Abstract

The dispersive effects of high-reflectivity broadband mirrors on femtosecond optical pulses have been analyzed for three different multilayer structures. In each case the high-reflectivity zone can be divided into two different regions symmetrically located around the mirror central frequency: high-dispersion and low-dispersion regions. The calculated temporal behavior of the reflected pulse shows high distortion of the pulse profile, a frequency chirp, and a broadening as high as a factor of 5.6, due to a single reflection, within the high-dispersion region. The use of these types of mirror should therefore be strictly limited to their low-dispersion side.

© 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. 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).
    [Crossref] [PubMed]
  3. R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, “Femtosecond Optical Pulses,” IEEE J. Quantum Electron. QE-19, 500 (1983).
    [Crossref]
  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. C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. Tomlinson, “Compression of Femtosecond Optical Pulses,” Appl. Phys. Lett. 40, 761 (1982).
    [Crossref]
  6. B. Nikolaus, D. Grischkowsky, “12× Pulse Compression Using Optical Fibers,” Appl. Phys. Lett. 42, 1 (1983).
    [Crossref]
  7. A. M. Weiner, J. G. Fujimoto, E. P. Ippen, “Compression and Shaping of Femtosecond Pulses,” in Technical Digest, Ultrafast Phenomena Conference, (Optical Society of America, Washington, D.C., 1984), paper TuA4.
  8. J.-M. Halbout, D. Grischkowsky, “12-fs Ultrashort Optical Pulse Compression at a High Repetition Rate,” Appl. Phys. Lett. 45, 1281 (1984).
    [Crossref]
  9. S. De Silvestri, P. Laporta, O. Svelto, “Analysis of Quarter-Wave Dielectric-Mirror Dispersion in Femtosecond Dye-Laser Cavities,” Opt. Lett. 2, 335 (1984).
    [Crossref]
  10. S. De Silvestri, P. Laporta, O. Svelto, “Effects of Cavity Dispersion on Femtosecond Mode-Locked Dye Laser,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, Eds. (Springer, New York, 1984), pp. 23–26.
    [Crossref]
  11. After the submission of our paper new experimental results on this subject were published;A. M. Weiner, J. G. Fujimoto, E. P. Ippen, “Femtosecond Time-Resolved Reflectometry Measurements of Multiple-Layer Dielectric Mirrors,” Opt. Lett. 10, 71 (1985).
    [Crossref] [PubMed]
  12. S. De Silvestri, P. Laporta, O. Svelto, “The Role of Cavity Dispersion in cw Mode-Locked Lasers,” IEEE J. Quantum Electron. QE-20, 533 (1984).
    [Crossref]
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 51–70.
  14. A. F. Turner, P. W. Baumeister, “Multilayer Mirrors with High Reflectance Over an Extended Spectral Region,” Appl. Opt. 5, 69 (1966).
    [Crossref] [PubMed]
  15. H. A. Macleod, Thin Film Optical Filters (Adam Hilger, London, 1969), pp. 88–110.
  16. M. S. Stix, E. P. Ippen, “Pulse Shaping in Passively Mode-Locked Ring Dye Lasers,” IEEE J. Quantum Electron. QE-19, 520 (1983).
    [Crossref]
  17. The duration of the incident (Gaussian) pulse is given throughout the paper as full width at half-maximum (FWHM), this being the usual notation. The broadening is instead expressed as the ratio between rms duration of incident and reflected pulses since, in the case of distorted reflected pulses, the FWHM is no longer meaningful.

1985 (1)

1984 (3)

S. De Silvestri, P. Laporta, O. Svelto, “The Role of Cavity Dispersion in cw Mode-Locked Lasers,” IEEE J. Quantum Electron. QE-20, 533 (1984).
[Crossref]

J.-M. Halbout, D. Grischkowsky, “12-fs Ultrashort Optical Pulse Compression at a High Repetition Rate,” Appl. Phys. Lett. 45, 1281 (1984).
[Crossref]

S. De Silvestri, P. Laporta, O. Svelto, “Analysis of Quarter-Wave Dielectric-Mirror Dispersion in Femtosecond Dye-Laser Cavities,” Opt. Lett. 2, 335 (1984).
[Crossref]

1983 (4)

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).
[Crossref] [PubMed]

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, “Femtosecond Optical Pulses,” IEEE J. Quantum Electron. QE-19, 500 (1983).
[Crossref]

B. Nikolaus, D. Grischkowsky, “12× Pulse Compression Using Optical Fibers,” Appl. Phys. Lett. 42, 1 (1983).
[Crossref]

M. S. Stix, E. P. Ippen, “Pulse Shaping in Passively Mode-Locked Ring Dye Lasers,” IEEE J. Quantum Electron. QE-19, 520 (1983).
[Crossref]

1982 (1)

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. 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]

1966 (1)

Balant, A. C.

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]

Baumeister, P. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 51–70.

De Silvestri, S.

S. De Silvestri, P. Laporta, O. Svelto, “The Role of Cavity Dispersion in cw Mode-Locked Lasers,” IEEE J. Quantum Electron. QE-20, 533 (1984).
[Crossref]

S. De Silvestri, P. Laporta, O. Svelto, “Analysis of Quarter-Wave Dielectric-Mirror Dispersion in Femtosecond Dye-Laser Cavities,” Opt. Lett. 2, 335 (1984).
[Crossref]

S. De Silvestri, P. Laporta, O. Svelto, “Effects of Cavity Dispersion on Femtosecond Mode-Locked Dye Laser,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, Eds. (Springer, New York, 1984), pp. 23–26.
[Crossref]

Diels, J.-C.

Dietel, W.

Fontaine, J. J.

Fork, R. L.

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, “Femtosecond Optical Pulses,” IEEE J. Quantum Electron. QE-19, 500 (1983).
[Crossref]

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. 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.

After the submission of our paper new experimental results on this subject were published;A. M. Weiner, J. G. Fujimoto, E. P. Ippen, “Femtosecond Time-Resolved Reflectometry Measurements of Multiple-Layer Dielectric Mirrors,” Opt. Lett. 10, 71 (1985).
[Crossref] [PubMed]

A. M. Weiner, J. G. Fujimoto, E. P. Ippen, “Compression and Shaping of Femtosecond Pulses,” in Technical Digest, Ultrafast Phenomena Conference, (Optical Society of America, Washington, D.C., 1984), paper TuA4.

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.-M. Halbout, D. Grischkowsky, “12-fs Ultrashort Optical Pulse Compression at a High Repetition Rate,” Appl. Phys. Lett. 45, 1281 (1984).
[Crossref]

B. Nikolaus, D. Grischkowsky, “12× Pulse Compression Using Optical Fibers,” Appl. Phys. Lett. 42, 1 (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-fs Ultrashort Optical Pulse Compression at a High Repetition Rate,” Appl. Phys. Lett. 45, 1281 (1984).
[Crossref]

Hirlimann, C. A.

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, “Femtosecond Optical Pulses,” IEEE J. Quantum Electron. QE-19, 500 (1983).
[Crossref]

Ippen, E. P.

After the submission of our paper new experimental results on this subject were published;A. M. Weiner, J. G. Fujimoto, E. P. Ippen, “Femtosecond Time-Resolved Reflectometry Measurements of Multiple-Layer Dielectric Mirrors,” Opt. Lett. 10, 71 (1985).
[Crossref] [PubMed]

M. S. Stix, E. P. Ippen, “Pulse Shaping in Passively Mode-Locked Ring Dye Lasers,” IEEE J. Quantum Electron. QE-19, 520 (1983).
[Crossref]

A. M. Weiner, J. G. Fujimoto, E. P. Ippen, “Compression and Shaping of Femtosecond Pulses,” in Technical Digest, Ultrafast Phenomena Conference, (Optical Society of America, Washington, D.C., 1984), paper TuA4.

Laporta, P.

S. De Silvestri, P. Laporta, O. Svelto, “Analysis of Quarter-Wave Dielectric-Mirror Dispersion in Femtosecond Dye-Laser Cavities,” Opt. Lett. 2, 335 (1984).
[Crossref]

S. De Silvestri, P. Laporta, O. Svelto, “The Role of Cavity Dispersion in cw Mode-Locked Lasers,” IEEE J. Quantum Electron. QE-20, 533 (1984).
[Crossref]

S. De Silvestri, P. Laporta, O. Svelto, “Effects of Cavity Dispersion on Femtosecond Mode-Locked Dye Laser,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, Eds. (Springer, New York, 1984), pp. 23–26.
[Crossref]

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters (Adam Hilger, London, 1969), pp. 88–110.

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, “12× Pulse Compression Using Optical Fibers,” Appl. Phys. Lett. 42, 1 (1983).
[Crossref]

Shank, C. V.

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, “Femtosecond Optical Pulses,” IEEE J. Quantum Electron. QE-19, 500 (1983).
[Crossref]

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. 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]

Stix, M. S.

M. S. Stix, E. P. Ippen, “Pulse Shaping in Passively Mode-Locked Ring Dye Lasers,” IEEE J. Quantum Electron. QE-19, 520 (1983).
[Crossref]

Stolen, R. H.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. Tomlinson, “Compression of Femtosecond Optical Pulses,” Appl. Phys. Lett. 40, 761 (1982).
[Crossref]

Svelto, O.

S. De Silvestri, P. Laporta, O. Svelto, “Analysis of Quarter-Wave Dielectric-Mirror Dispersion in Femtosecond Dye-Laser Cavities,” Opt. Lett. 2, 335 (1984).
[Crossref]

S. De Silvestri, P. Laporta, O. Svelto, “The Role of Cavity Dispersion in cw Mode-Locked Lasers,” IEEE J. Quantum Electron. QE-20, 533 (1984).
[Crossref]

S. De Silvestri, P. Laporta, O. Svelto, “Effects of Cavity Dispersion on Femtosecond Mode-Locked Dye Laser,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, Eds. (Springer, New York, 1984), pp. 23–26.
[Crossref]

Tomlinson, W.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. Tomlinson, “Compression of Femtosecond Optical Pulses,” Appl. Phys. Lett. 40, 761 (1982).
[Crossref]

Turner, A. F.

Weiner, A. M.

After the submission of our paper new experimental results on this subject were published;A. M. Weiner, J. G. Fujimoto, E. P. Ippen, “Femtosecond Time-Resolved Reflectometry Measurements of Multiple-Layer Dielectric Mirrors,” Opt. Lett. 10, 71 (1985).
[Crossref] [PubMed]

A. M. Weiner, J. G. Fujimoto, E. P. Ippen, “Compression and Shaping of Femtosecond Pulses,” in Technical Digest, Ultrafast Phenomena Conference, (Optical Society of America, Washington, D.C., 1984), paper TuA4.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 51–70.

Yen, R.

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, “Femtosecond Optical Pulses,” IEEE J. Quantum Electron. QE-19, 500 (1983).
[Crossref]

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. Tomlinson, “Compression of Femtosecond Optical Pulses,” Appl. Phys. Lett. 40, 761 (1982).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

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]

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. Tomlinson, “Compression of Femtosecond Optical Pulses,” Appl. Phys. Lett. 40, 761 (1982).
[Crossref]

B. Nikolaus, D. Grischkowsky, “12× Pulse Compression Using Optical Fibers,” Appl. Phys. Lett. 42, 1 (1983).
[Crossref]

J.-M. Halbout, D. Grischkowsky, “12-fs Ultrashort Optical Pulse Compression at a High Repetition Rate,” Appl. Phys. Lett. 45, 1281 (1984).
[Crossref]

IEEE J. Quantum Electron. (3)

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, “Femtosecond Optical Pulses,” IEEE J. Quantum Electron. QE-19, 500 (1983).
[Crossref]

S. De Silvestri, P. Laporta, O. Svelto, “The Role of Cavity Dispersion in cw Mode-Locked Lasers,” IEEE J. Quantum Electron. QE-20, 533 (1984).
[Crossref]

M. S. Stix, E. P. Ippen, “Pulse Shaping in Passively Mode-Locked Ring Dye Lasers,” IEEE J. Quantum Electron. QE-19, 520 (1983).
[Crossref]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

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]

Other (5)

S. De Silvestri, P. Laporta, O. Svelto, “Effects of Cavity Dispersion on Femtosecond Mode-Locked Dye Laser,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, Eds. (Springer, New York, 1984), pp. 23–26.
[Crossref]

A. M. Weiner, J. G. Fujimoto, E. P. Ippen, “Compression and Shaping of Femtosecond Pulses,” in Technical Digest, Ultrafast Phenomena Conference, (Optical Society of America, Washington, D.C., 1984), paper TuA4.

The duration of the incident (Gaussian) pulse is given throughout the paper as full width at half-maximum (FWHM), this being the usual notation. The broadening is instead expressed as the ratio between rms duration of incident and reflected pulses since, in the case of distorted reflected pulses, the FWHM is no longer meaningful.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 51–70.

H. A. Macleod, Thin Film Optical Filters (Adam Hilger, London, 1969), pp. 88–110.

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Figures (7)

Fig. 1
Fig. 1

Mirror phase shift Φ as a function of the normalized frequency ω/ω0. (a) Structure 1, two quarterwave stacks with slightly overlapping high-reflectivity zones on glass substrate: [A] [0.89H]-[0.89L 0.89H]9[L][1.14H 1.14L]9[1.14H][S]. (b) Structure 2, thirty-five-layer geometric progression stack on glass substrate: [A][q17H][q16L][q15H] … [qH][L][q−1H] … [q−15H][q−16L]-[q−17H][S]. The common ratio q is equal to 0.985. (c) Structure 3, thirty-five-layer arithmetic progression stack on glass substrate: [A][(1 −17k)H][(1 −16k)L][(1 −15k)H] … [(1 −k)H][L][(1 + k)H] … [(1 + 15k)H][(1 + 16k)L][(1 + 17k)H]·[S]. The common difference k is equal to 0.015. In each case, S denotes glass substrate (ns = 1.52) and A air. H and L are films of TiO2 (nH = 2.28) and SiO2 (nL = 1.45), one quarterwave thick at central wavelength λ0 = (2πc0)/ω0 = 550 nm.

Fig. 2
Fig. 2

Phase shift Φ as a function of the normalized frequency ω/ω0 (a) Structure 1, [A][1.14H][1.14L 1.14H]9[L][0.89H 0.89L]9[0.89H][S]. (b) Structure 2, [A][q−17H][q−16L][q−15H] … [q−1H][L][qH] …[q15H][q16L][q17H][S]. (c) Structure 3, [A][(1 + 17k)H][(1 + 16k)-L][(1 + 15k)H] … (1 + k)H][L][(1 − k)H] … [(1 − 15k)H][(1 −16k)L][(1 −17k)H][S]. Symbols have the same meaning as in Fig. 1.

Fig. 3
Fig. 3

Time delay of the reflected pulse for two different incident pulse durations (FHWM) as a function of the normalized carrier frequency ωL/ω0 (solid lines) in the case of mirror structures in Fig. 1: (a) structure 1; (b) structure 2; (c) structure 3. The mirror phase shifts Φ shown in Fig. 1 are also plotted as continuous functions for comparison purposes (dashed lines).

Fig. 4
Fig. 4

Broadening σri of the reflected pulse for two different incident pulse durations (FHWM) as a function of the normalized carrier frequency ωL/ω0 (solid lines) in the case of mirror structures in Fig. 1: (a) structure 1; (b) structure 2; (c) structure 3. The second derivatives of mirror phase shifts are also shown (dashed lines).

Fig. 5
Fig. 5

Broadening σri of the reflected pulse as a function of incident pulse duration (FHWM) for structure 1 in Fig. 1. The carrier frequency of the incident pulse is assumed to be that for which the broadening is maximum (ωL0= 0.93,590 nm). An expanded view is also shown.

Fig. 6
Fig. 6

Intensity shape I(t) of the reflected pulse as a function of time (solid lines): (a) structure 1 in Fig. 1, ωL/ω0 = 0.93 (590 nm); (b) structure 2 in Fig. 1, ωL/ω0 = 0.88 (625 nm); (c) structure 3 in Fig. 1, ωL/ω0 = 0.89 (620 nm). The incident Gaussian pulses are plotted as dashed lines. In each case, the carrier frequency of the incident pulse is assumed to be that for which the broadening is maximum.

Fig. 7
Fig. 7

Intensity shape I(t) of the reflected pulse (solid lines) as a function of time in the case of structure 1 in Fig. 1 for three different incident pulses: (a) FWHM = 15 fsec, ωL/ω0 = 0.90 (611 nm); (b) FWHM = 30 fsec, ωL/ω0 = 0.93 (590 nm); (c) FWHM = 50 fsec, ωL/ω0 = 0.93 (590nm). The corresponding incident Gaussian pulses are also plotted as dashed lines.

Tables (1)

Tables Icon

Table I Maximum Absolute Values of Φ″ in Low- and High-Dispersion Regions of the High-Reflectivity Zone for Different Types of Mirrors a

Equations (10)

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I ( t ) = | e ( t ) | 2 = 1 ( 2 π ) 2 G ( ω ) exp ( i ω t ) d ω .
G ( ω ) = A * ( ω ) A ( ω + ω ) d ω ,
A ( ω ) = a ( t ) exp ( i ω t ) d t .
t ̅ = 1 E t I ( t ) d t = i 2 π E [ d G ( ω ) d ω ] ω = 0 = i 2 π E A * ( ω ) A ( ω ) d ω ,
t 2 ¯ = 1 E t 2 I ( t ) d t = 1 2 π E [ d 2 G ( ω ) d ω 2 ] ω = 0 = 1 2 π E A * ( ω ) A ( ω ) d ω ,
σ = ( t 2 ¯ t ̅ 2 ) 1 / 2 .
t ̅ r = t ̅ i 1 2 π E | A i ( ω ) | 2 Φ ( ω + ω L ) d ω ,
t ̅ r 2 = t ̅ i 2 + 1 2 π E | A i ( ω ) | 2 Φ 2 ( ω + ω L ) d ω ,
t ̅ r = ( Φ + 1 8 Φ σ i 2 + 1 128 Φ v σ i 4 + ... ) ,
σ r = ( σ i 2 + 1 4 Φ 2 σ i 2 + 1 32 Φ 2 σ i 4 + 1 16 Φ Φ v σ i 4 + 5 768 Φ v 2 σ 6 + ... ) 1 / 2 ,

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