## Reinvestigation on the frequency dispersion of a grating-pair compressor |

Optics Express, Vol. 19, Issue 2, pp. 814-819 (2011)

http://dx.doi.org/10.1364/OE.19.000814

Acrobat PDF (718 KB)

### 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

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]

*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]

*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. 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]

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]

*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]

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

## 2. Treacy’s interpretation

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]

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]

*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

*γ*−

*θ*, respectively. They obey the following grating equation 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 where

*b*is the slant distance

*PABQ*is expected to be However in terms of Treacy’s consideration, Eq. (3) should be revised to be 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 2

*π*according to Eq. (1). Then, a total phase shift

*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 with the definition

*∂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], 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]

*π*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 2

*π*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

*R*,

*S*,

*T*in the

*n*th groove. The standard knowledge of diffraction gratings tell us that the reflected light field can be written as a single-unit diffraction term

*U*multiplied by an interference factor

*H*, which only allows for diffraction along the

*γ*direction governed by Eq. (1). Thus only the diffracted rays pointing to

*γ*in these secondary coherent sources contribute to

*U*. This permits treating the reflected field as a plane wave. So, the single-unit diffraction term

*U*at the equiphase front which is distant from the grating can be expressed as a Fraunhofer diffraction integral [26] where [

*x*

_{0},

*x*

_{1}] is the range on the equiphase front which just contains all the diffracted rays generated from the

*n*th groove,

*C*, a complex constant,

*F*(

*x*), the pupil function of a diffraction window restricted by a groove, as can also be approximated as a constant

*F*over the integration range,

*z*(

*x*), the path length of a ray from the incident front to a secondary source and to the equiphase front. Equation (6) can be converted to Here,

*z*is a path length including a special point

_{s}*S*on the surface of the groove, which we may call the mean-phase point (it depends on

*ω*). It makes

*ωz*/

_{s}*c*a resultant phase formed by all the diffracted rays on the groove, a result of the mean value theorem of complex integrals. The ray “2” then is a representative light ray on the

*n*th groove. All the mean-phase (

*S*) points on the grating are connected to make an effective grating surface. In a rare case, one groove may have two or more mean-phase points. We can arbitrarily choose one of these point as a representative. Above result implies that only the rays reflected by those representative mean-phase points are legal in geometrical optics.

*ω*diffracted by the two gratings, as is shown in Fig. 3. We can set the first diffraction point of an incident ray to be a “

*S*” position. However in general, the second diffraction point of this ray (

*B*) displaces a distance

*δ*≤

*d*leftwards from a nearest mean-phase point, for example

*S*. Then,

_{k}*PABQ*should be replaced by a legal path

*PA*+

*A′S*which comparatively has an advanced phase 2

_{k}Q′*πδ*/

*d*. Evidently,

*δ*depends on the frequency

*ω*. The total phase from the equiphase front

*P*to that of

*Q*then becomes Note that we need not to record an absolute phase but its parts depending on the frequency

*ω*. Let

*S*be a mean-phase location inside a groove which includes

_{n}*O*, where

*AO*is the same line as that in Fig. 1(a). Thus

*π*and

*ω*. Therefore,

*πδ*/

*d*, so Eq. (8) actually is the same as Eq. (4). We may call the phase correction

*R*(

*ω*) or −2

*πδ*/

*d*, referring to the analysis above, Fraunhofer correction. The correction has to be taken into account to regulate a ray tracing procedure in grating systems, if dispersion works. In terms of Fraunhofer correction, the confusion in Treacy’s model disappears at last.

*ω*to make its diffraction position on G2 in coincidence with that of a reference frequency

*ω*

_{0}, 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 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].

## 4. Conclusion

## Acknowledgments

## References and links

1. | D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. |

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. |

3. | C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. |

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. |

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. |

6. | B. Schenkel, R. Paschotta, and U. Keller, “Pulse compression with supercontinuum generation in microstructure fibers,” J. Opt. Soc. Am. B |

7. | O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. |

8. | G. Szabó and Z. Bor, “Broadband frequency doubler for femtosecond pulses,” Appl. Phys. B |

9. | A. V. Smith, “Group-velocity-matched three-wave mixing in birefringent crystals,” Opt. Lett. |

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 |

11. | T. Kobayashi, in |

12. | E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. |

13. | J. D. McMullen, “Analysis of compression of frequency chirped optical pulses,” Appl. Opt. |

14. | I. P. Christov and I. V. Tomov, “Large bandwidth pulse compression with diffraction gratings,” Opt. Commun. |

15. | S. D. Brorson and H. A. Haus, “Geometrical limitations in grating pair pulse compression,” Appl. Opt. |

16. | O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B |

17. | O. E. Martinez, “Matrix Formalism for Pulse Compressors,” IEEE J. Quantum Electron. |

18. | C. J. Mu, F. Tian, J. T. Bai, and X. Hou, “Matrix formulism calculation of dispersion in a grating compressor,” Acta Photonica Sinica |

19. | Z. G Zhang, T. Yagi, and T. Arisawa, “Ray-tracing model for stretcher dispersion calculation,” Appl. Opt. |

20. | Z. G Zhang and H. Sun, “Calculation and evaluation of dispersions in a femtosecond pulse amplification system,” Acta Physica Sinica |

21. | S. D. Brorson and H. A. Haus, “Diffraction gratings and geometrical optics,” J. Opt. Soc. Am. B |

22. | O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commnun. |

23. | S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express |

24. | J. J. Huang and L. Y. Zhang, “Transformation of few-cycle ultrashort pulsed Gaussian beams by an angular disperser,” J. Phys. B |

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, |

**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

- D. Strickland, and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464–2468 (1989). [CrossRef]
- G. Szabó, and Z. Bor, “Broadband frequency doubler for femtosecond pulses,” Appl. Phys. B 50, 51–54 (1990). [CrossRef]
- A. V. Smith, “Group-velocity-matched three-wave mixing in birefringent crystals,” Opt. Lett. 26, 719–721 (2001). [CrossRef]
- 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]
- 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]
- E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454–458 (1969). [CrossRef]
- J. D. McMullen, “Analysis of compression of frequency chirped optical pulses,” Appl. Opt. 18, 737–741 (1979). [CrossRef] [PubMed]
- I. P. Christov, and I. V. Tomov, “Large bandwidth pulse compression with diffraction gratings,” Opt. Commun. 58, 338–342 (1986). [CrossRef]
- S. D. Brorson, and H. A. Haus, “Geometrical limitations in grating pair pulse compression,” Appl. Opt. 27, 23–25 (1988). [CrossRef] [PubMed]
- O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986). [CrossRef]
- O. E. Martinez, “Matrix Formalism for Pulse Compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988). [CrossRef]
- 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).
- Z. G. Zhang, T. Yagi, and T. Arisawa, “Ray-tracing model for stretcher dispersion calculation,” Appl. Opt. 36, 3393–3399 (1997). [CrossRef] [PubMed]
- 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).
- S. D. Brorson, and H. A. Haus, “Diffraction gratings and geometrical optics,” J. Opt. Soc. Am. B 5, 247–248 (1988). [CrossRef]
- O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986). [CrossRef]
- 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]
- 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]
- 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.
- M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 1999), pp. 446–453.

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