Lorgeré, “Time-to-frequency Fourier transformation with photon echoes,” Opt. Lett. 26, 1397-1399 (2001).
[CrossRef]

Lorgeré, I.

G. Gorju, V. Crozatier, I. Lorgeré, J.-L. Le Gouët, and F. Bretenaker, “Wideband RF spectral analyzer based on spectral-spatial holography in Tm3+:YAG achieved with a highly stabilized frequency chirped laser,” J. Lumin. 127, 110-115 (2007).
[CrossRef]

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, and I. Lorgeré, “Wideband and high resolution coherent optical transients with a frequency agile laser oscillator,” Opt. Lett. 31, 3264-3266 (2006).
[CrossRef] [PubMed]

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, C.Gagnol, E. Ducloux, and I. Lorgeré, “Phase locking of a frequency agile laser oscillator,” Appl. Phys. Lett. 89, 261115 (2006).
[CrossRef]

V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. Le Gouët, and I. Lorgeré, “High resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450-1457 (2004).
[CrossRef]

L. Ménager, J.-L. Le Gouët, and I. Lorgeré, “Time-to-frequency Fourier transformation with photon echoes,” Opt. Lett. 26, 1397-1399 (2001).
[CrossRef]

Ménager, L.

Reibel, R. R.

Renner, C. J.

Tian, M.

Appl. Phys. Lett.

V. Crozatier, G. Gorju, J.-L. Le Gouët, F. Bretenaker, C.Gagnol, E. Ducloux, and I. Lorgeré, “Phase locking of a frequency agile laser oscillator,” Appl. Phys. Lett. 89, 261115 (2006).
[CrossRef]

IEEE J. Quantum Electron.

V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. Le Gouët, and I. Lorgeré, “High resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450-1457 (2004).
[CrossRef]

IEEE Trans. Audio Electroacoust.

L. I. Bluestein, “A linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. AU-18, 451-455 (1970).
[CrossRef]

J. Lumin.

G. Gorju, V. Crozatier, I. Lorgeré, J.-L. Le Gouët, and F. Bretenaker, “Wideband RF spectral analyzer based on spectral-spatial holography in Tm3+:YAG achieved with a highly stabilized frequency chirped laser,” J. Lumin. 127, 110-115 (2007).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Proc. IEEE

M. A. Jack, P. M. Grant, and J. H. Collins, “The theory design and applications of surface acoustic wave Fourier transform processors,” Proc. IEEE 68, 450-468 (1980).
[CrossRef]

Other

T. Böttger, “Laser frequency stabilization to spectral hole burning frequency reference in erbium-doped crystals: material and device optimization,” Ph.D.dissertation (Montana State University, 2002).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Chirp transform with photon echo: instantaneous frequency and intensity of the driving fields as a function of time. The dispersive filter is engraved by the first two overlapping pulses. Their frequency is scanned at rates r 1 and r 2 , over a spectral interval of width Δ. The probe pulse is chirped at rate r 3 over the same interval. The delay of the first two excitations is a function of frequency that varies from T = ( ν 1 ν 2 ) / r 2 at ν 1 to T + Δ / r 3 at ν 1 + Δ . The characteristic time evolution of the material response is given by the inverse spectral span Δ 1 . The figure illustrates the response to a rectangular probe of duration Δ / r 3 . With respect to the engraving fields, the response undergoes a Δ 2 / r 2 compression.

Fig. 2
Fig. 2

TBP and modulation bandwidth. The signal to be compressed, s ( t ) , is transposed as a single sideband on the optical carrier with the help of a Δ 0 -bandwidth modulator. Depending on whether s ( t ) is carried by E 3 ( t ) (upper box) or by E 2 ( t ) (lower box), the available TBP is given by Δ Δ 0 / r 3 or by Δ Δ 0 / r 2 . For a given TBP, the required modulator bandwidth is r 2 / r 3 smaller in the latter case.

Fig. 3
Fig. 3

Temporal compression and laser frequency errors. (a) SPE response normalized width Δ τ 1 / 2 (solid line) and normalized inverse coherently scanned interval Δ Γ / r 3 (dashed line) as a function of Δ Γ / r 3 . [(b) and (c)] Temporal profiles of the SPE response at Δ Γ / r 3 = 0 and 4, respectively.

Fig. 4
Fig. 4

Experimental setup. The chirped laser beam is split in two arms that counterpropagate to the Er 3 + : YSO . On one arm, the AO modulator AO2 transposes the signal on E 2 ( t ) . On the other arm, the shifter AO1 takes care of E 1 ( t ) and E 3 ( t ) time ordering and provides the chirp rate difference r 1 r 2 . E 1 ( t ) and E 3 ( t ) are cross-polarized with E 2 ( t ) . The SPE response is detected on photodiode PD.

Fig. 5
Fig. 5

Laser frequency time diagram. The AO modulation operates on top of the laser direct scan to provide the various chirp rates that are needed for engraving and probing. Both AOs offer a bandwidth of 35   MHz centered at 80 MHz. The 17 MHz-wide lower side of AO2 bandwidth is used to transpose s ( t ) on E 2 ( t ) . The 18 MHz-wide upper side of AO1 bandwidth is used to generate the small r 1 r 2 chirp rate difference needed to match r 3 .

Fig. 6
Fig. 6

Upper box: experimentally observed spectral distribution of the absorbing ions. The shaded area represents the 1.2 GHz-wide scanning range of the laser. Lower box: power Fourier transform of the spectral distribution over the 1.2 GHz-wide working interval.

Fig. 7
Fig. 7

Generation of a 1 μ s -long burst of ten pulses. The lower box displays the first pulse with a time scale magnification of ≈35. The pulse duration is close to the 1 ns expected value.

Fig. 8
Fig. 8

Generation of a 10 μ s -long burst of ten pulses. The lower box displays the first pulse with a time scale magnification of ≈500. The pulse duration is close to the 1 ns expected value. The dashed (respectively, dotted dashed) line represents the theoretically predicted decay accounting for the 6.7 × 10 3 s 1 relaxation rate of the optical dipole alone (respectively, accounting for both the optical dipole relaxation and the laser frequency errors at rate Γ = 1.1 × 10 5 s 1 ).

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

| s ̃ ( f = r t ) | = | ( s ( t ) e i π r t 2 ) e i π r t 2 | ,
E e ( t ) = E 1 ( t ) E 2 ( t ) E 3 ( t ) .
I e ( τ , Γ ) = I e ( 0 , 0 ) e Γ T 2 γ 2 + ( 2 π Δ τ ) 2 [ 1 e γ   cos ( 2 π Δ τ ) γ e γ sin ( 2 π Δ τ ) 2 π Δ τ ] ,
I e = ν 1 ν 1 + Δ d ν ν 1 ν 1 + Δ d ν E ̃ 1 ( ν ) E ̃ 2 ( ν ) E ̃ 2 ( ν ) E ̃ 1 ( ν ) E ̃ 3 ( ν ) E ̃ 3 ( ν ) e 2 i π ( ν ν ) t ,
E j ( t ) = E j 0 e 2 i π ν j t + i π ϵ j r j t 2 + i ϕ ( t ) ,
e i ϕ ( t ) i ϕ ( t ) = e ( 1 / 2 ) Γ | t t | .
E ̃ 1 ( ν ) E ̃ 2 ( ν ) = E 10 E 20 d t d t e 2 i π [ ( ν ν 1 ) t ( ν ν 2 ) t ] i π [ r 1 t 2 r 2 t 2 ] 1 / 2 Γ | t t | .
t , t u = t t ,     v = r 1 t r 2 t r 1 r 2 ,
E ̃ 1 ( ν ) E ̃ 2 ( ν ) = η r 3 E 10 E 20 e i π ( ν 2 ν 1 ) 2 / ( r 2 r 1 ) d f e i ( π / r 3 ) ( ν ν 1 f + r 3 T ) 2 L ( f ) ,
L ( f ) = Γ ( 2 π f ) 2 + ( Γ / 2 ) 2 .
E ̃ 3 ( ν ) E ̃ 3 ( ν ) = r 3 1 | E 30 | 2 d f e i ( π / r 3 ) [ ( ν ν 3 + f ) 2 ( ν ν 3 f ) 2 ] L ( f ) .
E ̃ 1 ( ν ) E ̃ 2 ( ν ) E ̃ 1 ( ν ) E ̃ 2 ( ν ) E ̃ 3 ( ν ) E ̃ 3 ( ν ) = η 2 | E 10 | 2 | E 20 | 2 | E 30 | 2 e 2 i π ( ν ν ) ( t 3 + T ) d f d f L ( f ) L ( f ) e i ( π / r 3 ) ( f 2 f 2 ) + 2 i π [ ( ν ν 1 ) / r 3 + T ] f 2 i π [ ( ν ν 1 ) / r 3 + T ] f d f L ( f ) e 2 i π ( ν ν ) f / r 3 ,
E ̃ 1 ( ν ) E ̃ 2 ( ν ) E ̃ 1 ( ν ) E ̃ 2 ( ν ) E ̃ 3 ( ν ) E ̃ 3 ( ν ) = η 2 | E 10 | 2 | E 20 | 2 | E 30 | 2 e 2 i π ( ν ν ) ( t 3 + T ) e ( Γ / 2 r 3 ) [ | ν ν 1 + r 3 T | + | ν ν 1 + r 3 T | + | ν ν | ] .
I e = η 2 | E 10 | 2 | E 20 | 2 | E 30 | 2 r 3 T Δ + r 3 T d ν r 3 T Δ + r 3 T d ν e 2 i π ( ν ν ) τ e ( ν + ν + | ν ν | ) Γ / ( 2 r 3 ) ,
I e ( τ , Γ ) = I e ( 0 , 0 ) e Γ T 2 γ 2 + ( 2 π Δ τ ) 2 [ 1 e γ   cos ( 2 π Δ τ ) γ e γ sin ( 2 π Δ τ ) 2 π Δ τ ] ,

Metrics