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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 2 — Jan. 17, 2011
  • pp: 814–819
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Reinvestigation on the frequency dispersion of a grating-pair compressor

Jin Jer Huang, Liu Yang Zhang, and Yu Qiang Yang  »View Author Affiliations


Optics Express, Vol. 19, Issue 2, pp. 814-819 (2011)
http://dx.doi.org/10.1364/OE.19.000814


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Abstract

The typical phase correction term introduced in a diffraction grating-pair is rediscussed. It shows that the correction causes a conceptual difficulty in geometrical optics. A study reveals that Fraunhofer diffraction explains the correction and only mean-phase light rays are allowed for diffraction analysis. Besides, an equivalent phase formulation without correction is recommended.

© 2011 Optical Society of America

1. Introduction

In ultrafast optics, angular dispersive elements like diffraction gratings wield a great influence in pulse generation, amplification, shaping and compression, which are frequently applied in the well-known chirped pulse amplification [1

1. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]

, 2

2. A. Dubietis, G. Jonu sauskas, and A. Piskarskas, “Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal,” Opt. Commun. 88, 437–440 (1992). [CrossRef]

], compression of self-phase modulated or supercontinuum-generation pulses in fibers [3

3. C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761–763 (1982). [CrossRef]

, 4

4. R. L. Fork, C. H. Brito 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]

, 5

5. R. E. Kennedy, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “High-peak-power femtosecond pulse compression with polarization-maintaining ytterbium-doped fiber amplification,” Opt. Lett. 32, 1199–1201 (2007). [CrossRef] [PubMed]

, 6

6. B. Schenkel, R. Paschotta, and U. Keller, “Pulse compression with supercontinuum generation in microstructure fibers,” J. Opt. Soc. Am. B 22, 687–693 (2005). [CrossRef]

], pulse-front matched or group-velocity matched parametric processes [7

7. O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464–2468 (1989). [CrossRef]

, 8

8. G. Szabó and Z. Bor, “Broadband frequency doubler for femtosecond pulses,” Appl. Phys. B 50, 51–54 (1990). [CrossRef]

, 9

9. A. V. Smith, “Group-velocity-matched three-wave mixing in birefringent crystals,” Opt. Lett. 26, 719–721 (2001). [CrossRef]

, 10

10. S. Ashihara, T. Shimura, and K. Kuroda, “Group-velocity matched second-harmonic generation in tilted quasi-phase-matched gratings,” J. Opt. Soc. Am. B 20, 853–856 (2003). [CrossRef]

, 11

11. T. Kobayashi, in Femtosecond Optical Frequency Comb: Principle, Operation and Applications, edited by J. Ye and S. T. Cundiff (Springer, Berlin, 2005), pp. 133–175. [CrossRef]

]. Now it is well understood that the pulse stretch and compression by a grating-pair comes from a strong second order frequency dispersion imposed on incident ultrashort pulses [12

12. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454–458 (1969). [CrossRef]

, 13

13. J. D. McMullen, “Analysis of compression of frequency chirped optical pulses,” Appl. Opt. 18, 737–741 (1979). [CrossRef] [PubMed]

, 14

14. I. P. Christov and I. V. Tomov, “Large bandwidth pulse compression with diffraction gratings,” Opt. Commun. 58, 338–342 (1986). [CrossRef]

, 15

15. S. D. Brorson and H. A. Haus, “Geometrical limitations in grating pair pulse compression,” Appl. Opt. 27, 23–25 (1988). [CrossRef] [PubMed]

], which can be derived by a phase function ϕ(ω), as was first shown by Treacy [12

12. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454–458 (1969). [CrossRef]

]. To catch a right result, Treacy added a phase correction term R(ω) to an initial eikonal phase, which was claimed as “a consistent definition for the waves in the grating system”. Treacy’s model, for the second and higher order dispersions, has been verified in many experiments through optical pulse stretch and compression [1

1. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]

, 2

2. A. Dubietis, G. Jonu sauskas, and A. Piskarskas, “Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal,” Opt. Commun. 88, 437–440 (1992). [CrossRef]

, 3

3. C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761–763 (1982). [CrossRef]

, 4

4. R. L. Fork, C. H. Brito 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]

] and theoretically confirmed by the Fresnel-Kirchhoff integration and the matrix optics [16

16. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986). [CrossRef]

, 17

17. O. E. Martinez, “Matrix Formalism for Pulse Compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988). [CrossRef]

, 18

18. C. J. Mu, F. Tian, J. T. Bai, and X. Hou, “Matrix formulism calculation of dispersion in a grating compressor,” Acta Photonica Sinica 31, 1116–1119 (2002)[In Chinese].

]. However, Treacy’s interpretation on the correction has not convinced followers due to an inadequacy in his reasoning. The confusion therein is that the original phase term without the correction is seemingly equivalent to that with in the background of geometrical optics, nevertheless, the two terms deliver different dispersive relations. This makes one easy to misuse ϕ(ω) [19

19. Z. G Zhang, T. Yagi, and T. Arisawa, “Ray-tracing model for stretcher dispersion calculation,” Appl. Opt. 36, 3393–3399 (1997). [CrossRef] [PubMed]

, 20

20. Z. G Zhang and H. Sun, “Calculation and evaluation of dispersions in a femtosecond pulse amplification system,” Acta Physica Sinica 50, 1080–1086 (2000)[In Chinese].

]. Although Brorson and Haus tried to understand the group-delay time rendered by Treacy’s model through a general Fermat’s principle [21

21. S. D. Brorson and H. A. Haus, “Diffraction gratings and geometrical optics,” J. Opt. Soc. Am. B 5, 247–248 (1988). [CrossRef]

], their analysis did not find a way to obviate the correction if a phase function has to be introduced. It is quite necessary to clear the confusion owing to the significance of grating devices in fundamental optics and applications. In present paper, we will explore the physics beneath the phase correction and clarify the use of phase terms involving grating-pairs.

2. Treacy’s interpretation

Consider a planar ultrashort optical pulse impinging into a parallel grating-pair. The first grating will cause angular dispersion or pulse-front tilt [22

22. O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commnun. 59, 229–232 (1986). [CrossRef]

, 23

23. S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004). [CrossRef] [PubMed]

, 24

24. J. J. Huang and L. Y. Zhang, “Transformation of few-cycle ultrashort pulsed Gaussian beams by an angular disperser,” J. Phys. B 43, 175601 (2010). [CrossRef]

], and the second one counteracts the tilt but creates spatial chirping and frequency dispersion [16

16. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986). [CrossRef]

]. A simple geometrical scheme is illustrated in Fig. 1(a) where two rays belonging respectively to angular frequency ω and neighboring frequency ω′ are sampled for a comparison. The incident and reflective angles of the ω component on the first grating are γ and γθ, respectively. They obey the following grating equation
sin(γθ)+sin(γ)=2πmcωd
(1)
where d is the grating constant, c is the light speed in vacuum, m is the order of interference. In the following part, we will consider a practical case m = 1. Accordingly, the length p of the ray path PABQ for ω can be expressed as
p=b[1+cos(θ)]
(2)
where b is the slant distance AB¯ between the two gratings. Thus, the phase through the path PABQ is expected to be
ϕ0(ω)=ωcb[1+cos(θ)].
(3)
However in terms of Treacy’s consideration, Eq. (3) should be revised to be
ϕT(ω)=ωcb[1+cos(θ)]+R(ω)
(4)
where R(ω) is the phase correction, the necessity of which is illustrated in Fig. 1(b) where two rays with the same frequency are displaced transversely. DD′ and EE′ therein represent the equiphase fronts before and after diffraction by the second grating (G2), respectively. Since the path length of DBE does not match that of D′B′E′, a phase compensation is needed. When we slowly move the ray “1” towards the ray “2” without changing its propagation direction, the point B′ will thus move to B along the line BB′ on the surface of G2. If B′ crosses one groove for a distance equal to d, the relative phase change through D′B′E′ is 2π according to Eq. (1). Then, a total phase shift 2πBB¯/d can be predicted between the light paths DBE and D′B′E′. Furthermore, we can fix B′ at the point O which makes AO [see Fig. 1(a)] normal to the surface of G2. With respect to the reference ray [dashed green line in Fig. 1(a)] through O, the phase shift can be expressed as
R(ω)=2πGdtan(γθ)
(5)
with the definition GAO¯. Above interpretation is the main idea shown by Treacy. One can refer to the original paper if details are required. The validity of Eqs. (4), (5) can be simply checked as follows. In general, a group-delay time is derived from ∂ϕ/∂ω which can be rewritten as p/u where u is the group velocity. Equation (3) in the configuration γ > θ will give rise to a group velocity greater than c, whereas Eq. (4) yields u = c as is easily accepted since every frequency component of an ultrashort pulse propagates in a constant speed c [25

25. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin in Optics of Femtosecond Laser Pulses , edited by Y. Atanov (AIP, New York, 1992), pp. 205–206.

], moreover it is a result of Fermat’s principle [21

21. S. D. Brorson and H. A. Haus, “Diffraction gratings and geometrical optics,” J. Opt. Soc. Am. B 5, 247–248 (1988). [CrossRef]

].

Fig. 1 Grating-pair arrangement for pulse compression (a) and schematic diagram of Treacy’s phase correction (b). Light path calculation starts and ends at the equiphase plane Z = 0 for the incident and output rays where the frequency ω′ is slightly smaller than ω. DD′, EE′, FF′, HH′ represent symmetrically sampled equiphase planes.

However by the geometrical optics, the relative phase of the two rays in Fig. 1(b) only differ in a number of 2π induced by G2 since they start at the same equiphase (DD′) and arrive at the same one (EE′). Therefore, R(ω) can be canceled out because a couple of 2π do not contribute to the dispersion. So one is easy to doubt the validity of R(ω). On the other hand, if we add two other equiphases of FF′ and HH′ in Fig. 1(b) and make the four equiphase fronts to form a centrosymmetric configuration (symmetric point pairs are A/B, E/F, D/H and their primed ones), obviously an opposite phase compensation should be made between the two light paths F′A′H′ and FAH. The path length of F′A′BE is actually equal to that of FAB′E′, as does not require any phase correction for the the two rays! Now it seems that we are trapped in an awkward dilemma, thus R(ω) demands a right physical interpretation.

3. Fraunhofer correction

Fig. 2 Principle of the diffraction grating where blazed grooves are simply shown for an analysis of the phase variation of a pencil of diffracted rays in one unit. Phase difference exist between the three sampled ray pairs 1 − 1, 2 − 2, 3 − 3. The incident and diffracted angles are the same as those of the second grating in Fig. 1(a).

Fig. 3 Schematic show of a diffracted ray between two parallel gratings in which mean-phase points (denoted by a subscript S ) are illustrated as a reference for phase correction. Here, an illegal light path PABQ should be replaced by a correct representative path PA + A′SkQ′ where A locates at a mean-phase point.

Actually, there is a way to circumvent the correction issue. If we parallel shift the ray of ω to make its diffraction position on G2 in coincidence with that of a reference frequency ω0, then Fraunhofer correction will not appear. The method is shown in Fig. 4. We just make that the light path PACR of ω is substituted with the path PA + A′BQ. Then the phase becomes
ϕ(ω)=ωcb0[cos(θ)+cos(δθ)],b0=b(ω=ω0).
(9)
Since A and B are two fixed points formed by the reference light ray, Eq. (9) does not depend on the validity of the light path AB (It will differ in a constant phase when a representative path is chosen). If we move the reference point O in Fig. 3 to B in Fig. 4, Eq. (9) will turn into Treacy’s expression, i.e. Eq. (4). We hereby recommend Eq. (9) in applications for which one needs not trying to understand Fraunhofer correction. Besides, Eq. (9) has a distinct advantage for phase estimation in a system with multiple grating-pairs. Certainly if one starts a work from the group-delay time ∂ϕ/∂ω = p/c, the correction does not have to be considered too [25

25. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin in Optics of Femtosecond Laser Pulses , edited by Y. Atanov (AIP, New York, 1992), pp. 205–206.

].

Fig. 4 Light paths of two frequencies inside a grating-pair similar to those in Fig. 1(a), where δθ is the angle between the diffracted directions of a ray ω and a referred ray ω0.

4. Conclusion

Treacy’s model for a grating-pair pulse compressor is reconsidered and one conceptual confusion in phase correction is analyzed. Analysis shows that the original interpretation can not account for the phase correction, which however, is due to Fraunhofer diffraction. It follows that only those light rays reflected at mean-phase positions on the grating surface stand to reason in optics for a coherent beam containing many frequencies. Nevertheless, the correction is negligible for a monochromatic wave since it is equivalent to a constant phase shift.

Acknowledgments

This work was supported by the tiptop-talent fund and science foundation for youth scholars (No. 2009YF032) from Harbin University of Science and Technology in China.

References and links

1.

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]

2.

A. Dubietis, G. Jonu sauskas, and A. Piskarskas, “Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal,” Opt. Commun. 88, 437–440 (1992). [CrossRef]

3.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761–763 (1982). [CrossRef]

4.

R. L. Fork, C. H. Brito 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]

5.

R. E. Kennedy, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “High-peak-power femtosecond pulse compression with polarization-maintaining ytterbium-doped fiber amplification,” Opt. Lett. 32, 1199–1201 (2007). [CrossRef] [PubMed]

6.

B. Schenkel, R. Paschotta, and U. Keller, “Pulse compression with supercontinuum generation in microstructure fibers,” J. Opt. Soc. Am. B 22, 687–693 (2005). [CrossRef]

7.

O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464–2468 (1989). [CrossRef]

8.

G. Szabó and Z. Bor, “Broadband frequency doubler for femtosecond pulses,” Appl. Phys. B 50, 51–54 (1990). [CrossRef]

9.

A. V. Smith, “Group-velocity-matched three-wave mixing in birefringent crystals,” Opt. Lett. 26, 719–721 (2001). [CrossRef]

10.

S. Ashihara, T. Shimura, and K. Kuroda, “Group-velocity matched second-harmonic generation in tilted quasi-phase-matched gratings,” J. Opt. Soc. Am. B 20, 853–856 (2003). [CrossRef]

11.

T. Kobayashi, in Femtosecond Optical Frequency Comb: Principle, Operation and Applications, edited by J. Ye and S. T. Cundiff (Springer, Berlin, 2005), pp. 133–175. [CrossRef]

12.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454–458 (1969). [CrossRef]

13.

J. D. McMullen, “Analysis of compression of frequency chirped optical pulses,” Appl. Opt. 18, 737–741 (1979). [CrossRef] [PubMed]

14.

I. P. Christov and I. V. Tomov, “Large bandwidth pulse compression with diffraction gratings,” Opt. Commun. 58, 338–342 (1986). [CrossRef]

15.

S. D. Brorson and H. A. Haus, “Geometrical limitations in grating pair pulse compression,” Appl. Opt. 27, 23–25 (1988). [CrossRef] [PubMed]

16.

O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986). [CrossRef]

17.

O. E. Martinez, “Matrix Formalism for Pulse Compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988). [CrossRef]

18.

C. J. Mu, F. Tian, J. T. Bai, and X. Hou, “Matrix formulism calculation of dispersion in a grating compressor,” Acta Photonica Sinica 31, 1116–1119 (2002)[In Chinese].

19.

Z. G Zhang, T. Yagi, and T. Arisawa, “Ray-tracing model for stretcher dispersion calculation,” Appl. Opt. 36, 3393–3399 (1997). [CrossRef] [PubMed]

20.

Z. G Zhang and H. Sun, “Calculation and evaluation of dispersions in a femtosecond pulse amplification system,” Acta Physica Sinica 50, 1080–1086 (2000)[In Chinese].

21.

S. D. Brorson and H. A. Haus, “Diffraction gratings and geometrical optics,” J. Opt. Soc. Am. B 5, 247–248 (1988). [CrossRef]

22.

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commnun. 59, 229–232 (1986). [CrossRef]

23.

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004). [CrossRef] [PubMed]

24.

J. J. Huang and L. Y. Zhang, “Transformation of few-cycle ultrashort pulsed Gaussian beams by an angular disperser,” J. Phys. B 43, 175601 (2010). [CrossRef]

25.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin in Optics of Femtosecond Laser Pulses , edited by Y. Atanov (AIP, New York, 1992), pp. 205–206.

26.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), pp. 446–453.

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(320.0320) Ultrafast optics : Ultrafast optics
(320.5520) Ultrafast optics : Pulse compression

ToC Category:
Ultrafast Optics

History
Original Manuscript: December 1, 2010
Revised Manuscript: December 23, 2010
Manuscript Accepted: December 23, 2010
Published: January 5, 2011

Citation
Jin Jer Huang, Liu Yang Zhang, and Yu Qiang Yang, "Reinvestigation on the frequency dispersion of a grating-pair compressor," Opt. Express 19, 814-819 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-814


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References

  1. D. Strickland, and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]
  2. A. Dubietis, “G. Jonu sauskas, and A. Piskarskas, “Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal,” Opt. Commun. 88, 437–440 (1992). [CrossRef]
  3. C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761–763 (1982). [CrossRef]
  4. R. L. Fork, C. H. Brito 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]
  5. R. E. Kennedy, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “High-peak-power femtosecond pulse compression with polarization-maintaining ytterbium-doped fiber amplification,” Opt. Lett. 32, 1199–1201 (2007). [CrossRef] [PubMed]
  6. B. Schenkel, R. Paschotta, and U. Keller, “Pulse compression with supercontinuum generation in microstructure fibers,” J. Opt. Soc. Am. B 22, 687–693 (2005). [CrossRef]
  7. O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464–2468 (1989). [CrossRef]
  8. G. Szabó, and Z. Bor, “Broadband frequency doubler for femtosecond pulses,” Appl. Phys. B 50, 51–54 (1990). [CrossRef]
  9. A. V. Smith, “Group-velocity-matched three-wave mixing in birefringent crystals,” Opt. Lett. 26, 719–721 (2001). [CrossRef]
  10. S. Ashihara, T. Shimura, and K. Kuroda, “Group-velocity matched second-harmonic generation in tilted quasiphase-matched gratings,” J. Opt. Soc. Am. B 20, 853–856 (2003). [CrossRef]
  11. T. Kobayashi, in Femtosecond Optical Frequency Comb: Principle, Operation and Applications, edited by J. Ye and S. T. Cundiff (Springer, Berlin, 2005), pp. 133–175. [CrossRef]
  12. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454–458 (1969). [CrossRef]
  13. J. D. McMullen, “Analysis of compression of frequency chirped optical pulses,” Appl. Opt. 18, 737–741 (1979). [CrossRef] [PubMed]
  14. I. P. Christov, and I. V. Tomov, “Large bandwidth pulse compression with diffraction gratings,” Opt. Commun. 58, 338–342 (1986). [CrossRef]
  15. S. D. Brorson, and H. A. Haus, “Geometrical limitations in grating pair pulse compression,” Appl. Opt. 27, 23–25 (1988). [CrossRef] [PubMed]
  16. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986). [CrossRef]
  17. O. E. Martinez, “Matrix Formalism for Pulse Compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988). [CrossRef]
  18. C. J. Mu, F. Tian, J. T. Bai, and X. Hou, “Matrix formulism calculation of dispersion in a grating compressor,” Acta Photonica Sinica 31, 1116–1119 (2002) (In Chinese).
  19. Z. G. Zhang, T. Yagi, and T. Arisawa, “Ray-tracing model for stretcher dispersion calculation,” Appl. Opt. 36, 3393–3399 (1997). [CrossRef] [PubMed]
  20. Z. G. Zhang, and H. Sun, “Calculation and evaluation of dispersions in a femtosecond pulse amplification system,” Acta Physica Sinica 50, 1080–1086 (2000) (In Chinese).
  21. S. D. Brorson, and H. A. Haus, “Diffraction gratings and geometrical optics,” J. Opt. Soc. Am. B 5, 247–248 (1988). [CrossRef]
  22. O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986). [CrossRef]
  23. S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004). [CrossRef] [PubMed]
  24. J. J. Huang, and L. Y. Zhang, “Transformation of few-cycle ultrashort pulsed Gaussian beams by an angular disperser,” J. Phys. B 43, 175601 (2010). [CrossRef]
  25. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, in Optics of Femtosecond Laser Pulses, edited by Y. Atanov (AIP, New York, 1992), pp. 205–206.
  26. M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 1999), pp. 446–453.

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