Abstract

We show theoretically and confirm experimentally that, by propagating a frequency-swept ultrashort pulse through an ensemble of two-level systems, it is possible to measure the time-dependent relative phase of the pulse with respect to that of the impact-excited oscillating resonant polarization. The phase is obtained from the intensity modulation of the pulse as measured by a cross-correlation technique.

© 1985 Optical Society of America

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References

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  1. R. L. Fork, B. I. Greene, C. V. Shank, “Generation of optical pulses shorter than 0.1 psec by colliding pulse mode locking,” Appl. Phys. Lett. 38, 671 (1981).
    [CrossRef]
  2. See, for example, J.-M. Halbout, C. L. Tang, “Generation of 55-fsec optical pulses,” IEEE J. Quantum Electron. QE-19, 487 (1983);W. Dietel, J. J. Fontaine, J. C. Diels, “Intracavity pulse compression with glass: a new method of generating pulses shorter than 60 fsec,” Opt. Lett. 8, 4 (1983);see also Ref. 7.
    [CrossRef] [PubMed]
  3. J. A. Valdmanis, R. L. Fork, “Generation of optical pulses shorter than 30 fsec in a laser-balancing passive mode-locking with soliton-like pulse shaping,” J. Opt. Soc. Am. A. 1, 1337 (A)(1984).
  4. H. Nakatsuka, D. Grischkowsky, A. C. Balant, “Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion,” Phys. Rev. Lett. 47, 910 (1981).
    [CrossRef]
  5. B. Nikolaus, D. Grischkowsky, “90-fs tunable optical pulses obtained by two-stage pulse compression,” Appl. Phys. Lett. 43, 228 (1983).
    [CrossRef]
  6. C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761 (1982).
    [CrossRef]
  7. J. G. Fujimoto, A. M. Weiner, E. P. Ippen, “Generation and measurement of optical pulses as short as 16 fs,” Appl. Phys. Lett. 44, 832 (1984).
    [CrossRef]
  8. J.-M. Halbout, D. Grischkowsky, “12-femtosecond ultrashort optical pulse compression at a high repetition rate,” Appl. Phys. Lett. 45, 1281 (1984).
    [CrossRef]
  9. J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552 (1984).
    [CrossRef]
  10. J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
    [CrossRef]
  11. M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604 (1970).
    [CrossRef]
  12. We note that this approach has the limitation that the time variations in the input-pulse intensity must be slower than the time scale on which the phase of the pulse changes.
  13. The vapor cell had two 3-mm-thick windows, which did not have a significant effect on the results. In general, the linearity of the system allows one conceptually to think of both windows being at the cell input, in which case they will have the effect of changing the frequency sweep rate of the input pulse by a small amount. In the experimental and the theoretical cases considered here, this effect is negligible.
  14. D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555 (1984).
    [CrossRef]

1984 (5)

J. A. Valdmanis, R. L. Fork, “Generation of optical pulses shorter than 30 fsec in a laser-balancing passive mode-locking with soliton-like pulse shaping,” J. Opt. Soc. Am. A. 1, 1337 (A)(1984).

J. G. Fujimoto, A. M. Weiner, E. P. Ippen, “Generation and measurement of optical pulses as short as 16 fs,” Appl. Phys. Lett. 44, 832 (1984).
[CrossRef]

J.-M. Halbout, D. Grischkowsky, “12-femtosecond ultrashort optical pulse compression at a high repetition rate,” Appl. Phys. Lett. 45, 1281 (1984).
[CrossRef]

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552 (1984).
[CrossRef]

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

1983 (2)

B. Nikolaus, D. Grischkowsky, “90-fs tunable optical pulses obtained by two-stage pulse compression,” Appl. Phys. Lett. 43, 228 (1983).
[CrossRef]

See, for example, J.-M. Halbout, C. L. Tang, “Generation of 55-fsec optical pulses,” IEEE J. Quantum Electron. QE-19, 487 (1983);W. Dietel, J. J. Fontaine, J. C. Diels, “Intracavity pulse compression with glass: a new method of generating pulses shorter than 60 fsec,” Opt. Lett. 8, 4 (1983);see also Ref. 7.
[CrossRef] [PubMed]

1982 (1)

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

1981 (2)

R. L. Fork, B. I. Greene, C. V. Shank, “Generation of optical pulses shorter than 0.1 psec by colliding pulse mode locking,” Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

H. Nakatsuka, D. Grischkowsky, A. C. Balant, “Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion,” Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

1970 (1)

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604 (1970).
[CrossRef]

Auston, D. H.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

Balant, A. C.

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552 (1984).
[CrossRef]

H. Nakatsuka, D. Grischkowsky, A. C. Balant, “Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion,” Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

Cheung, K. P.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

Crisp, M. D.

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604 (1970).
[CrossRef]

Diels, J. C.

J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
[CrossRef]

Dietel, W.

J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
[CrossRef]

Dopel, E.

J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
[CrossRef]

Fontaine, J.

J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
[CrossRef]

Fork, R. L.

J. A. Valdmanis, R. L. Fork, “Generation of optical pulses shorter than 30 fsec in a laser-balancing passive mode-locking with soliton-like pulse shaping,” J. Opt. Soc. Am. A. 1, 1337 (A)(1984).

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

R. L. Fork, B. I. Greene, C. V. Shank, “Generation of optical pulses shorter than 0.1 psec by colliding pulse mode locking,” Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

Fujimoto, J. G.

J. G. Fujimoto, A. M. Weiner, E. P. Ippen, “Generation and measurement of optical pulses as short as 16 fs,” Appl. Phys. Lett. 44, 832 (1984).
[CrossRef]

Greene, B. I.

R. L. Fork, B. I. Greene, C. V. Shank, “Generation of optical pulses shorter than 0.1 psec by colliding pulse mode locking,” Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

Grischkowsky, D.

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552 (1984).
[CrossRef]

J.-M. Halbout, D. Grischkowsky, “12-femtosecond ultrashort optical pulse compression at a high repetition rate,” Appl. Phys. Lett. 45, 1281 (1984).
[CrossRef]

B. Nikolaus, D. Grischkowsky, “90-fs tunable optical pulses obtained by two-stage pulse compression,” Appl. Phys. Lett. 43, 228 (1983).
[CrossRef]

H. Nakatsuka, D. Grischkowsky, A. C. Balant, “Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion,” Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

Halbout, J.-M.

J.-M. Halbout, D. Grischkowsky, “12-femtosecond ultrashort optical pulse compression at a high repetition rate,” Appl. Phys. Lett. 45, 1281 (1984).
[CrossRef]

See, for example, J.-M. Halbout, C. L. Tang, “Generation of 55-fsec optical pulses,” IEEE J. Quantum Electron. QE-19, 487 (1983);W. Dietel, J. J. Fontaine, J. C. Diels, “Intracavity pulse compression with glass: a new method of generating pulses shorter than 60 fsec,” Opt. Lett. 8, 4 (1983);see also Ref. 7.
[CrossRef] [PubMed]

Ippen, E. P.

J. G. Fujimoto, A. M. Weiner, E. P. Ippen, “Generation and measurement of optical pulses as short as 16 fs,” Appl. Phys. Lett. 44, 832 (1984).
[CrossRef]

Kleinman, D. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

McMichael, I. C.

J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
[CrossRef]

Nakatsuka, H.

H. Nakatsuka, D. Grischkowsky, A. C. Balant, “Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion,” Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

Nikolaus, B.

B. Nikolaus, D. Grischkowsky, “90-fs tunable optical pulses obtained by two-stage pulse compression,” Appl. Phys. Lett. 43, 228 (1983).
[CrossRef]

Rothenberg, J. E.

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552 (1984).
[CrossRef]

Rudolph, V.

J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
[CrossRef]

Shank, C. V.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

R. L. Fork, B. I. Greene, C. V. Shank, “Generation of optical pulses shorter than 0.1 psec by colliding pulse mode locking,” Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

Simoni, F.

J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
[CrossRef]

Stolen, R. H.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

Tang, C. L.

See, for example, J.-M. Halbout, C. L. Tang, “Generation of 55-fsec optical pulses,” IEEE J. Quantum Electron. QE-19, 487 (1983);W. Dietel, J. J. Fontaine, J. C. Diels, “Intracavity pulse compression with glass: a new method of generating pulses shorter than 60 fsec,” Opt. Lett. 8, 4 (1983);see also Ref. 7.
[CrossRef] [PubMed]

Tomlinson, W. J.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

Torti, R.

J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
[CrossRef]

Valdmanis, J. A.

J. A. Valdmanis, R. L. Fork, “Generation of optical pulses shorter than 30 fsec in a laser-balancing passive mode-locking with soliton-like pulse shaping,” J. Opt. Soc. Am. A. 1, 1337 (A)(1984).

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

Vanherzeele, H.

J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
[CrossRef]

Weiner, A. M.

J. G. Fujimoto, A. M. Weiner, E. P. Ippen, “Generation and measurement of optical pulses as short as 16 fs,” Appl. Phys. Lett. 44, 832 (1984).
[CrossRef]

Wilhelmi, B.

J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
[CrossRef]

Yen, R.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

Appl. Phys. Lett. (5)

R. L. Fork, B. I. Greene, C. V. Shank, “Generation of optical pulses shorter than 0.1 psec by colliding pulse mode locking,” Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

B. Nikolaus, D. Grischkowsky, “90-fs tunable optical pulses obtained by two-stage pulse compression,” Appl. Phys. Lett. 43, 228 (1983).
[CrossRef]

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

J. G. Fujimoto, A. M. Weiner, E. P. Ippen, “Generation and measurement of optical pulses as short as 16 fs,” Appl. Phys. Lett. 44, 832 (1984).
[CrossRef]

J.-M. Halbout, D. Grischkowsky, “12-femtosecond ultrashort optical pulse compression at a high repetition rate,” Appl. Phys. Lett. 45, 1281 (1984).
[CrossRef]

IEEE J. Quantum Electron. (1)

See, for example, J.-M. Halbout, C. L. Tang, “Generation of 55-fsec optical pulses,” IEEE J. Quantum Electron. QE-19, 487 (1983);W. Dietel, J. J. Fontaine, J. C. Diels, “Intracavity pulse compression with glass: a new method of generating pulses shorter than 60 fsec,” Opt. Lett. 8, 4 (1983);see also Ref. 7.
[CrossRef] [PubMed]

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

J. A. Valdmanis, R. L. Fork, “Generation of optical pulses shorter than 30 fsec in a laser-balancing passive mode-locking with soliton-like pulse shaping,” J. Opt. Soc. Am. A. 1, 1337 (A)(1984).

Phys. Rev. A (1)

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604 (1970).
[CrossRef]

Phys. Rev. Lett. (3)

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

H. Nakatsuka, D. Grischkowsky, A. C. Balant, “Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion,” Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552 (1984).
[CrossRef]

Other (3)

J. C. Diels, W. Dietel, E. Dopel, J. Fontaine, I. C. McMichael, V. Rudolph, F. Simoni, R. Torti, H. Vanherzeele, B. Wilhelmi, “Colliding pulse femtosecond lasers and applications to the measurement of optical parameters,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 30.
[CrossRef]

We note that this approach has the limitation that the time variations in the input-pulse intensity must be slower than the time scale on which the phase of the pulse changes.

The vapor cell had two 3-mm-thick windows, which did not have a significant effect on the results. In general, the linearity of the system allows one conceptually to think of both windows being at the cell input, in which case they will have the effect of changing the frequency sweep rate of the input pulse by a small amount. In the experimental and the theoretical cases considered here, this effect is negligible.

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Figures (9)

Fig. 1
Fig. 1

Input pulses showing intensity (solid lines) and relative instantaneous frequency Ωi (dashed lines) versus time. Relative instantaneous frequency, given in wave numbers (cm−1), is equal to the resonance frequency of the two-level medium for Ωi = 0 and t = 0.

Fig. 2
Fig. 2

Calculated output pulses, corresponding to the input pulses of Fig. 1, versus reduced time.

Fig. 3
Fig. 3

Measured relative phase Φ(t) (dots) determined by counting the maxima and minima on the output pulses of Fig. 2 and Φ(t) (solid lines) from the input pulses of Fig. 1.

Fig. 4
Fig. 4

Relative instantaneous frequencies Ωi (dots) obtained from Fig. 3 and Ωi (solid lines) determined from input pulses of Fig. 1.

Fig. 5
Fig. 5

Schematic diagram of the experimental apparatus. PM, photomultiplier tube.

Fig. 6
Fig. 6

(a) Measured cross correlation (resolution, 0.4 psec) of the input pulse to the Na cell. (b) Measured cross correlation of the output pulse (solid line) from the Na cell.

Fig. 7
Fig. 7

(a) Calculated output pulse from the Na cell. (b) Measured cross correlation of the output pulse (solid line) compared with the cross correlation of the calculated pulse (dashed line), with a 0.4-psec probing pulse.

Fig. 8
Fig. 8

(a) Calculated output pulse from the Na cell assuming that only the D1 line is present. (b) Calculated output pulse assuming that only the D2 line is present. (c) Calculated output pulse obtained by summation of the effects of the D1 and D2 lines. The dashed lines indicate when the frequency of the pulse is resonant with the D2 lines.

Fig. 9
Fig. 9

(a) Measured output pulse from the Na cell shown on an expanded scale with the maxima and minima marked by dots. (b) Relative phase Φ(t) obtained from direct counting of the maxima and minima (dots) in (a). The dashed curve is the phase expected from a linearly swept pulse relative to the D1 line, and the solid curve relative to the D2 line.

Equations (8)

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

E ( z = 0 , t ) = Re [ H ( 0 , t ) exp ( i ω o t ) ] ,
E ¯ ( 0 , Ω ) = 1 2 π exp ( i Ω t ) E ( 0 , t ) d t .
E ( z , t ) = Re { exp ( i ω 0 t ) E ¯ ( 0 , Ω ) exp ( i Ω t ) × exp ( α z / 2 ) exp [ i n z ( Ω + ω 0 ) / c ] d Ω } ,
E out = E in + E rad ,
E in = E 0 exp [ i ( ω o τ + Φ ( τ ) + β ) ] ,
E rad = a E 0 exp [ i ( ω o τ + β ) ] .
I out | a exp [ i ( ω o τ + β ) ] + exp [ i ( ω o τ + Φ ( τ ) + β ) ] | 2 = | a + exp [ i Φ ( τ ) ] | 2 .
| a | N μ 2 z / b ,

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