Abstract

In case of tunable ultrashort-pulse lasers, besides the prism material, separation, and insertion, one may choose the central wavelength providing a better high-order dispersion compensation. Because of errors that accumulate in numerical iterations, the analytic approach may be preferable for high-order dispersion examination. Analytic expressions for the fourth- and the fifth-order dispersions of crossed prisms pairs, both at arbitrary and Brewster-angle incidences are presented. Examples are given for e-sapphire slabs and for one or two pairs of fused-silica prisms. Results are shown providing compensation of both the second- and the third-order dispersions, by keeping as low as possible the higher-order ones.

© 2003 Optical Society of America

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References

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  1. R. L. Fork, O. E. Martinez, J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150–152 (1984).
    [CrossRef] [PubMed]
  2. J. P. Gordon, R. L. Fork, “Optical resonator with negative dispersion,” Opt. Lett. 9, 153–155 (1984).
    [CrossRef] [PubMed]
  3. R. L. Fork, C. H. Brito Cruz, P. C. Becker, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
    [CrossRef] [PubMed]
  4. D. E. Spence, P. N. Kean, W. Sibbett, “60-fsec pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 16, 42–44 (1991).
    [CrossRef] [PubMed]
  5. F. Krausz, Ch. Spielmann, T. Brabec, E. Wintner, A. J. Schmidt, “Generation of 33-fs optical pulses from a solid-state laser,” Opt. Lett. 17, 204–206 (1992).
    [CrossRef] [PubMed]
  6. M. H. Ober, E. Sorokin, I. Sorokina, F. Krausz, E. Wintner, I. A. Shcherbakov, “Subpicosecond mode locking of a Nd3+-doped garnet laser,” Opt. Lett. 17, 1364–1366 (1992).
    [CrossRef]
  7. B. Proctor, F. Wise, “Quartz prism sequence for reduction of cubic phase in a mode-locked Ti:Al2O3 laser,” Opt. Lett. 17, 1295–1297 (1992).
    [CrossRef] [PubMed]
  8. P. LiKamWa, B. H. T. Chai, A. Miller, “Self-mode-locked Cr3+:LiCaAlF6 laser,” Opt. Lett. 17, 1438–1440 (1992).
    [CrossRef]
  9. N. H. Rizvi, P. M. W. French, J. R. Taylor, “Generation of 33-fs pulses from a passively mode-locked Cr3+:LiSrAlF6 laser,” Opt. Lett. 17, 1605–1607 (1992).
    [CrossRef] [PubMed]
  10. J. M. Jacobson, K. Naganuma, H. A. Haus, J. G. Fujimoto, A. G. Jacobson, “Femtosecond pulse generation in a Ti:Al2O3 laser by using second- and third-order intracavity dispersion,” Opt. Lett. 17, 1608–1610 (1992).
    [CrossRef] [PubMed]
  11. P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, A. J. Schmidt, “Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero group-delay dispersion,” Opt. Lett. 18, 54–56 (1993).
    [CrossRef] [PubMed]
  12. I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. Zhou, C. P. Huang, “Fourth-order dispersion-limited solitary pulses,” Opt. Lett. 19, 1465–1467 (1994).
    [CrossRef] [PubMed]
  13. Z. Zhang, K. Torizuka, T. Itatani, K. Kobayashi, T. Sugaya, T. Nakagawa, “Self-starting mode-locked femtosecond forsterite laser with a semiconductor saturable-absorber mirror,” Opt. Lett. 22, 1006–1008 (1997).
    [CrossRef] [PubMed]
  14. R. E. Sherriff, “Analytic expressions for group-delay dispersion and cubic dispersion in arbitrary prism sequences,” J. Opt. Soc. Am. B 15, 1224–1230 (1998).
    [CrossRef]
  15. Optics and Coatings Catalog (1994), CVI Laser Corporation, 200 Dorado Place SE, P.O. Box 11308, Albuquerque, N.M. 87192.
  16. R. Szipöcs, K. Ferencz, Ch. Spielmann, F. Krausz, “Chirped multilayer coatings for broadband dispersion control in femtosecond lasers,” Opt. Lett. 19, 201–203 (1994).
    [CrossRef] [PubMed]
  17. A. C. Tien, R. Chang, J. Wang, “Adjustable negative group-velocity dispersion in graded-index lenses,” Opt. Lett. 17, 1177–1179 (1992).
    [CrossRef] [PubMed]

1998 (1)

1997 (1)

1994 (2)

1993 (1)

1992 (7)

1991 (1)

1987 (1)

1984 (2)

Becker, P. C.

Brabec, T.

Brito Cruz, C. H.

Chai, B. H. T.

Chang, R.

Christov, I. P.

Curley, P. F.

Ferencz, K.

Fork, R. L.

French, P. M. W.

Fujimoto, J. G.

Gordon, J. P.

Haus, H. A.

Huang, C. P.

Itatani, T.

Jacobson, A. G.

Jacobson, J. M.

Kapteyn, H. C.

Kean, P. N.

Kobayashi, K.

Krausz, F.

LiKamWa, P.

Martinez, O. E.

Miller, A.

Murnane, M. M.

Naganuma, K.

Nakagawa, T.

Ober, M. H.

Proctor, B.

Rizvi, N. H.

Schmidt, A. J.

Shank, C. V.

Shcherbakov, I. A.

Sherriff, R. E.

Sibbett, W.

Sorokin, E.

Sorokina, I.

Spence, D. E.

Spielmann, Ch.

Sugaya, T.

Szipöcs, R.

Taylor, J. R.

Tien, A. C.

Torizuka, K.

Wang, J.

Wintner, E.

Wise, F.

Zhang, Z.

Zhou, J.

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

Opt. Lett. (15)

R. Szipöcs, K. Ferencz, Ch. Spielmann, F. Krausz, “Chirped multilayer coatings for broadband dispersion control in femtosecond lasers,” Opt. Lett. 19, 201–203 (1994).
[CrossRef] [PubMed]

A. C. Tien, R. Chang, J. Wang, “Adjustable negative group-velocity dispersion in graded-index lenses,” Opt. Lett. 17, 1177–1179 (1992).
[CrossRef] [PubMed]

R. L. Fork, O. E. Martinez, J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150–152 (1984).
[CrossRef] [PubMed]

J. P. Gordon, R. L. Fork, “Optical resonator with negative dispersion,” Opt. Lett. 9, 153–155 (1984).
[CrossRef] [PubMed]

R. L. Fork, C. H. Brito Cruz, P. C. Becker, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
[CrossRef] [PubMed]

D. E. Spence, P. N. Kean, W. Sibbett, “60-fsec pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 16, 42–44 (1991).
[CrossRef] [PubMed]

F. Krausz, Ch. Spielmann, T. Brabec, E. Wintner, A. J. Schmidt, “Generation of 33-fs optical pulses from a solid-state laser,” Opt. Lett. 17, 204–206 (1992).
[CrossRef] [PubMed]

M. H. Ober, E. Sorokin, I. Sorokina, F. Krausz, E. Wintner, I. A. Shcherbakov, “Subpicosecond mode locking of a Nd3+-doped garnet laser,” Opt. Lett. 17, 1364–1366 (1992).
[CrossRef]

B. Proctor, F. Wise, “Quartz prism sequence for reduction of cubic phase in a mode-locked Ti:Al2O3 laser,” Opt. Lett. 17, 1295–1297 (1992).
[CrossRef] [PubMed]

P. LiKamWa, B. H. T. Chai, A. Miller, “Self-mode-locked Cr3+:LiCaAlF6 laser,” Opt. Lett. 17, 1438–1440 (1992).
[CrossRef]

N. H. Rizvi, P. M. W. French, J. R. Taylor, “Generation of 33-fs pulses from a passively mode-locked Cr3+:LiSrAlF6 laser,” Opt. Lett. 17, 1605–1607 (1992).
[CrossRef] [PubMed]

J. M. Jacobson, K. Naganuma, H. A. Haus, J. G. Fujimoto, A. G. Jacobson, “Femtosecond pulse generation in a Ti:Al2O3 laser by using second- and third-order intracavity dispersion,” Opt. Lett. 17, 1608–1610 (1992).
[CrossRef] [PubMed]

P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, A. J. Schmidt, “Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero group-delay dispersion,” Opt. Lett. 18, 54–56 (1993).
[CrossRef] [PubMed]

I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. Zhou, C. P. Huang, “Fourth-order dispersion-limited solitary pulses,” Opt. Lett. 19, 1465–1467 (1994).
[CrossRef] [PubMed]

Z. Zhang, K. Torizuka, T. Itatani, K. Kobayashi, T. Sugaya, T. Nakagawa, “Self-starting mode-locked femtosecond forsterite laser with a semiconductor saturable-absorber mirror,” Opt. Lett. 22, 1006–1008 (1997).
[CrossRef] [PubMed]

Other (1)

Optics and Coatings Catalog (1994), CVI Laser Corporation, 200 Dorado Place SE, P.O. Box 11308, Albuquerque, N.M. 87192.

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Figures (4)

Fig. 1
Fig. 1

(a) One pair of crossed prisms that are used at minimum deviation and oriented so that the rays enter and leave at Brewster’s angle. (b) Nomenclature of the transit angles for a single prism.

Fig. 2
Fig. 2

Variations of derivatives d k Φ/dω k , with k = 2–5, against the light wavelength λ: (a)–(d) for a fused-silica slab with the ray-path length through it varying from 2.5 to 10 mm in increments of 2.5 mm (curves 1 to 4) and (e)–(h) for a pair of fused-silica prisms that are used at minimum deviation and Brewster-angle incidence at the respective wavelength, with L varying from 100 to 400 mm in increments of 100 mm (curves 1 to 4).

Fig. 3
Fig. 3

Pairs of values (l a , L) providing d3Φ/dω3 = 0 for the overall system that is formed by an e-sapphire slab, with the ray-path length through it of 2 mm, beside (a) one pair and (b) two pairs of fused-silica prisms at minimum deviation and Brewster-angle incidence, with the central wavelength varying from 0.775 to 0.875 μm in increments of 0.025 μm (curves 1 to 5). The same in (c), but one pair of fused-silica prisms with fixed apex angle, α = 60°, at the Brewster-angle incidence for the respective central wavelength.

Fig. 4
Fig. 4

Variations of derivatives d k Φ/dω k against L, with k equal to 2, 4, and 5: (a)–(c) for the system comprising one pair of crossed prisms with parameters of Fig. 3(a) providing d3Φ/dω3 = 0 and (d)–(f) for the system comprising two pairs of crossed prisms at minimum deviation and Brewster-angle incidence with parameters of Fig. 3(b).

Equations (25)

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d2Φdω2ωc=λ32πc2d2Pdλ2λc,
d3Φdω3ωc=-λ44π2c33 d2Pdλ2+λ d3Pdλ3λc,
d4Φdω4ωc=λ58π3c412 d2Pdλ2+8λ d3Pdλ3+λ2d4Pdλ4λc,
d5Φdω5ωc=-λ616π4c560 d2Pdλ2+60λ d3Pdλ3+15λ2d4Pdλ4+λ3d5Pdλ5λc.
dP/dλ=dP/dβdβ/dndn/dλ.
dβ/dn=-2,
d2β/dn2=-22n-n-3,
d3β/dn3=-24+12n2-6n-2+3n-4+3n-6,
d4β/dn4=-624n+40n3-24n-1+6n-5-9n-7-5n-9,
d5β/dn5=2112-3096n2+112n4-24n-2+36n-4-6n-6+n-8+18n-10+7n-12.
dkP/dλk=XkL sin β+YkL cos β,
X2=-β,
X3=β3-β,
X4=6β2β-βν,
X5=-β5+10β3β+10β2β+15ββ2-βν.
Y2=-β2,
Y3=-3ββ,
Y4=β4-3β2-4ββ,
Y5=10ββ3-β-5ββν.
τ1=tan r,  τ2=tan ψ.
βI=-sin θ+τ1 cos θ/cos ψ,
βII=-τ2βI+τ12n-1βI,
βIII=1+τ22βI3-2τ2βI+τ12n-1βII+τ12n-23+2τ12βI,
βIV=-2τ21+τ22βI4-2τ2βI+τ12n-1βIII+51+τ22βI2-2τ2βII+2τ12n-23+2τ12βII-2τ12n-36+9τ12+4τ14βI,
βV=21+4τ22+3τ24βI5-2τ2βI+τ12n-1βIV+71+τ22βI2-6τ2βII+3τ12n-23+2τ12βIII+61+τ222βII-3τ2βI2βI-τ12n-36+9τ12+4τ14βII+6τ12n-410+25τ12+24τ14+8τ16βI.

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