Abstract

Techniques are described for synthesizing a statistically well-characterized radio-frequency noise spectrum and imposing this noise spectrum on a well-stabilized laser beam by means of extracavity phase and frequency modulation. The basis of the noise spectrum is the Gaussian noise-voltage spectrum derived from shot noise. The Gaussian property is carefully preserved at every step, ensuring that correlation functions to all orders can be represented in terms of the lowest-order correlation function, which, in turn, is the Fourier transform of the power spectrum, according to the Wiener–Khintchine theorem. The power spectrum can be measured and controlled, in a range that has Lorentzian and Gaussian forms as limiting cases, by controlling the bandwidth and amplitude of the noise voltage at the source. The objective of this methodology is the definitive experimental study of nonlinear-optical absorption processes, all such processes requiring knowledge of higher-order correlation functions for complete definition.

© 1988 Optical Society of America

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References

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  1. D. S. Elliott, R. Roy, and S. J. Smith, Phys. Rev. A 26, 12 (1982).
    [Crossref]
  2. D. S. Elliott, R. Roy, and S. J. Smith, in Spectral Line Shapes, K. Burnett, ed. (de Gruyter, Berlin1983),Vol. 2, pp. 989–998.
  3. D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. A 32, 887 (1985).
    [Crossref] [PubMed]
  4. M. W. Hamilton, D. S. Elliott, K. Arnett, S. J. Smith, M. Dziemballa, and P. Zoller, Phys. Rev. A 36, 178 (1987).
    [Crossref] [PubMed]
  5. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).
  6. S. O. Rice, Bell Tel. J. 23, 282 (1944).
    [Crossref]
  7. M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945).
    [Crossref]
  8. B. R. Mollow, Phys. Rev. 175, 1555 (1968).
    [Crossref]
  9. F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
    [Crossref]
  10. P. Avan and C. Cohen-Tannoudji, J. Phys. B 10, 171 (1977).
    [Crossref]
  11. H. Metcalf, M. Hamilton, J. Brandenberger, and K. Arnett, Department of Physics, State University of New York, Stony Brook, New York 11794 (personal communication);see also K. Arnett, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1988).
  12. P. Zoller, in Multiphoton Processes, P. Lambropoulos and S. J. Smith, eds. (Springer-Verlag, Berlin, 1984), pp. 68–75.
    [Crossref]
  13. See, for example, L. A. Westling, M. G. Raymer, and J. J. Snyder, J. Opt. Soc. Am. B 1, 150 (1984).
    [Crossref]
  14. P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, and M. D. Levenson, J. Phys. B 11, 1733 (1978).
    [Crossref]
  15. S. E. Moody, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1975).
  16. S. E. Moody and M. Lambropoulos, Phys. Rev. A 15, 1497 (1977).
    [Crossref]
  17. P. B. Hogan, S. J. Smith, A. T. Georges, and P. Lambropoulos, Phys. Rev. A 18, 587 (1978).
    [Crossref]
  18. R. M. Whitely and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
    [Crossref]
  19. A. T. Georges and P. Lambropoulos, Phys. Rev. A 20, 991 (1979).
    [Crossref]
  20. D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. Lett. 53, 439 (1984).
    [Crossref]
  21. M. W. Hamilton, D. S. Elliott, K. Arnett, and S. J. Smith, Phys. Rev. A 33, 778 (1986).
    [Crossref] [PubMed]
  22. A. I. Burshstein, Sov. Phys. JETP 21, 567 (1966).
  23. G. S. Agarwal, Phys. Rev. Lett. 37, 1383 (1976).
    [Crossref]
  24. J. H. Eberly, Phys. Rev. Lett. 37, 1387 (1976).
    [Crossref]
  25. H. Haken, Handb. Phys. 25, 2 (1970).
  26. D. Middleton, Phil. Mag. 42, 689 (1951);An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  27. See, for example, A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).
  28. L. Hollberg, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1983).
  29. Reference Data for Radio Engineers/ITT, 6th ed.(Sams, Indianapolis, Ind., 1975).

1987 (1)

M. W. Hamilton, D. S. Elliott, K. Arnett, S. J. Smith, M. Dziemballa, and P. Zoller, Phys. Rev. A 36, 178 (1987).
[Crossref] [PubMed]

1986 (1)

M. W. Hamilton, D. S. Elliott, K. Arnett, and S. J. Smith, Phys. Rev. A 33, 778 (1986).
[Crossref] [PubMed]

1985 (1)

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. A 32, 887 (1985).
[Crossref] [PubMed]

1984 (2)

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. Lett. 53, 439 (1984).
[Crossref]

See, for example, L. A. Westling, M. G. Raymer, and J. J. Snyder, J. Opt. Soc. Am. B 1, 150 (1984).
[Crossref]

1982 (1)

D. S. Elliott, R. Roy, and S. J. Smith, Phys. Rev. A 26, 12 (1982).
[Crossref]

1979 (1)

A. T. Georges and P. Lambropoulos, Phys. Rev. A 20, 991 (1979).
[Crossref]

1978 (2)

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, and M. D. Levenson, J. Phys. B 11, 1733 (1978).
[Crossref]

P. B. Hogan, S. J. Smith, A. T. Georges, and P. Lambropoulos, Phys. Rev. A 18, 587 (1978).
[Crossref]

1977 (2)

S. E. Moody and M. Lambropoulos, Phys. Rev. A 15, 1497 (1977).
[Crossref]

P. Avan and C. Cohen-Tannoudji, J. Phys. B 10, 171 (1977).
[Crossref]

1976 (3)

R. M. Whitely and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[Crossref]

G. S. Agarwal, Phys. Rev. Lett. 37, 1383 (1976).
[Crossref]

J. H. Eberly, Phys. Rev. Lett. 37, 1387 (1976).
[Crossref]

1970 (1)

H. Haken, Handb. Phys. 25, 2 (1970).

1968 (1)

B. R. Mollow, Phys. Rev. 175, 1555 (1968).
[Crossref]

1966 (1)

A. I. Burshstein, Sov. Phys. JETP 21, 567 (1966).

1965 (1)

F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
[Crossref]

1951 (1)

D. Middleton, Phil. Mag. 42, 689 (1951);An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

1945 (1)

M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945).
[Crossref]

1944 (1)

S. O. Rice, Bell Tel. J. 23, 282 (1944).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, Phys. Rev. Lett. 37, 1383 (1976).
[Crossref]

Agostini, P.

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, and M. D. Levenson, J. Phys. B 11, 1733 (1978).
[Crossref]

Arecchi, F. T.

F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
[Crossref]

Arnett, K.

M. W. Hamilton, D. S. Elliott, K. Arnett, S. J. Smith, M. Dziemballa, and P. Zoller, Phys. Rev. A 36, 178 (1987).
[Crossref] [PubMed]

M. W. Hamilton, D. S. Elliott, K. Arnett, and S. J. Smith, Phys. Rev. A 33, 778 (1986).
[Crossref] [PubMed]

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. A 32, 887 (1985).
[Crossref] [PubMed]

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. Lett. 53, 439 (1984).
[Crossref]

H. Metcalf, M. Hamilton, J. Brandenberger, and K. Arnett, Department of Physics, State University of New York, Stony Brook, New York 11794 (personal communication);see also K. Arnett, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1988).

Avan, P.

P. Avan and C. Cohen-Tannoudji, J. Phys. B 10, 171 (1977).
[Crossref]

Brandenberger, J.

H. Metcalf, M. Hamilton, J. Brandenberger, and K. Arnett, Department of Physics, State University of New York, Stony Brook, New York 11794 (personal communication);see also K. Arnett, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1988).

Burshstein, A. I.

A. I. Burshstein, Sov. Phys. JETP 21, 567 (1966).

Cohen-Tannoudji, C.

P. Avan and C. Cohen-Tannoudji, J. Phys. B 10, 171 (1977).
[Crossref]

Dziemballa, M.

M. W. Hamilton, D. S. Elliott, K. Arnett, S. J. Smith, M. Dziemballa, and P. Zoller, Phys. Rev. A 36, 178 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. H. Eberly, Phys. Rev. Lett. 37, 1387 (1976).
[Crossref]

Elliott, D. S.

M. W. Hamilton, D. S. Elliott, K. Arnett, S. J. Smith, M. Dziemballa, and P. Zoller, Phys. Rev. A 36, 178 (1987).
[Crossref] [PubMed]

M. W. Hamilton, D. S. Elliott, K. Arnett, and S. J. Smith, Phys. Rev. A 33, 778 (1986).
[Crossref] [PubMed]

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. A 32, 887 (1985).
[Crossref] [PubMed]

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. Lett. 53, 439 (1984).
[Crossref]

D. S. Elliott, R. Roy, and S. J. Smith, Phys. Rev. A 26, 12 (1982).
[Crossref]

D. S. Elliott, R. Roy, and S. J. Smith, in Spectral Line Shapes, K. Burnett, ed. (de Gruyter, Berlin1983),Vol. 2, pp. 989–998.

Georges, A. T.

A. T. Georges and P. Lambropoulos, Phys. Rev. A 20, 991 (1979).
[Crossref]

P. B. Hogan, S. J. Smith, A. T. Georges, and P. Lambropoulos, Phys. Rev. A 18, 587 (1978).
[Crossref]

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, and M. D. Levenson, J. Phys. B 11, 1733 (1978).
[Crossref]

Haken, H.

H. Haken, Handb. Phys. 25, 2 (1970).

Hamilton, M.

H. Metcalf, M. Hamilton, J. Brandenberger, and K. Arnett, Department of Physics, State University of New York, Stony Brook, New York 11794 (personal communication);see also K. Arnett, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1988).

Hamilton, M. W.

M. W. Hamilton, D. S. Elliott, K. Arnett, S. J. Smith, M. Dziemballa, and P. Zoller, Phys. Rev. A 36, 178 (1987).
[Crossref] [PubMed]

M. W. Hamilton, D. S. Elliott, K. Arnett, and S. J. Smith, Phys. Rev. A 33, 778 (1986).
[Crossref] [PubMed]

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. A 32, 887 (1985).
[Crossref] [PubMed]

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. Lett. 53, 439 (1984).
[Crossref]

Hogan, P. B.

P. B. Hogan, S. J. Smith, A. T. Georges, and P. Lambropoulos, Phys. Rev. A 18, 587 (1978).
[Crossref]

Hollberg, L.

L. Hollberg, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1983).

Lambropoulos, M.

S. E. Moody and M. Lambropoulos, Phys. Rev. A 15, 1497 (1977).
[Crossref]

Lambropoulos, P.

A. T. Georges and P. Lambropoulos, Phys. Rev. A 20, 991 (1979).
[Crossref]

P. B. Hogan, S. J. Smith, A. T. Georges, and P. Lambropoulos, Phys. Rev. A 18, 587 (1978).
[Crossref]

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, and M. D. Levenson, J. Phys. B 11, 1733 (1978).
[Crossref]

Levenson, M. D.

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, and M. D. Levenson, J. Phys. B 11, 1733 (1978).
[Crossref]

Metcalf, H.

H. Metcalf, M. Hamilton, J. Brandenberger, and K. Arnett, Department of Physics, State University of New York, Stony Brook, New York 11794 (personal communication);see also K. Arnett, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1988).

Middleton, D.

D. Middleton, Phil. Mag. 42, 689 (1951);An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Mollow, B. R.

B. R. Mollow, Phys. Rev. 175, 1555 (1968).
[Crossref]

Moody, S. E.

S. E. Moody and M. Lambropoulos, Phys. Rev. A 15, 1497 (1977).
[Crossref]

S. E. Moody, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1975).

Papoulis, A.

See, for example, A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

Raymer, M. G.

Rice, S. O.

S. O. Rice, Bell Tel. J. 23, 282 (1944).
[Crossref]

Roy, R.

D. S. Elliott, R. Roy, and S. J. Smith, Phys. Rev. A 26, 12 (1982).
[Crossref]

D. S. Elliott, R. Roy, and S. J. Smith, in Spectral Line Shapes, K. Burnett, ed. (de Gruyter, Berlin1983),Vol. 2, pp. 989–998.

Smith, S. J.

M. W. Hamilton, D. S. Elliott, K. Arnett, S. J. Smith, M. Dziemballa, and P. Zoller, Phys. Rev. A 36, 178 (1987).
[Crossref] [PubMed]

M. W. Hamilton, D. S. Elliott, K. Arnett, and S. J. Smith, Phys. Rev. A 33, 778 (1986).
[Crossref] [PubMed]

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. A 32, 887 (1985).
[Crossref] [PubMed]

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. Lett. 53, 439 (1984).
[Crossref]

D. S. Elliott, R. Roy, and S. J. Smith, Phys. Rev. A 26, 12 (1982).
[Crossref]

P. B. Hogan, S. J. Smith, A. T. Georges, and P. Lambropoulos, Phys. Rev. A 18, 587 (1978).
[Crossref]

D. S. Elliott, R. Roy, and S. J. Smith, in Spectral Line Shapes, K. Burnett, ed. (de Gruyter, Berlin1983),Vol. 2, pp. 989–998.

Snyder, J. J.

Stroud, C. R.

R. M. Whitely and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[Crossref]

Uhlenbeck, G. E.

M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945).
[Crossref]

Wang, M. C.

M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945).
[Crossref]

Westling, L. A.

Wheatley, S. E.

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, and M. D. Levenson, J. Phys. B 11, 1733 (1978).
[Crossref]

Whitely, R. M.

R. M. Whitely and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[Crossref]

Yariv, A.

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

Zoller, P.

M. W. Hamilton, D. S. Elliott, K. Arnett, S. J. Smith, M. Dziemballa, and P. Zoller, Phys. Rev. A 36, 178 (1987).
[Crossref] [PubMed]

P. Zoller, in Multiphoton Processes, P. Lambropoulos and S. J. Smith, eds. (Springer-Verlag, Berlin, 1984), pp. 68–75.
[Crossref]

Bell Tel. J. (1)

S. O. Rice, Bell Tel. J. 23, 282 (1944).
[Crossref]

Handb. Phys. (1)

H. Haken, Handb. Phys. 25, 2 (1970).

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

J. Phys. B (2)

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, and M. D. Levenson, J. Phys. B 11, 1733 (1978).
[Crossref]

P. Avan and C. Cohen-Tannoudji, J. Phys. B 10, 171 (1977).
[Crossref]

Phil. Mag. (1)

D. Middleton, Phil. Mag. 42, 689 (1951);An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Phys. Rev. (1)

B. R. Mollow, Phys. Rev. 175, 1555 (1968).
[Crossref]

Phys. Rev. A (8)

D. S. Elliott, R. Roy, and S. J. Smith, Phys. Rev. A 26, 12 (1982).
[Crossref]

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. A 32, 887 (1985).
[Crossref] [PubMed]

M. W. Hamilton, D. S. Elliott, K. Arnett, S. J. Smith, M. Dziemballa, and P. Zoller, Phys. Rev. A 36, 178 (1987).
[Crossref] [PubMed]

S. E. Moody and M. Lambropoulos, Phys. Rev. A 15, 1497 (1977).
[Crossref]

P. B. Hogan, S. J. Smith, A. T. Georges, and P. Lambropoulos, Phys. Rev. A 18, 587 (1978).
[Crossref]

R. M. Whitely and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[Crossref]

A. T. Georges and P. Lambropoulos, Phys. Rev. A 20, 991 (1979).
[Crossref]

M. W. Hamilton, D. S. Elliott, K. Arnett, and S. J. Smith, Phys. Rev. A 33, 778 (1986).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

G. S. Agarwal, Phys. Rev. Lett. 37, 1383 (1976).
[Crossref]

J. H. Eberly, Phys. Rev. Lett. 37, 1387 (1976).
[Crossref]

D. S. Elliott, M. W. Hamilton, K. Arnett, and S. J. Smith, Phys. Rev. Lett. 53, 439 (1984).
[Crossref]

F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
[Crossref]

Rev. Mod. Phys. (1)

M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945).
[Crossref]

Sov. Phys. JETP (1)

A. I. Burshstein, Sov. Phys. JETP 21, 567 (1966).

Other (8)

See, for example, A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

L. Hollberg, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1983).

Reference Data for Radio Engineers/ITT, 6th ed.(Sams, Indianapolis, Ind., 1975).

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

D. S. Elliott, R. Roy, and S. J. Smith, in Spectral Line Shapes, K. Burnett, ed. (de Gruyter, Berlin1983),Vol. 2, pp. 989–998.

H. Metcalf, M. Hamilton, J. Brandenberger, and K. Arnett, Department of Physics, State University of New York, Stony Brook, New York 11794 (personal communication);see also K. Arnett, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1988).

P. Zoller, in Multiphoton Processes, P. Lambropoulos and S. J. Smith, eds. (Springer-Verlag, Berlin, 1984), pp. 68–75.
[Crossref]

S. E. Moody, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1975).

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Figures (13)

Fig. 1
Fig. 1

The double-peaked double-optical-resonance spectrum of the Rabi split 3 2P1/2 state in sodium, taken from Ref. 17. Of interest is the asymmetry reversal that occurs with progressively greater detuning Δa of the saturating laser field.

Fig. 2
Fig. 2

A constant-amplitude, fluctuating frequency signal with a rms deviation frequency of approximately 40 MHz, on a scale of 2.5 sec/division, is taken from a display on a 350-MHz bandwidth oscilloscope. The limited frequency response of the oscilloscope is responsible for some loss of amplitude from the faster fluctuations.

Fig. 3
Fig. 3

Calculated laser power spectra corresponding to b/2π = 7 MHz and β/2π = 1, 10, 100 MHz, using Eq. (22) for the power spectrum of a laser frequency modulated with Gaussian noise.

Fig. 4
Fig. 4

Simplified block diagram of the frequency-modulation apparatus, showing the principal functional steps.

Fig. 5
Fig. 5

Noise voltage power spectrum measured with a rf spectrum analyzer. This noise source was used for frequency modulation (AOM technique) for frequencies as great as 6 MHz. A similar, wider band noise source was used for phase modulation.

Fig. 6
Fig. 6

Configuration of an active network designed for shaping the noise power spectrum; C’s, capacitors.

Fig. 7
Fig. 7

Measured probability distribution of the random noise voltage produced by the avalanche-diode noise generator. The measurement was carried out with a wideband (800-MHz) transient digitizer. This distribution is based on 1200 digitized waveforms, each yielding 512 data points. The smooth line is a fitted Gaussian function. The wings are shown magnified by a factor of 100.

Fig. 8
Fig. 8

A typical tuning curve of the VCO. The operating point was ∼9 V, corresponding to a slope of ∼80 MHz/V. The rms noise signal voltage is typically less than 100 mV; the portion of the tuning curve used is essentially linear.

Fig. 9
Fig. 9

Schematic diagram of the frequency-modulation apparatus.

Fig. 10
Fig. 10

Schematic diagram of the phase-modulation apparatus.

Fig. 11
Fig. 11

Configuration of the 200-MHz cross-over network indicated in Fig. 10: L’s, inductors; C’s, capacitors.

Fig. 12
Fig. 12

A block diagram of the complete apparatus used to study the effects of fluctuations on the 3S–5S two-photon absorption profile in atomic sodium. The fast photodiode and rf spectrum analyzer are used to determine the power spectrum of the laser field, and the phase modulator and optical cavity are used for calibrating the frequency scan of the laser.

Fig. 13
Fig. 13

A power spectrum, recorded as described in connection with Fig. 12, is shown for the parameters b/2π = 7 MHz and β/2π = 80 MHz. The pure Lorentzian and Gaussian forms with the same FWHM are shown for comparison. The small peaks on the left are artifacts introduced in the heterodyne detection process.

Tables (1)

Tables Icon

Table 1 Examples of the Fractional Deviation of the Even Moments of the Noise Distribution from Those Required of a Gaussian Processa

Equations (47)

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

E ( t ) = E ( t ) exp { i [ ω 0 t + ϕ ( t ) ] } ,
E ( t ) = E 0 exp { i [ ω 0 t + ϕ ( t ) ] } .
E ( t ) = E ( t ) exp ( i ω 0 t ) .
E ( t ) = E 0 exp { i [ ω 0 t + ϕ ( t ) ] } .
ϕ ( t ) = t 0 t ω ( t ) d t .
R ω ( τ ) = ω ( t ) ω ( t + τ ) = 2 b δ ( τ ) ,
R ω ( τ ) = ω ( t ) ω ( t + τ ) = b β exp ( β | τ | ) ,
P ω ( ω ) = 2 b [ 1 + ( ω / β ) 2 ] 1 ,
R E ( τ ) = ½ E ( t ) E * ( t + τ ) .
R E ( τ ) = E 0 2 2 exp ( i ω 0 τ ) exp { i [ ϕ ( t + τ ) ϕ ( t ) ] } .
exp { i [ ϕ ( t + τ ) ϕ ( t ) ] } = exp { ½ [ ϕ ( t + τ ) ϕ ( t ) ] 2 } .
[ ϕ ( t + τ ) ϕ ( t ) ] 2 = [ t t + τ ω ( t ) d t ] 2 .
[ ϕ ( t + τ ) ϕ ( t ) ] 2 = τ / 2 τ / 2 ω ( t ) d t τ / 2 τ / 2 ω ( t ) d t
[ ϕ ( t + τ ) ϕ ( t ) ] 2 = τ / 2 τ / 2 τ / 2 τ / 2 ω ( t ) ω ( t ) d t d t .
[ ϕ ( t + τ ) ϕ ( t ) ] 2 = τ τ ω ( 0 ) ω ( T ) ( τ | T | ) d T .
R E ( τ ) = E 0 2 2 exp ( i ω 0 τ ) exp [ 1 2 τ τ R ω ( t ) ( τ | t | ) d t ] .
P ( ω ) = e i ω τ R ( τ ) d τ .
R ( τ ) = 1 2 π e i ω τ P ( ω ) d ω .
P E ( ω ) = E 0 2 2 d τ exp [ i ( ω ω 0 ) τ ] × exp [ 1 2 τ τ R ω ( t ) ( τ | t | ) d t ] .
[ ϕ ( t + τ ) ϕ ( t ) ] 2 = τ τ d t ( τ | t | ) 1 2 π e i ω t P ω ( ω ) d ω .
[ ϕ ( t + τ ) ϕ ( t ) ] 2 = 2 π d ω P ω ( ω ) ( sin ω τ / 2 ω ) 2 .
P E ( ω ) = E 0 2 2 d τ exp [ i ( ω ω 0 ) τ ] × exp [ 1 π d ω P ω ( ω ) ( sin ω τ / 2 ω ) 2 ] ,
1 2 τ τ b β exp { β | t | } ( τ | t | ) d t ,
b { | τ | + [ exp ( β | τ | ) 1 ] / β } .
P E ( ω ) = E 0 2 2 d τ exp [ i ( ω ω 0 ) τ ] × exp ( b { | τ | + [ exp ( β | τ | ) 1 ] / β } ) .
P E ( ω ) β b = E 0 2 2 d τ exp [ i ( ω ω 0 ) τ ] exp ( b | τ | ) = E 0 2 2 2 b ( ω ω 0 ) 2 + b 2 .
b { | τ | [ exp ( β | τ | ) 1 ] / β } b β τ 2 2 .
P E ( ω ) β b = E 0 2 2 d τ exp [ i ( ω ω 0 ) τ ] exp ( b β τ 2 2 ) = E 0 2 2 ( 2 π b β ) 1 / 2 exp [ ( ω ω 0 ) 2 2 b β ] ,
P ϕ ( ω ) = ω 2 P ω ( ω ) .
[ ϕ ( t + τ ) ϕ ( t ) ] 2 .
2 [ ϕ 2 ( 0 ) ϕ ( τ ) ϕ ( 0 ) ] .
ϕ 2 ( 0 ) = 1 2 π P ϕ ( ω ) d ω
ϕ ( τ ) ϕ ( 0 ) = 1 2 π P ϕ ( ω ) e i ω τ d ω ,
[ ϕ ( t + τ ) ϕ ( t ) ] 2 = 2 π d ω P ϕ ( ω ) sin 2 ( ω τ / 2 ) .
P ω ( ω ) = 2 b [ 1 + ( ω / β ) 2 ] 1 .
P ϕ ( ω ) = 2 b ω 2 [ 1 + ( ω / β ) 2 ] 1 .
ϕ ˙ ( t ) = D V ( t ) ,
2 λ a sin θ = λ / n ,
P EOM = 1 2 π P ϕ ( ω ) d ω ,
P EOM = 2 b π [ 1 ω L 1 β ( π 2 tan 1 ω L β ) ] .
ω ( t 1 ) ω ( t 2 ) ω ( t 2 n ) = ω ( t i ) ω ( t j ) ω ( t k ) ω ( t l ) ,
ω ( t 1 ) ω ( t 2 ) ω ( t 2 n 1 ) = 0 .
V ( t ) V ( t + τ ) = 2 b δ ( τ ) .
V ( t ) V ( t + τ ) = b β exp ( β | τ | ) .
ϕ ˙ ( t ) = D V ( t ) ,
b = D 2 b .
V 2 n ( 2 n ) ! n ! 2 n V 2 n V 2 n ,

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