Abstract

We give a simple general formula for the total angular dispersion due to multiple arbitrary dispersive elements in a series. It is simply the sum of the individual elements’ angular dispersions but with each divided by the total spatial magnification afterward (or, equivalently, multiplied by the total angular magnification afterward).

© 2010 Optical Society of America

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Errata

Vikrant Chauhan, Jacob Cohen, and Rick Trebino, "Simple dispersion law for arbitrary sequences of dispersive optics: erratum," Appl. Opt. 50, 4339-4339 (2011)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-50-22-4339

References

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  1. R. J. Niefer and J. B. Atkinson, “The design of achromatic prism beam expanders for pulsed dye lasers,” Opt. Commun. 67, 139–143 (1988).
    [CrossRef]
  2. D. C. Hanna, P. A. Kärkkäinen, and R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
    [CrossRef]
  3. I. Shoshan and U. P. Oppenheim, “The use of a diffraction grating as a beam expander in a dye laser cavity,” Opt. Commun. 25, 375–378 (1978).
    [CrossRef]
  4. R. Trebino, “Achromatic N-prism beam expanders: optimal configurations,” Appl. Opt. 24, 1130–1138 (1985).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. F. J. Duarte and J. A. Piper, “Prism preexpanded grazing-incidence grating cavity for pulsed dye lasers,” Appl. Opt. 20, 2113–2116 (1981).
    [CrossRef] [PubMed]
  7. R. L. Fork, C. H. B. Cruz, P. C. Becker, and C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  9. R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150–152 (1984).
    [CrossRef] [PubMed]
  10. O. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in 1.3–1.6 μm region,” IEEE J. Quantum Electron. 23, 59–64 (1987).
    [CrossRef]
  11. S. Akturk, X. Gu, M. Kimmel, and R. Trebino, “Extremely simple single-prism ultrashort-pulse compressor,” Opt. Express 14, 10101–10108 (2006).
    [CrossRef] [PubMed]
  12. S. Kane and J. Squier, “Grism-pair stretcher–compressor system for simultaneous second- and third-order dispersion compensation in chirped-pulse amplification,” J. Opt. Soc. Am. B 14, 661–665 (1997).
    [CrossRef]
  13. E. A. Gibson, D. M. Gaudiosi, H. C. Kapteyn, R. Jimenez, S. Kane, R. Huff, C. Durfee, and J. Squier, “Efficient reflection grisms for pulse compression and dispersion compensation of femtosecond pulses,” Opt. Lett. 31, 3363–3365 (2006).
    [CrossRef] [PubMed]
  14. V. Chauhan, P. Bowlan, J. Cohen, and R. Trebino, “Single-diffraction-grating and grism pulse compressors,” J. Opt. Soc. Am. B 27, 619–624 (2010).
    [CrossRef]
  15. F. J. Duarte and J. A. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
    [CrossRef]
  16. F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957).
  17. G. Chartier, Introduction to Optics (Springer, 2005).
  18. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  19. I. R. Kenyon, Light Fantastic (Oxford University, 2008), pp. 640–645.
  20. J. K. Robertson, Introduction to Physical Optics, 3rd ed., University Physics Series (Van Nostrand, 1941).
  21. G. S. Monk, Light: Principles and Experiments, 2nd ed.(Dover, 2000).
  22. J. R. Meyer-Arendt, Introduction to Classical and Modern Optics, 4th ed. (Benjamin Cummings, 1994), pp. 480–482.
  23. R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman Group, 1974), pp. 677–685.
  24. A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
    [CrossRef]

2010

2006

1997

S. Kane and J. Squier, “Grism-pair stretcher–compressor system for simultaneous second- and third-order dispersion compensation in chirped-pulse amplification,” J. Opt. Soc. Am. B 14, 661–665 (1997).
[CrossRef]

1994

1990

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

1988

R. J. Niefer and J. B. Atkinson, “The design of achromatic prism beam expanders for pulsed dye lasers,” Opt. Commun. 67, 139–143 (1988).
[CrossRef]

1987

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

O. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in 1.3–1.6 μm region,” IEEE J. Quantum Electron. 23, 59–64 (1987).
[CrossRef]

1985

1984

1983

1982

F. J. Duarte and J. A. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
[CrossRef]

1981

1978

I. Shoshan and U. P. Oppenheim, “The use of a diffraction grating as a beam expander in a dye laser cavity,” Opt. Commun. 25, 375–378 (1978).
[CrossRef]

1975

D. C. Hanna, P. A. Kärkkäinen, and R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
[CrossRef]

Akturk, S.

Atkinson, J. B.

R. J. Niefer and J. B. Atkinson, “The design of achromatic prism beam expanders for pulsed dye lasers,” Opt. Commun. 67, 139–143 (1988).
[CrossRef]

Becker, P. C.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Bowlan, P.

Chartier, G.

G. Chartier, Introduction to Optics (Springer, 2005).

Chauhan, V.

Cohen, J.

Cruz, C. H. B.

Diels, J. C.

Dietel, W.

Duarte, F. J.

Durfee, C.

Fontaine, J. J.

Fork, R. L.

Gaudiosi, D. M.

Gibson, E. A.

Gordon, J. P.

Gu, X.

Hanna, D. C.

D. C. Hanna, P. A. Kärkkäinen, and R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
[CrossRef]

Huff, R.

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957).

Jimenez, R.

Kane, S.

E. A. Gibson, D. M. Gaudiosi, H. C. Kapteyn, R. Jimenez, S. Kane, R. Huff, C. Durfee, and J. Squier, “Efficient reflection grisms for pulse compression and dispersion compensation of femtosecond pulses,” Opt. Lett. 31, 3363–3365 (2006).
[CrossRef] [PubMed]

S. Kane and J. Squier, “Grism-pair stretcher–compressor system for simultaneous second- and third-order dispersion compensation in chirped-pulse amplification,” J. Opt. Soc. Am. B 14, 661–665 (1997).
[CrossRef]

Kapteyn, H. C.

Kärkkäinen, P. A.

D. C. Hanna, P. A. Kärkkäinen, and R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
[CrossRef]

Kenyon, I. R.

I. R. Kenyon, Light Fantastic (Oxford University, 2008), pp. 640–645.

Kimmel, M.

Kostenbauder, A. G.

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman Group, 1974), pp. 677–685.

Martinez, O.

O. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in 1.3–1.6 μm region,” IEEE J. Quantum Electron. 23, 59–64 (1987).
[CrossRef]

Martinez, O. E.

Meyer-Arendt, J. R.

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics, 4th ed. (Benjamin Cummings, 1994), pp. 480–482.

Monk, G. S.

G. S. Monk, Light: Principles and Experiments, 2nd ed.(Dover, 2000).

Niefer, R. J.

R. J. Niefer and J. B. Atkinson, “The design of achromatic prism beam expanders for pulsed dye lasers,” Opt. Commun. 67, 139–143 (1988).
[CrossRef]

Oppenheim, U. P.

I. Shoshan and U. P. Oppenheim, “The use of a diffraction grating as a beam expander in a dye laser cavity,” Opt. Commun. 25, 375–378 (1978).
[CrossRef]

Piper, J. A.

F. J. Duarte and J. A. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
[CrossRef]

F. J. Duarte and J. A. Piper, “Prism preexpanded grazing-incidence grating cavity for pulsed dye lasers,” Appl. Opt. 20, 2113–2116 (1981).
[CrossRef] [PubMed]

Robertson, J. K.

J. K. Robertson, Introduction to Physical Optics, 3rd ed., University Physics Series (Van Nostrand, 1941).

Shank, C. V.

Shoshan, I.

I. Shoshan and U. P. Oppenheim, “The use of a diffraction grating as a beam expander in a dye laser cavity,” Opt. Commun. 25, 375–378 (1978).
[CrossRef]

Squier, J.

E. A. Gibson, D. M. Gaudiosi, H. C. Kapteyn, R. Jimenez, S. Kane, R. Huff, C. Durfee, and J. Squier, “Efficient reflection grisms for pulse compression and dispersion compensation of femtosecond pulses,” Opt. Lett. 31, 3363–3365 (2006).
[CrossRef] [PubMed]

S. Kane and J. Squier, “Grism-pair stretcher–compressor system for simultaneous second- and third-order dispersion compensation in chirped-pulse amplification,” J. Opt. Soc. Am. B 14, 661–665 (1997).
[CrossRef]

Trebino, R.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Wyatt, R.

D. C. Hanna, P. A. Kärkkäinen, and R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

O. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in 1.3–1.6 μm region,” IEEE J. Quantum Electron. 23, 59–64 (1987).
[CrossRef]

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

J. Opt. Soc. Am. B

S. Kane and J. Squier, “Grism-pair stretcher–compressor system for simultaneous second- and third-order dispersion compensation in chirped-pulse amplification,” J. Opt. Soc. Am. B 14, 661–665 (1997).
[CrossRef]

V. Chauhan, P. Bowlan, J. Cohen, and R. Trebino, “Single-diffraction-grating and grism pulse compressors,” J. Opt. Soc. Am. B 27, 619–624 (2010).
[CrossRef]

Opt. Commun.

F. J. Duarte and J. A. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
[CrossRef]

R. J. Niefer and J. B. Atkinson, “The design of achromatic prism beam expanders for pulsed dye lasers,” Opt. Commun. 67, 139–143 (1988).
[CrossRef]

I. Shoshan and U. P. Oppenheim, “The use of a diffraction grating as a beam expander in a dye laser cavity,” Opt. Commun. 25, 375–378 (1978).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

D. C. Hanna, P. A. Kärkkäinen, and R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
[CrossRef]

Other

F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957).

G. Chartier, Introduction to Optics (Springer, 2005).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

I. R. Kenyon, Light Fantastic (Oxford University, 2008), pp. 640–645.

J. K. Robertson, Introduction to Physical Optics, 3rd ed., University Physics Series (Van Nostrand, 1941).

G. S. Monk, Light: Principles and Experiments, 2nd ed.(Dover, 2000).

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics, 4th ed. (Benjamin Cummings, 1994), pp. 480–482.

R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman Group, 1974), pp. 677–685.

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Figures (3)

Fig. 1
Fig. 1

Prism, beam, and their relevant angles; dashed lines are normals to the prism faces.

Fig. 2
Fig. 2

Sequence of dispersive optics with their respective dispersions and magnifications.

Fig. 3
Fig. 3

Schematic of a single-prism–grating pulse compressor.

Equations (20)

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D = [ d n d λ sin ( ϕ ) cos ( θ ) / cos ( γ ) cos ( β ) ] + d n d λ sin ( β ) cos ( ϕ ) ,
D = d 1 m 2 + d 2 .
D total = D 1 M 2 M 3 M N + D 2 M 3 M 4 M N + + D N ,
K = [ M B 0 E 0 1 / M 0 D D M / λ 0 H 1 I 0 0 0 1 ] .
M = cos γ cos θ .
D = sin γ sin θ f 0 cos γ ,
K grating = [ M 0 0 0 0 1 / M 0 D D M / λ 0 0 1 0 0 0 0 1 ] .
[ M N 0 0 0 0 1 / M N 0 D N D N M N / λ 0 0 1 0 0 0 0 1 ] [ M 1 0 0 0 0 1 / M 1 0 D 1 D 1 M 1 / λ 0 0 1 0 0 0 0 1 ] = [ n = 1 N M n 0 0 0 0 1 / n = 1 N M n 0 n = 1 N D n / p = n + 1 N M p n = 1 N D n p = 1 n M p / λ 0 0 1 0 0 0 0 1 ] .
D total = n = 1 N D n p = n + 1 N M p ,
[ M N B N 0 E N 0 1 / M N 0 D N D N M N / λ 0 H N 1 I N 0 0 0 1 ] [ M 1 B 1 0 E 1 0 1 / M 1 0 D 1 D 1 M 1 / λ 0 H 1 1 I 1 0 0 0 1 ] = [ n = 1 N M n B total 0 E total 0 1 / n = 1 N M n 0 n = 1 N D n / p = n + 1 N M p n = 1 N D n p = 1 n M p / λ 0 H total 1 I total 0 0 0 1 ] .
M total = M 1 M 2 M 3 M N ,
D total = D 1 M 2 M 3 M N + D 2 M 3 M 4 M N + + D N .
D total = D 1 μ 2 μ 3 μ N + D 2 μ 3 μ 4 μ N + + D N .
[ M B 0 E 0 1 / M 0 D D M / λ 0 H 1 I 0 0 0 1 ] [ M B 0 E 0 1 / M 0 D D M / λ 0 H 1 I 0 0 0 1 ] .
M = M prism 1 M grating M prism 2 .
D = D prism 1 M grating M prism 2 + D grating M prism 2 + D prism 2 .
M 1 D 1 = D 2 = M 3 D 3 = D 4 ,
M 1 = 1 M 2 = M 3 = 1 M 4 .
D tot = D 1 M 2 M 3 M 4 + D 2 M 3 M 4 + D 3 M 4 + D 4 .
GDD = 2 D 2 L λ 0 ,

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