Abstract

We present here instabilities observed in the transverse profile of a continuous-wave light beam that crosses a sample of sodium vapor twice by means of a feedback mirror. The wave front and the intensity of the beam are spatially modulated by the intensity-dependent dispersive and absorptive action of the vapor. Experimental evidence indicates that the vapor acts as a phase-conjugate mirror, thus providing an active cavity when it is coupled with the feedback mirror. The instabilities develop as a consequence of sideband generation at the resonant modes of this cavity. The transverse-mode profiles of these sidebands are analyzed experimentally.

© 1988 Optical Society of America

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References

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  1. H. M. Gibbs, M. W. Derstine, K. Tai, J. F. Valley, J. V. Moloney, F. A. Hopf, M. LeBerre, E. Ressayre, and A. Tallet, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 340.
  2. J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974);M. LeBerre, E. Ressayre, A. Tallet, and F. P. Matter, J. Opt. Soc. Am. B 2, 956 (1985)
    [CrossRef]
  3. M. LeBerre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, M. C. Rushford, and N. Peyghambarian, J. Opt. Soc. Am. B 1, 591 (1984)
    [CrossRef]
  4. J. Au Yeung, D. Fekete, D. M. Pepper, and A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979)
    [CrossRef]
  5. R. C. Lind and D. G. Steel, Opt. Lett. 6, 554 (1981)
    [CrossRef] [PubMed]
  6. J. R. R. Leite, P. Simoneau, D. Bloch, S. LeBoiteaux, and M. Ducloy, Europhys. Lett. 2, 747 (1986):
    [CrossRef]
  7. N. Ioli, F. Strumia, and A. Moretti, J. Opt. Soc. Am. 61, 1251 (1971)
    [CrossRef]
  8. See “Television,” in Encyclopedia Britannica, 15th ed. (U. Chicago Press, Chicago, Ill., Vol. 18), p. 105
  9. A. E. Siegman, P. A. Belanger, and A. Hardy, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983),p. 465.
    [CrossRef]
  10. This is a common process; indeed, we see quasi-periodicity in the spectrum of our (single-line) Ar laser at any time when there is more than one transverse mode for the same longitudinal order. We use a specially calibrated aperture in the Ar laser to kill these transverse modes because, in our experimental arrangement, the Na vapor shows a great ability to amplify the fast intensity modulation of the dye laser induced by this mode beating.

1986 (1)

J. R. R. Leite, P. Simoneau, D. Bloch, S. LeBoiteaux, and M. Ducloy, Europhys. Lett. 2, 747 (1986):
[CrossRef]

1984 (1)

1981 (1)

1979 (1)

J. Au Yeung, D. Fekete, D. M. Pepper, and A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979)
[CrossRef]

1974 (1)

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974);M. LeBerre, E. Ressayre, A. Tallet, and F. P. Matter, J. Opt. Soc. Am. B 2, 956 (1985)
[CrossRef]

1971 (1)

Ashkin, A.

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974);M. LeBerre, E. Ressayre, A. Tallet, and F. P. Matter, J. Opt. Soc. Am. B 2, 956 (1985)
[CrossRef]

Au Yeung, J.

J. Au Yeung, D. Fekete, D. M. Pepper, and A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979)
[CrossRef]

Belanger, P. A.

A. E. Siegman, P. A. Belanger, and A. Hardy, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983),p. 465.
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974);M. LeBerre, E. Ressayre, A. Tallet, and F. P. Matter, J. Opt. Soc. Am. B 2, 956 (1985)
[CrossRef]

Bloch, D.

J. R. R. Leite, P. Simoneau, D. Bloch, S. LeBoiteaux, and M. Ducloy, Europhys. Lett. 2, 747 (1986):
[CrossRef]

Derstine, M. W.

H. M. Gibbs, M. W. Derstine, K. Tai, J. F. Valley, J. V. Moloney, F. A. Hopf, M. LeBerre, E. Ressayre, and A. Tallet, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 340.

Ducloy, M.

J. R. R. Leite, P. Simoneau, D. Bloch, S. LeBoiteaux, and M. Ducloy, Europhys. Lett. 2, 747 (1986):
[CrossRef]

Fekete, D.

J. Au Yeung, D. Fekete, D. M. Pepper, and A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979)
[CrossRef]

Gibbs, H. M.

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, M. C. Rushford, and N. Peyghambarian, J. Opt. Soc. Am. B 1, 591 (1984)
[CrossRef]

H. M. Gibbs, M. W. Derstine, K. Tai, J. F. Valley, J. V. Moloney, F. A. Hopf, M. LeBerre, E. Ressayre, and A. Tallet, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 340.

Hardy, A.

A. E. Siegman, P. A. Belanger, and A. Hardy, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983),p. 465.
[CrossRef]

Hopf, F. A.

H. M. Gibbs, M. W. Derstine, K. Tai, J. F. Valley, J. V. Moloney, F. A. Hopf, M. LeBerre, E. Ressayre, and A. Tallet, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 340.

Ioli, N.

LeBerre, M.

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, M. C. Rushford, and N. Peyghambarian, J. Opt. Soc. Am. B 1, 591 (1984)
[CrossRef]

H. M. Gibbs, M. W. Derstine, K. Tai, J. F. Valley, J. V. Moloney, F. A. Hopf, M. LeBerre, E. Ressayre, and A. Tallet, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 340.

LeBoiteaux, S.

J. R. R. Leite, P. Simoneau, D. Bloch, S. LeBoiteaux, and M. Ducloy, Europhys. Lett. 2, 747 (1986):
[CrossRef]

Leite, J. R. R.

J. R. R. Leite, P. Simoneau, D. Bloch, S. LeBoiteaux, and M. Ducloy, Europhys. Lett. 2, 747 (1986):
[CrossRef]

Lind, R. C.

Moloney, J. V.

H. M. Gibbs, M. W. Derstine, K. Tai, J. F. Valley, J. V. Moloney, F. A. Hopf, M. LeBerre, E. Ressayre, and A. Tallet, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 340.

Moretti, A.

Pepper, D. M.

J. Au Yeung, D. Fekete, D. M. Pepper, and A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979)
[CrossRef]

Peyghambarian, N.

Ressayre, E.

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, M. C. Rushford, and N. Peyghambarian, J. Opt. Soc. Am. B 1, 591 (1984)
[CrossRef]

H. M. Gibbs, M. W. Derstine, K. Tai, J. F. Valley, J. V. Moloney, F. A. Hopf, M. LeBerre, E. Ressayre, and A. Tallet, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 340.

Rushford, M. C.

Siegman, A. E.

A. E. Siegman, P. A. Belanger, and A. Hardy, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983),p. 465.
[CrossRef]

Simoneau, P.

J. R. R. Leite, P. Simoneau, D. Bloch, S. LeBoiteaux, and M. Ducloy, Europhys. Lett. 2, 747 (1986):
[CrossRef]

Steel, D. G.

Strumia, F.

Tai, K.

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, M. C. Rushford, and N. Peyghambarian, J. Opt. Soc. Am. B 1, 591 (1984)
[CrossRef]

H. M. Gibbs, M. W. Derstine, K. Tai, J. F. Valley, J. V. Moloney, F. A. Hopf, M. LeBerre, E. Ressayre, and A. Tallet, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 340.

Tallet, A.

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, M. C. Rushford, and N. Peyghambarian, J. Opt. Soc. Am. B 1, 591 (1984)
[CrossRef]

H. M. Gibbs, M. W. Derstine, K. Tai, J. F. Valley, J. V. Moloney, F. A. Hopf, M. LeBerre, E. Ressayre, and A. Tallet, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 340.

Valley, J. F.

H. M. Gibbs, M. W. Derstine, K. Tai, J. F. Valley, J. V. Moloney, F. A. Hopf, M. LeBerre, E. Ressayre, and A. Tallet, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 340.

Yariv, A.

J. Au Yeung, D. Fekete, D. M. Pepper, and A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979)
[CrossRef]

Europhys. Lett. (1)

J. R. R. Leite, P. Simoneau, D. Bloch, S. LeBoiteaux, and M. Ducloy, Europhys. Lett. 2, 747 (1986):
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Au Yeung, D. Fekete, D. M. Pepper, and A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979)
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974);M. LeBerre, E. Ressayre, A. Tallet, and F. P. Matter, J. Opt. Soc. Am. B 2, 956 (1985)
[CrossRef]

Other (4)

H. M. Gibbs, M. W. Derstine, K. Tai, J. F. Valley, J. V. Moloney, F. A. Hopf, M. LeBerre, E. Ressayre, and A. Tallet, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 340.

See “Television,” in Encyclopedia Britannica, 15th ed. (U. Chicago Press, Chicago, Ill., Vol. 18), p. 105

A. E. Siegman, P. A. Belanger, and A. Hardy, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983),p. 465.
[CrossRef]

This is a common process; indeed, we see quasi-periodicity in the spectrum of our (single-line) Ar laser at any time when there is more than one transverse mode for the same longitudinal order. We use a specially calibrated aperture in the Ar laser to kill these transverse modes because, in our experimental arrangement, the Na vapor shows a great ability to amplify the fast intensity modulation of the dye laser induced by this mode beating.

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Figures (16)

Fig. 1
Fig. 1

Experimental setup: La, lens to reduce the beam astigmatism; D1 and D2, diaphragms; P.B.S.’s, polarizing beam splitters; L1 and L2, doublet lenses; FM, feedback mirror (R ≥ 99%); PD’s, photodiodes; F.P., Fabry–Perot interferometer. The vacuum cell is slightly tilted, and the light reflected by the input window is detected to stabilize the input power.

Fig. 2
Fig. 2

Na flow cell. Nonmagnetic stainless steel is used throughout except for the window W, the rubber gaskets (indicated in black), and the copper gasket. The oven is suspended in the center of a Varian 2.75-in. (∼7-cm) cross by means of three narrow rods (not shown) anchored to the bottom flange. The thermal dissipation is low, and it takes about 1 h to cool the oven from the operating temperature to below 100° C. A water-cooling jacket encircles the cross pipes. The dashed line in the oven represents the mesh used to hold the Na (liquid above 93° C).

Fig. 3
Fig. 3

Typical spectra. The peaks are normally 40 dB above the background. Here only the fundamental frequency is represented in the sketches; however, several higher harmonics may have a comparable intensity. The vertical arrows indicate the approximate range of existence of the various phenomena versus the argon pressure. (a) Limit cycle. (b) Quasi-periodicity: two close peaks generate a number of side peaks. (c) The structure of (b) may transform into a broad peak when the pressure is decreased. The limit of resolution of the experiment does not allow for a clear analysis of this behavior; however, at the lower pressure, the background level rises, indicating that chaos may really develop. (d) Period doubling. This phenomenon is observed at low pressure when the lens L2 is translated (about 20 μm) from concentric alignment. (e) Period-doubled broad peak

Fig. 4
Fig. 4

Instabilities in output power versus input power. The light intensity in one point of the beam profile is represented on the y axis (in arbitrary units). (a) The photodiode detects the central spot of the backward beam. This picture shows the transition from one limit cycle to another through quasi-periodicity. The x axis represents Pin (50 mW/division), N ≅ 3 × 1012 atm/cm3, αl ≅ 1, νLν0 = −3.2 GHz, L = 226.6 cm, FP(L1) = 3.3 cm (i.e., the focal plane of lens Li is 3.3 cm to the right of the cell center), FP(L2) = −1.2 cm (1.2 cm to the left of the cell center), and P = 4 Torr Ar. (b) Symmetric oscillations seen by a detector in the central spot of the backward beam. The x axis represents Pin (50 mW/division), P = 0.3 Torr Ar, L = 226.6 cm, νLν0 = −4.6 GHz, αl = 0.13, Δ ≅ 600, N = 7 × 1012 atoms/cm3, FP(L1) = 3.3 cm, and FP(L2) = −1.3 cm. (c) Time evolution (10 nsec/division) with the same conditions as in (b) except Pin = 345 mW. The oscillation frequency is 31.1 MHz.

Fig. 5
Fig. 5

Sequence of power spectra of the photodiode signal at various Pin. Experimental conditions are the same as for Fig. 4(a). Values of Pin are (a) 145 mW, (b) 325 mW, (c) 370 mW, and (d) 440 mW.

Fig. 6
Fig. 6

The far-field beam profiles (b)–(d) observed through the Fabry–Perot interferometer at the frequencies indicated in (a) by the arrows. The Fabry–Perot free spectral range is 1.5 GHz, νLν0 = −1.6 GHz, N ≅ 2 × 1012 atoms/cm3, L = 90 cm, P = 0.3 Torr Ar, FP(L1) = 3.3 cm, and FP(L2) = −1.3 cm.

Fig. 7
Fig. 7

Sketches of the far-field beam profiles. The arrows in (a), (c), and (d) indicate that the oscillation occurs between the two extreme profiles shown; the arrows in (b) show the direction of rotation. S denotes symmetric oscillations.

Fig. 8
Fig. 8

P1D oscillation: (a) dc profile, (b) ac profile, and (c) and (d) stroboscopic views; the system switches spontaneously between (c) and (d). This corresponds to a change of the direction of rotation. N = 7 × 1012 atoms/cm3, νLν0 = −1.6 GHz, αL > 11, P = 0.3 Torr Ar, FP(L1) 3.3 cm, FP(L2) = −1.3 cm, and Pin = 450 mW

Fig. 9
Fig. 9

P3D oscillation: (a) dc profile, (b) ac profile, and (c) stroboscopic view. L = 226.6 cm, N = 6 × 1012 atoms/cm3, νLν0 = −1.63 GHz, Pin = 477 mW, P = 0.3 Torr Ar, FP(L1) = 3.3 cm, and FP(L2) = −1.3 cm.

Fig. 10
Fig. 10

Symmetric oscillations for same conditions as for Figs. 4(c) and 4(d): (a) dc profile; (b) ac profile; (c)–(e) stroboscopic views at three different delays of the reference signal.

Fig. 11
Fig. 11

Results for nonconcentric lens alignment. This profile replaces the P1D oscillation when the lens is translated in the x direction about 80 μm. Other conditions are the same as for Fig. 8.

Fig. 12
Fig. 12

Asymmetry in the input beam: (a) ac profile; (b) stroboscope view. L = 226.6 cm, N = 7 × 1013 atoms/cm3, νLν0 = −3.9 GHz, P = 4 Torr Ar, FP(L1) = 3.3 cm, FP(L2) = −1.3 cm, and Pin = 340 mW. The oscillation frequency is 29.15 MHz

Fig. 13
Fig. 13

Beating of TEM00 and TEM0±1 modes: three-dimensional plot of power as a function of r and ϕ at a particular time. In the computation the power of each of the TEM0±1 modes is 0.3 of the TEM00 power. The beam waist is the same for all three modes

Fig. 14
Fig. 14

Phase diagram at N = 6 × 1012 atoms/cm3. The matched regions show the extent of the P1D and P2D oscillations. The dashed line in the P1D region gives the loci of the maximum oscillation amplitude. The almost-straight solid line gives the boundary for the appearance of a ring in the dc profile. The dashed line between the P1D and P3D regions gives the loci of an ≅ 64-MHz oscillation. The numbers in the figure give the fundamental oscillation frequency in megahertz. L = 226.6 cm, P = 0.3 Torr Ar, FP(L1) = 3.3 cm, and FP(L2) = −1.3 cm.

Fig. 15
Fig. 15

Phase diagram at N = 9 × 1012 atoms/cm3. The notation and the other conditions are given in Fig. 14. The gray regions have broad-spectrum dynamics. Here the P2D oscillations may be replaced by a broad-spectrum oscillation for a slightly different alignment.

Fig. 16
Fig. 16

Oscillation period (Tosc) versus twice the delay time (4L/c = 2tR). The lower points refer to the right-hand scale and show a several-nanosecond displacement from 4L/c.

Equations (3)

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E 0 = A 0 exp [ i ω t ( r / w ) 2 ] + c . c . ;
E ± = A ± ( 2 r / w ) l exp { i [ ( ω ± δ ω ) t ± l ϕ ] ( r / ω ) 2 } + c . c . ,
[ | A 0 | 2 + ( | A + | 2 + | A | 2 ) ( 2 r / w ) 2 l ] exp [ 2 ( r / w ) 2 ] ,

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