Ray-tracing model for stretcher dispersion calculation - OSA Publishing

Ray-tracing model for stretcher dispersion calculation Zhigang Zhang, Takashi Yagi, and Takashi Arisawa

We propose a ray-tracing model that provides a clear physical picture and simple formulas for grating pair stretcher dispersion calculations. With this model we can easily demonstrate why and to what extent the stretcher and compressor are opposite quantitatively without using a Fourier transform. The dispersion calculation shows that the spherical aberration in the stretcher decreases fourth-order dispersion compared with an aberration-free stretcher. In a chirped pulse amplification system, this fourth order can help to reduce residual fourth-order dispersion. The effect of the finite beam size and the misalignment are also considered. © 1997 Optical Society of America Key words: Ultrafast phenomena, ultrashort pulses, chirped pulse amplification, stretcher.

1. Introduction

Well-developed dispersion compensation techniques have made 8 –10-fs laser pulses routinely available from Kerr-lens mode-locked Ti:sapphire lasers.1– 4 Recently from 30-fs to sub-20-fs amplified pulses have been achieved with the chirped pulse amplification ~CPA! system.5–10 The key point of the technique for recompression of femtosecond pulses after CPA is to remove the extra dispersion that was obtained during stretching and amplification. As the aberration of a spherical mirror in a conventional grating stretcher system is considered as one of the error sources, some researchers have developed aberration-free stretchers.9,11 On the other hand, recent studies have indicated a tendency to remove this kind of stretcher from the CPA system.5,6 The other error source is the material dispersion that accumulates when the pulses pass through the gain material, the polarizer, and the Pockels cell switch in the amplifier. The material dispersion shifts the ratio of third-order to second-order dispersion to a smaller value, making it necessary to adjust the gratWhen this research was done Z. Zhang and T. Yagi were with the Laser Laboratory, Institute of Research and Innovation, 1201 Takada, Kashiwa, Chiba 277, Japan. Z. Zhang is now with the Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, Ibaraki 305, Japan. T. Arisawa is with the Kansai Research Establishment, Applied Photon Research Center, 608-9 Higashi, Shiokojicho, Shimogyo-ku, Kyoto 600, Japan. Received 18 April 1996; revised manuscript received 21 October 1996. 0003-6935y97y153393-07$10.00y0 © 1997 Optical Society of America

ing orientation in the compressor to fit the new ratio.12 However, by doing this, a large fourth order will remain, preventing the pulse from being compressed further. For this reason, regenerative amplifiers have been replaced by multiple pass amplifiers in some recent research5,7 in order to reduce the total material length in the amplifier. Although these modifications resulted in successful sub-20 –30-fs pulse amplification, they severely restrict the pulse energy and the pumping efficiency unless multiple stage amplifiers are employed. Nevertheless, for the applications that require moderate pulse widths ~50 –100 fs! and higher than millijoule levels of energy, the CPA systems that consist of a grating stretcher and a regenerative amplifier are not replaceable. Actually, with a carefully designed stretcher ~using mirror aberration to cancel the fourth order!, one can use the CPA system to produce amplified pulses as short as 18 fs.9,10 The grating–lens or grating–mirror stretcher has been generally admitted to be the conjugation of a grating pair compressor when the stretcher has the same grating groove density, orientation, and separation. This concept is based on Fourier transform of a grating through a telescope,13 which is not straightforward. The telescope that can be used for Fourier transforms is a perfect Kirchhoff–Fresnel diffraction system in which the chromatic and spherical aberrations are neglected. For an ultrabroadband femtosecond pulse to be stretched by such a stretcher, the aberrations are not negligible. To evaluate the dispersion in a grating–mirror stretcher, ray tracing for each wavelength component becomes necessary. However, previous ray tracings were addressed on 20 May 1997 y Vol. 36, No. 15 y APPLIED OPTICS

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Fig. 1. Single-mirror stretcher and compressor model for dispersion calculation.

the basis of numerical calculation for a particular grating–mirror arrangement, so that the general implications of the model cannot be easily drawn. Here we propose a ray-tracing model with which we can write the stretcher dispersion in terms of a compressor in analytical form, so that we can easily demonstrate why a stretcher functions in an opposite way as a compressor and to what extent it is aberration free. Our calculations show how much dispersion error would occur from a folded stretcher and how the misalignment of a folding mirror would affect these errors. The model also allows us to calculate the finite beam size. 2. Stretcher Model A.

Stretcher with a Single Mirror

We begin with the grating pair compressor shown in Fig. 1, where two gratings ~Grating 1 and Grating 2! are arranged in parallel with a perpendicular distance G. A spherical mirror, which is not necessarily needed for the compressor, is placed such that the incident spot at grating 1 is the spherical center of the mirror. We tried to avoid using the Littrow angle of incidence because it could introduce an additional vertical chirp.9,14 When the incident beam follows the path PABS, the longer wavelength components experience a delay ~see Fig. 2!. In this case, the grating pair is known as a compressor. The group delay, when evaluated in the A–S plane ~perpendicular to the paper!, is given by15 t5

dF p b~1 1 cos u! 5 5 , dv c c

(1)

where p is the optical ray path length for the quasimonochromatic wave, F is the phase of the wave, b 5 AB is the slant distance between the gratings, and c is the speed of light. The incident angle g and the diffraction angle g 2 u follow the grating equation sin g 1 sin~g 2 u! 5 lyd,

(2)

where l is the wavelength and d is the groove space of the gratings. The group delay dispersion ~i 5 2!, the third-order dispersion ~i 5 3!, and the fourth3394

APPLIED OPTICS y Vol. 36, No. 15 y 20 May 1997

Fig. 2. Demonstration of a grating pair as either a compressor or a stretcher. ll and ls represent the long and short wavelength components, respectively.

order dispersion ~i 5 4! can be obtained by

SD

diF di21 p . i 5 dv dvi21 c

(3)

The derivatives of u with respect to v can be obtained from grating Eq. ~2!. Likewise, it is easy to demonstrate that, if the beam follows the path PACBQ, the shorter wavelength components will experience a delay ~see also Fig. 2!. Thus in such an arrangement, the grating pair acts as a stretcher. Note that AC must be equal to R to obtain this function with R as the radius of the mirror. Because at the P–Q plane the ray path length p is p 5 PA 1 ACB 1 BE 1 EQ 5 4R 2 b~1 1 cos u!,

(4)

where PA 5 R, ACB 5 2R 2 b,

(5) (6)

BE 5 ~R 2 b!cos u,

(7)

EQ 5 R~1 2 cos u!,

(8)

The group delay for the stretcher can be obtained easily as t5

dF 4R b~1 1 cos u! 5 2 . dv c c

(9)

Compared with Eq. ~1!, we see that the group delay of the stretcher is the same as that of the compressor except for the opposite sign and a constant. Because the constant vanishes during derivation, the group delay dispersion and the consequent higher-order dispersions have the same values as those of the compressor, with the sole exception of an opposite sign. Note that no approximations are made for deriving Eq. ~9!, therefore this single-mirror stretcher configuration is free of spherical aberration errors.

B.

Stretcher with a Folded Telescope

A single-mirror stretcher can easily be extended to model the most commonly used grating–telescope stretcher that was first described by Martinez10 as a compressor in the 1.3–1.6-mm region. Figure 3 shows our grating–telescope stretcher model in a folded configuration. In comparison with our previous results, the axis of the system has a constant angle u0 with respect to a horizontal reference line, where u0 is the difference between the angles of incidence and diffraction at the central wavelength. For an arbitrary ray diffracted by a grating with an angle u1 to the system axis, the angle of diffraction becomes g 2 u 5 g 2 ~u0 1 u1!.

(10) Fig. 3. Stretcher model with a folded telescope.

As the system is folded, the diffraction and the collimation gratings can be shared. However we prefer to regard them as two individual gratings, stacked one above the other. The distances from the mirror to the diffraction grating and from the mirror to the collimation grating are s1 and sc, respectively. The ray tracing begins at P and is incident on the grating with an angle g. The diffracted beam of a wavelength with an initial angle to the u1 axis bounces between the spherical and folding mirrors, following PB, BD, DE, EH, HI, back to the collimation grating with an angle of u4 to the axis, and exits the system at Q. The total ray path length is then p 5 p0 1 p1 1 p2 1 p3 1 p4 1 p5,

where

S

G5R 12

D

sin f4 sc 2 cos~g 2 u0! sin u4 R

(14)

is the perpendicular distance between the collimation grating and the image of the diffraction grating. Note that G and s4 are no longer constants because of the aberration. In the paraxial approximation where the aberration is negligible, the image position of the diffractoin grating should be s4 5 R 2 s1.

(11)

(15)

We then get the slant distance where pi ’s are defined as p0 5 PB 5 l0,

(12a)

p1 5 BD 5 l1,

(12b)

p2 5 DEH 5 l3 2 l2,

(12c)

p3 5 HI 5 HA 2 AI 5 l4 2 b,

(12d)

p4 5 IJ 5 p3 cos~u0 1 u4!,

(12e)

p5 5 JQ 5 LO 2 KM 2 MO 5 R 2 @l4 cos~u 1 u4! 1 ~R 2 s4!cos u0#.

b0 5 R 2 s1 2 sc 5 2f 2 s1 2 sc

D

sin f4 sc cos~g 2 u0! 5R 12 2 sin u4 R cos~g 2 u0 2 u4! (13)

(16)

(17)

is the only scaling parameter that has been used in previous studies9,13 for evaluation of stretcher dispersions. Based on the use of ray tracing, we obtain

F S D G

p0 5 l0 5 R 1 2 1 2

cos~g 2 u0! b 5 ~s4 2 sc! cos~g 2 u0 2 u4!

G 5 , cos~g 2 u0 2 u4!

cos~g 2 u0! . cos~g 2 u0 2 u4!

Particularly for the ray within the axis ~the central wavelength!,

(12f )

The ray-tracing parameter li ’s are defined in the Appendix and can be obtained by successive ray tracings. b 5 AI is supposed to be the slant distance between the collimation grating and the image of the diffraction grating located at s4:

S

b 5 ~R 2 s1 2 sc!

p1 5 R

s1 cos u0 , R

sin~u1 2 f1! , sin u1

F

(18a)

(18b)

G

p2 5 R

sin~u3 2 f3! sin~u1 2 f1! 1 , sin u3 sin u2

(18c)

p3 5 R

sin~u3 2 f3! 2 b, sin u4

(18d)

where ui and fi are the angle parameters that we used in ray tracing ~see the Appendix!. Substituting pi ’s into Eq. ~11!, we have the total optical 20 May 1997 y Vol. 36, No. 15 y APPLIED OPTICS

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path length

S HF S D G F D S DG S D J

p5R 22 12 1 1 3

1 s1 cos u0 1 sin~u1 2 f1! R sin u1

1 1 1 1 sin~u3 2 f3! 1 sin u2 sin u3 sin u4

sin f4 sin f4 sc cos u0 2 1 2 2 sin u4 sin u4 R

cos~g 2 u0! @1 1 cos~u0 1 u4!# . cos~g 2 u0 2 u4!

(19)

Unlike Eq. ~4!, it is not obvious from Eq. ~19! that the stretcher is opposite the compressor, because most terms are functions of the diffraction angle and therefore of the wavelength. It is these terms that are the sources of phase distortion. The terms within the braces are independent of the mirror radius, which allows us to define constant C, aberration term A, and dispersion term D, respectively, as

S D

C522 12

F

s1 cos u0, R

S

A 5 sin~u1 2 f1!

S

S

D

D

DG

sin f4 1 cos u0, sin u4

(21)

cos~g 2 u0! sin f4 sc 2 cos u4 R cos~g 2 u0 2 u4!

3 @1 1 cos~u0 1 u4!# 5

du1 is of the order of 1022 when s1 5 0.4R. This implies that the diffraction grating should be positioned as close as possible to the mirror. In an extreme case ~aberration free!, s1 5 0, so that dAy du1 3 0. The system transits to the single-mirror configuration shown in Fig. 1. One can also obtain the same result by substituting s1 5 0 into the ray-tracing formulas ~see the Appendix!

1 1 1 1 sin~u3 2 f3! sin u1 sin u2

1 1 3 1 sin u3 sin u4 D5 12

(20)

Fig. 4. Derivative of the aberration term dAydu1 with respect to the relative diffraction angle u1 for various grating positions.

(22)

b @1 1 cos~u0 1 u4!#. R

f1 5 u1,

(24a)

f2 5 2f1 5 2u1,

(24b)

u2 5 2u1,

(24c)

f4 5 2f3 5 0,

(24d)

u4 5 u1.

(24e)

Then Eq. ~25! gives a result similar to that in Eq. ~4! except for the lack of a constant:

Thus, the optical path length in a telescope stretcher is simplified to

p 5 R~4 2 cos u0! 2 b@1 1 cos~u0 1 u1!#.

p 5 R~C 1 A 2 D!.

When s1 5 sc 5 Ry2 ~not shown in Fig. 4! and the rays are paraxial, we have an approximation of b 5 0, and then the system should be a constant delay line in which

(23)

Note that C, A, and D are dimensionless parameters. To derive the same result as in Eq. ~4!, aberration term A must be independent of the wavelength. Unfortunately it is not easy to show that A is a constant. We could use dAydu1 to determine to what degree A is a constant. Figure 4 shows the calculated dAydu1 term as a function of u1 at various grating positions. The value of dAydu1 should be zero for an aberrationfree stretcher; however, it increases continuously with the offset angle u1, which is understandable because only small off-axis angles can maintain paraxial rays for an imaging system. This indicates that the grating groove density should be reduced when one uses a stretcher to expand a very short pulse. The value of dAydu1 is significantly higher at additional positions of the diffraction grating from the mirror. For example, at position s1 5 0.05R, dAydu1 is of the order of 1025, whereas dAy 3396


p 5 3R 2 R cos u0. C.

(25)

(26)

Stretcher with Finite Incident Beam Size

The above stretcher model can be applied to the case of a finite beam size. One can stipulate that the finite beam size results in an s1 shift across the beam. A simple geometric manipulation gives the maximum shift Ds1 as Ds1 5

s cos~g 2 u0 2 u1! , 2 cos g sin u1

(27)

with s as the full incident beam size. The optical path length of a ray at the edge of the incident beam

Fig. 5. Stretcher model with a misaligned folding mirror. ε represents the misalignment angle.

can be calculated by replacing s1 with s1 1 Ds1 in Eq. ~27!. D.

Stretcher with Small Folding Mirror Misalignment

In practice, a real stretcher may not be perfectly aligned, because the spectrum trace is so large and the central wavelength is difficult to identify by a viewer. We assume that the first trace is aligned critically, whereas the additional traces may have a small misalignment because of the folding mirror. If this small angle misalignment of the folding mirror were ε, as defined in Fig. 5, the optical path would become

HF S D G F D S S

p5R 12 12 1 2 3 1

S

1 s1 cos u0 1 sin~u1 2 f1! R sin u1

1 1 1 1 sin~u39 2 f39! 1 sin u2 sin u39 sin u49

D

DG

sin f49 sin f49 sc cos u0 2 1 2 2 sin u49 sin u49 R

cos~g 2 u0! @1 1 cos~u0 1 u49!# cos~g 2 u0 2 u49!

S

D

S

DJ

1 sin f2 cos ε sin u3 12 1 2 sin u2 cos~u3 1 ε! sin u39

,

(28)

where the angles u3,49 and f3,49 are those we obtained by ray tracing through u39 5 u3 1 2ε.

(29)

Introducing misalignment term Am yields Am 5

S

D

S

D

1 sin f2 cos ε sin u3 12 . 1 2 sin u2 cos~u3 1 ε! sin u39

(30)

The total optical path length can be written as pm 5 R~C 1 A 1 Am 2 D!.

(31)

Fig. 6. Normalized group delay dispersion ~single pass, s0 5 sc 5 Ry4, 1400-linesymm grating groove density! as a function of wavelength for misalignment angles of 22°, 0°, and 12°. The dashed curve represents the group delay dispersion of the aberration-free stretcher.

Obviously when ε 3 0, Am 3 0, resulting in pm 3 p, which means that the system returns to the wellaligned case. 3. Calculation Examples and Discussions

The model we have presented delivers an easy derivation of dispersion in every order in a phase expansion series and a comparison of the dispersion with an aberration-free stretcher. Prior to a discussion, we want to interpret the higher-order dispersion in analytical geometry. If we begin with a second-order term in the phase expansion series and draw it as a curve with respect to optical frequency, the third and fourth orders will actually be the slope and the curvature of the second-order curve at that particular wavelength. Thus evaluation of the dispersion errors between a practical stretcher and an aberration-free stretcher provides the comparison between the slopes and curvatures of their second-order curves. Consequently, the dispersion compensation technique in a CPA system can also be considered as a second-order curve matching both slopes and curvatures. Figure 6 shows the second-order dispersion curves of a grating–telescope stretcher normalized to the mirror radius as a function of wavelength for various folding angles, and these are compared with an aberration-free stretcher ~i.e., A 5 constant, u4 5 u1!. The calculations were made for a grating groove density of 1400 linesymm, an incident angle of g 5 44°, and a grating position s1 5 sc 5 Ry4. Apparently, when the system is perfectly aligned at the central wavelength, the second and third orders are in good agreement with that of an aberration-free stretcher. However, there are indeed some visible differences at longer and shorter wavelengths. In other words, the second-order dispersion in such a stretcher has a smaller curvature, indicating a smaller fourth-order dispersion. The direct result of this mismatch in longer and shorter wavelengths is the long and low level wings in the compressed pulse, if the pulse spectrum bandwidth is sufficiently large and does not 20 May 1997 y Vol. 36, No. 15 y APPLIED OPTICS

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considered here, 64°, the spectral trace displacement on the mirror surface can be as large as a few centimeters, which is easy to locate and correct. Therefore, we can neglect the effect of the misalignment on the spatial varying spectrum. For a finite beam size of a few millimeters, for example, the calculation shows that the beam path error is less than 1025 with respect to a radius of curvature of 1000 mm. It may be negligible in the first instance.

Fig. 7. Output angle deviation from an incident angle for misalignment angles of 24°, 22°, 0°, 12°, and 14°.

pass through an amplifier. However in a practical CPA system, this fourth-order error can help to reduce the residual positive fourth order that results from the smaller diffraction angle in the grating compressor, which is necessary to compensate for thirdorder distortion.9,12 The detailed calculations for a whole CPA system can be done precisely to demonstrate this point, but they are beyond the scope of this paper. When a folding mirror is misaligned ~62° in Fig. 6!, not only the second order but also the higher orders of dispersion deviate from those of an aberration-free stretcher but in different directions. One can easily compensate the second order by varying the grating separation in the compressor, however, the higher orders are not readily compensable. We also examined the collimations at the output plane. The deviation angle with respect to the incident angle is calculated and shown in Fig. 7. The deviation angle is never zero, but it is small in the case of perfect alignment. When misalignment occurs, this angle becomes large but is still well below 1°. In addition, for the largest misalignment angle

4. Conclusions

We have developed a stretcher ray-tracing model and provided a set of simple but precise formulas for evaluation of its dispersion. This model is straightforward and is consistent with the physical meaning of a compressor. This model shows that the spherical aberration would result in a decreased fourth-order dispersion compared with an aberration-free stretcher. The residual positive fourth order in a CPA system can be partially reduced because of this smaller fourth order in the stretcher. The misalignment of the folding mirror of the stretcher also produces either larger or smaller higher-order dispersion errors, leaving the possibility of further reducing the higher-order dispersion by varying the folding angle in the stretcher. For a practical folded stretcher, a three-dimensional model is necessary, particularly for extremely ultrashort pulse amplification. This research is in progress.

Appendix

For the reader’s convenience, the folded stretcher must be opened to observe the clear traces, as shown in Fig. 8. The ray path starting from P is incident on the grating at B and is diffracted and bounced by the

Fig. 8. Ray-tracing paths for an open grating– telescope stretcher. Dashed curve and lines represent the image part of the mirror, grating, and paths when the telescope is folded. 3398


spherical mirror at D to H to form the first trace. The result of this trace is a virtual image point B9. By defining s1 5 BC, s2 5 CB9, l1 5 BD, and l2 5 B9D, we obtain the following formulas for the first trace:

S D

sin f1 5 1 2

s1 sin u1, R

(A1a)

f2 5 2f1,

(A1b)

u2 5 u1 2 2f1,

(A1c)

s2 sin f2 , 512 R sin u2

(A1d)

l1 5 R


(A1e)

l2 5 R

sin~u1 2 f1! . sin u2

(A1f )

The second trace is from B9H to HA9, where A9 is the real image of B9. Note that in general A9 and B are not the same points although they are close. By defining s3 5 B9O, s4 5 OA9, l3 5 B9H, and l4 5 HA9, we have the second trace formulas: u3 5 2u2,

(A2a)

s3 s2 511 , R R

(A2b)

S D

sin f3 5 1 2

s3 sin u3, R

(A2c)

f4 5 2f3,

(A2d)

u4 5 u3 2 2f3,

(A2e)

s4 sin f4 , 512 R sin u4

(A2f )

l3 5 R


(A2g)

l4 5 R

sin~u3 2 f3! . sin u4

(A2h)

The authors thank Y. Pang and E. Gabl of ClarkMXR, Inc. for the stimulating discussions.

References 1. P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, and A. J. Schmidt, “Operation of a femtosecond solitary laser in the vicinity of zero group-delay dispersion,” Opt. Lett. 18, 54 –56 ~1993!. 2. M. T. Asaki, C-P. Huang, D. Garvey, J. Zhou, H. C. Kapteyn, and M. M. Murnane, “Generation of 11-fs pulses from a selfmode-locked Ti:sapphire laser,” Opt. Lett. 18, 977–979 ~1993!. 3. B. Proctor and F. Wise, “Generation of 13-fs pulses from a modelocked Ti:Al2O3 laser with reduced third order dispersion,” Appl. Phys. Lett. 62, 470 – 472 ~1993!. 4. A. Stingl, M. Lenzner, Ch. Spielmann, F. Krausz, and R. Szipo¨cs, “Sub-10-fs mirror-dispersion-controlled Ti:sapphire laser,” Opt. Lett. 20, 602– 604 ~1995!. 5. M. Lenzner, Ch. Spielmann, E. Wintner, F. Krausz, and A. J. Schmidt, “Sub-20-fs, kilohertz-repetition-rate Ti:sapphire amplifier,” Opt. Lett. 20, 1397–1399 ~1995!. 6. T. Joo, Y. Jia, and G. R. Fleming, “Ti:sapphire regenerative amplifier for ultrashort high-power multikilohertz pulses without an external stretcher,” Opt. Lett. 20, 389 –391 ~1995!. 7. J. Zhou, C. P. Huang, M. M. Murnane, and H. C. Kapteyn, “Amplification of 26-fs, 2-TW pulses near the gain-narrowing limit in Ti:sapphire,” Opt. Lett. 20, 64 – 66 ~1995!. 8. K. Wynne, G. D. Reid, and R. M. Hochstrasser, Regenerative amplification of 30 fs pulses in Ti:sapphire at 5 kHz,” Opt. Lett. 19, 895– 897 ~1994!. 9. B. E. Lemoff and C. P. J. Barty, “Quintic-phase-limited, spatially uniform expansion and recompression of ultrashort optical pulses,” Opt. Lett. 18, 1651–1653 ~1993!. 10. C. P. J. Barty, T. Guo, C. Le Blanc, F. Raksi, C. Rose-Petruck, J. Squier, K. R. Wilson, V. V. Yakovlev, and K. Yamakawa, “Generation of 18-fs, multiterawatt pulses by regenerative pulse shaping and chirped-pulse amplification,” Opt. Lett. 21, 668 – 670 ~1996!. 11. G. Cheriaux, P. Rousseau, F. Salin, J. P. Chambaret, B. Walker, and L. F. Dimauro, “Aberration-free stretcher design for ultra-short pulse amplification,” Opt. Lett. 21, 414 – 416 ~1996!. 12. J. V. Rudd, G. Korn, S. Kane, J. Squier, G. Mourou, and P. Bado, “Chirped-pulse amplification of 55-fs pulses at 1-kHz repetition rate in a Ti:Al2O3 regenerative amplifier,” Opt. Lett. 18, 2044 –2046 ~1993!. 13. O. E. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in the 1.3–1.6 mm region,” IEEE J. Quantum Electron. QE-23, 59 – 64 ~1987!. 14. Z. Zhang, S. Harayama, T. Yagi, and T. Arisawa, “Vertical chirp in grating stretcher and compressor,” Appl. Phys. Lett. 67, 176 –178 ~1995!. 15. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454 – 458 ~1969!.

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