evolution of the frequency chirp of gaussian pulses.pdf

8/11/2019 Evolution of the frequency chirp of Gaussian pulses.pdf

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Evolution of the frequency chirp of Gaussianpulses and beams when passing through a pulse

compressor

Derong Li

1, 3

, Xiaohua Lv

1

*, Pamela Bowlan

2

,Rui Du1, Shaoqun Zeng

1, Qingming Luo

1

1Britton Chance Center for Biomedical Photonics, Wuhan National Laboratory for Optoelectronics, HuazhongUniversity of Science & Technology, Wuhan, 430074, China

2Georgia Institute of Technology, School of Physics 837 State Street NW, Atlanta, Georgia 30332 USA3Key Lab for Biomedical Informatics and Health Engineering, Institute of Biomedical and Health Engineering,

Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen, 518055, China*[email protected]

Abstract: The evolution of the frequency chirp of a laser pulse inside aclassical pulse compressor is very different for plane waves and Gaussianbeams, although after propagating through the last (4th) dispersive element,the two models give the same results. In this paper, we have analyzed theevolution of the frequency chirp of Gaussian pulses and beams using amethod which directly obtains the spectral phase acquired by the

compressor. We found the spatiotemporal couplings in the phase to be thefundamental reason for the difference in the frequency chirp acquired by aGaussian beam and a plane wave. When the Gaussian beam propagates, anadditional frequency chirp will be introduced if any spatiotemporalcouplings (i.e. angular dispersion, spatial chirp or pulse front tilt) arepresent. However, if there are no couplings present, the chirp of theGaussian beam is the same as that of a plane wave. When the Gaussianbeam is well collimated, the introduced frequency chirp predicted by theplane wave and Gaussian beam models are in closer agreement. This workimproves our understanding of pulse compressors and should be helpful foroptimizing dispersion compensation schemes in many applications offemtosecond laser pulses.

2009 Optical Society of America

OCIS codes:(320.0320) Ultrafast optics; (320.5520) Pulse compression; (260.2030) Dispersion

References and links

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2. R. L. Fork, O. E. Martinez, and J. P. Gordon, Negative dispersion using pairs of prisms, Opt. Lett. 9(5), 150152 (1984).

3. J. Squier, F. Salin, G. Mourou, and D. Harter, 100-fs pulse generation and amplification in Ti:AI2O3, Opt. Lett.16(5), 324326 (1991).

4. C. L. Blanc, G. Grillon, J. P. Chambaret, A. Migus, and A. Antonetti, Compact and efficient multipassTi:sapphire system for femtosecond chirped-pulse amplification at the terawatt level, Opt. Lett. 18(2), 140142(1993).

5. O. E. Martinez, 3000 times grating compressor with positive group velocity dispersion: application to fibercompensation in 1.3-1.6 um region, IEEE J. Quantum Electron. 23(1), 5964 (1987).

6. S. Zeng, D. Li, X. Lv, J. Liu, and Q. Luo, Pulse broadening of the femtosecond pulses in a Gaussian beampassing an angular disperser, Opt. Lett. 32(9), 11801182 (2007).

7. D. Li, X. Li, S. Zeng, and Q. Luo, A generalized analysis of femtosecond laser pulse broadening after angulardispersion, Opt. Express 16(1), 237247 (2008).

8. D. Li, S. Zeng, Q. Luo, P. Bowlan, V. Chauahan, and R. Trebino, Propagation dependence of chirp in Gaussianpulses and beams due to angular dispersion, Opt. Lett. 34(7), 962964 (2009).

9. S. Szatmari, G. Kuhnle, and P. Simon, Pulse compression and traveling wave excitation scheme using a singledispersive element, Appl. Opt. 29(36), 53725379 (1990).

10. S. Szatmri, P. Simon, and M. Feuerhake, Group velocity dispersion compensated propagation of short pulses indispersive media, Opt. Lett. 21(15), 11561158 (1996).

#112210 - $15.00 USD Received 1 Jun 2009; revised 4 Sep 2009; accepted 5 Sep 2009; published 10 Sep 2009

(C) 2009 OSA 14 September 2009 / Vol. 17, No. 19 / OPTICS EXPRESS 17070


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11. S. Akturk, X. Gu, M. Kimmel, and R. Trebino, Extremely simple single-prism ultrashort- pulse compressor,Opt. Express 14(21), 1010110108 (2006).

12. M. Nakazawa, T. Nakashima, and H. Kubota, Optical pulse compression using a TeO2 acousto-optical lightdeflector, Opt. Lett. 13(2), 120122 (1988).

13. S. Zeng, X. Lv, C. Zhan, W. R. Chen, W. Xiong, S. L. Jacques, and Q. Luo, Simultaneous compensation forspatial and temporal dispersion of acousto-optical deflectors for two-dimensional scanning with a single prism,Opt. Lett. 31(8), 10911093 (2006).

14. Y. Kremer, J. F. Lger, R. Lapole, N. Honnorat, Y. Candela, S. Dieudonn, and L. Bourdieu, A spatio-temporally compensated acousto-optic scanner for two-photon microscopy providing large field of view, Opt.Express 16(14), 1006610076 (2008).

15. O. E. Martinez, Grating and prism compressors in the case of finite beam size, J. Opt. Soc. Am. B 3(7), 929934 (1986).

16. O. E. Martinez, Pulse distortions in tilted pulse schemes for ultrashort pulses, Opt. Commun. 59(3), 229232(1986).

17. Z. L. Horvth, Z. Benk, A. P. Kovcs, H. A. Hazim, and Z. Bor, Propagation of femtosecond pulses throughlenses, gratings, and slits, Opt. Eng. 32(10), 24912500 (1993).

18. K. Varj, A. P. Kovcs, K. Osvay, and G. Kurdi, Angular dispersion of femtosecond pulses in a Gaussianbeam, Opt. Lett. 27(22), 20342036 (2002).

19. J. C. Diels, and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).20. C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, Temporal aberrations due to

misalignments of a stretcher-compressor system and compensation, IEEE J. Quantum Electron. 30(7), 16621670 (1994).

21. K. Osvay, A. P. Kovcs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatri, Angular dispersion and temporalchange of femtosecond pulses from misaligned pulse compressors, IEEE J. Sel. Top. Quantum Electron. 10(1),213220 (2004).

22. K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, Measurement of non-

compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,Opt. Commun. 248(1-3), 201209 (2005).

23. A. E. Siegman, Lasers, (University Science, Mill Valley, CA, 1986).24. X. Gu, S. Akturk, and R. Trebino, Spatial chirp in ultrafast optics, Opt. Commun. 242(4-6), 599604 (2004).25. S. Akturk, M. Kimmel, P. OShea, and R. Trebino, Measuring spatial chirp in ultrashort pulses using single-shot

Frequency-Resolved Optical Gating, Opt. Express 11(1), 6878 (2003).26. S. Akturk, X. Gu, E. Zeek, and R. Trebino, Pulse-front tilt caused by spatial and temporal chirp, Opt. Express

12(19), 43994410 (2004).27. S. Akturk, X. Gu, P. Gabolde, and R. Trebino, The general theory of first-order spatio-temporal distortions of

Gaussian pulses and beams, Opt. Express 13(21), 86428661 (2005).28. P. Gabolde, D. Lee, S. Akturk, and R. Trebino, Describing first-order spatio-temporal distortions in ultrashort

pulses using normalized parameters, Opt. Express 15(1), 242251 (2007).

1. Introduction

A classical pulse compressor (commonly including four gratings or prisms, or a pair ofgratings and prisms with a double-pass configuration) [1,2] can stretch or compress a

femtosecond laser pulse by introducing variable amounts of positive or negative frequencychirp (also referred to as just the chirp). Pulse compressors are ubiquitous in ultrafast opticsbecause of their ability to tailor the pulses temporal duration, which is essential for makingand maintaining intense, short pulses. Important applications include chirped-pulseamplification (CPA) and material-dispersion compensation, the latter of which is necessaryfor generating ultrashort pulses [35]. Since the duration of the pulse after the compressordepends on its group-delay dispersion, it is vital to able to accurately calculate this quantity[68].

In some practical applications, the classical 4-dispersive element compressor is not alwayssuitable, and instead, a single angular dispersion element [911] or a pair of angulardispersion elements (also just referred to as an element) in a single-pass configuration [1214] are used to for dispersion control. In these cases, spatiotemporal couplings (meaning thatthere are x-, or equivalently x-tcross terms in the field) such as angular dispersion, spatialchirp, and pulse front tilt are present in the output pulse, making it even more difficult tocalculate the chirp that is added to the pulse by the compressor.

To model compressors, either a plane wave [1,2,912], or a Gaussian beam [58,1518])model is usually used. Martinez first showed that these two models sometimes give verydifferent results. Namely, the propagation dependence of the chirp of a pulse whilepropagating inside a classical pulse compressor is very different for the two types of beams.




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Interestingly, however, after propagating through the 4th dispersive element (i.e., once all ofthe angular dispersion is removed), both models predict the same results [15]. Similarly it hasbeen shown several times, that, when a pulse passes through a single angular disperser, thechirp of a Gaussian beam increases nonlinearly with propagation distance away from thedisperser, while that of a plane wave increases linearly [8]. Though compressors are verycommonly used, it seems that there is still more to learn about how they affect ultrashortpulses.

Usually, the diffraction integral is used to investigate the propagation dependence of anultrashort pulse, but this approach is quite complex making it difficult to use to understandhow compressors work and why they effect plane waves differently than they do Gaussianbeams [15]. Actually, the evolution of the chirp is entirely caused by a spatio-spectral phase(x,z,) that is added to the pulse by the compressor (since the chirp is defined as the secondorder derivative of the pulses spectral phase with respect to angular frequency) [19].Therefore, it is sufficient to determine the chirps evolution by calculating this phase changethat is acquired by the pulse due to propagation through the compressor.

In this paper, we have derived expressions for the propagation dependence of thefrequency chirp of a femtosecond Gaussian laser pulse after passing through each element of aclassical four-element compressor. These expressions were derived by calculating the phasechange acquired by the Gaussian beam after propagating through each element of thecompressor. These calculations reveal the physical mechanism by which chirp is acquired byan ultrashort pulse in a pulse compressor. As the effects of misalignment of the dispersiveelements are very complex [2022], we consider only perfectly aligned compressors in thispaper.

2. Phase acquired by a Gaussian pulse in a compressor

A steadily propagating electromagnetic field in free space is governed by the scalarapproximation of the Helmholtz equation. The plane wave is the simplest solution to thisequation, and the Gaussian beam is also a special solution which is obtained if the slowlyvarying amplitude (SVA) approximation is made. The Gaussian beam is a very good modelfor realistic laser beams, so it is frequently adopted for modeling optics experiments. Thephase of a Gaussian beam propagating in free space is given by [23]:

2

1( , ) tan ( ).

2 ( )R

kr zr z kz

R z z

= + (1)

where k is the wave-number, z is the propagation distance away from the beam waist on theaxis, r= (x

2+y

2)

1/2is the distance from the z axis,R(z) =z+zR

2/zis the radius of curvature of

the wave front, zR= kwo2/2 is the Rayleigh range of the Gaussian beam, and w0is the beam

waist size. Equation (1) describes the phase shift of a Gaussian beam at the point (r, z) relativeto the original point (0, 0). In this equation, the first term is the geometrical phase shift whichis the same as the phase of a plane wave. The second term represents a phase shift relative tothe radial position. This is due to the finite beam size of the Gaussian beam. No such termexists in the expression for plane waves because they are infinite in space. The third term isthe Gouy phase shift which is relative to the geometrical phase shift, and is also unique to aGaussian beam [23].




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Fig. 1. (a) A classical pulse compressor, which consists of four identical prisms (or otherangular dispersers, such as gratings). (b) Optical path diagram of a femtosecond laser pulse

passing through a classical pulse compressor. For two adjacent elements, the spacings are L1,L2,L3, respectively. Due to angular dispersion, in the system, different spectral componentshave different paths. Taking the entrance vertices (O1 ~O4) of each element as the originalpoint, four reference distances can be established: they arex1-z1throughx4-z4. The distances forany spectral component are defined as x1-z1 throughx4-z4. In this figure, the dashed linebetween the second and third element is the wave front of the pulse. When the alignment isperfect, 1= 3and L1= L3. BW is the beam waist, and d is the distance between the beamwaist and the first dispersive element.

As shown in Fig. 1, in order to study the chirp evolution of a femtosecond Gaussian laserpulse propagating through a pulse compressor, we first need to define all of the systemparameters. For a single angular dispersion element, the deflection angle of a femtosecondlaser pulse after passing through the element is described by [15,16]:

( , ) .

= + = +

(2)

where is the angular magnification, is the angular dispersion, is the angle of incidence,and is the frequency. For the four elements in a perfectly aligned pulse compressor, theparameters must obey the following equations [15]:

12 2

1 1

1, ,

= = (3)

3 1 3 1, , = = (4)

34 4

3 3

1, .

= = (5)

where the subscripts on each parameter correspond to the number of the element. We will use and to denote these parameters for the first element. It is assumed that the angulardispersion only occurs in the x-zplane and the Gaussian beam still follows the propagationrules of free space in the y direction. In this case, we omit the information for the y axis inorder to simplify our calculation. When an arbitrary spectral component of a femtosecondGaussian laser pulse passes through the first element and arrives at the point (x1, z1), the




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waist position is at a distance d/2+ z1relative to the apex of the element for the opposite

propagation direction for which the equivalent Rayleigh range is zR1= zR/2, where d is the

distance between the beam waist and the first dispersive element [18]. Thus the phaseexpression for the Gaussian beam can be written as:

2 2

2 11 1

1 1 1 1 2

11

/( , , ) ( / ) tan ( ).

2 ( / )R

kx d zx z k d z

zR d z

+= + +

+ (6)

In the same way, when an arbitrary spectral component passes through the second, thirdand fourth element and arrives at the locations (x2, z2), (x3, z3) and (x4, z4), thecorresponding waist position is equivalent to d+

2z1+z2, d/

2+z1+z2/2+z3,and d+

2z1 + z2 + 2z3 + z4 with respect to the apex of the first element for the opposite

propagation direction, and the corresponding Rayleigh ranges are of zR2= zR,zR3= zR/2, and

zR4=zR. Then the corresponding phase expressions for the Gaussian beam are:

2 2

2 12 1 2

2 2 2 1 2 2

21 2

( , , ) ( ) tan ( ),2 ( )

R

kx d z zx z k d z z

zR d z z

+ += + + +

+ + (7)

22 2 3

3 3 3 1 2 3 2 2

1 2 3

2 2

1 1 2 3

3

( , , ) ( / / )2 ( / / )

/ /tan ( ),R

kxx z k d z z z

R d z z z

d z z zz

= + + + ++ + +

+ + +

(8)

2

2 2 4

4 4 4 1 2 3 4 2 2

1 2 3 4

2 2

1 1 2 3 4

4

( , , ) ( )2 ( )

tan ( ).R

kxx z k d z z z z

R d z z z z

d z z z z

z

= + + + + ++ + + +

+ + + +

(9)

where the subscripts of the phase functions represent the number of elements. From the aboveanalysis, we obtain the spatio-spectral phase of the Gaussian beam at any position whenpassing through a pulse compressor. The corresponding frequency chirp can be obtained bycalculating the second order derivative of the phase with respect to the frequency .

3. Chirp evolution of a Gaussian pulse in a compressor

As shown in Fig. 1, after passing through the first angular dispersion element, each spectralcomponent of the femtosecond laser pulse is separated in space, and propagates a differentdistance. The reference spectral component propagates a distance z1, while an arbitraryspectral component propagates a distance z1, which can be related toz1using the angle 1.The relevant formulae are [17]:

1 1 1 1 1cos sin ,z z x = + (10)

1 1 1 1 1sin cos .x z x = + (11)

And the corresponding first and second order derivatives are:

'

10,z = (12)

2'' 1

1 1,

dz z

d

=

(13)

' 1

1 1,

dx z

d

= (14)




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2

'' 1

1 1 2.

dx z

d

= (15)

where = d1/d is the angular dispersion introduced by the element. The second order

derivative of the expression for1 with respect to spectral frequency (which describes the

introduced chirp of the Gaussian beam passing through the first element) can then be obtained

as follows:2 2 2 2

'' 2 21 1 11 1 12 2 2 2

1 1

( )( ) [1 ].

( / ) ( )G

R

k z z d zz k z k z

R d z d z z

+= + =

+ + + (16)

Comparing the above expression with the phase function1

(Eq. (6)), we can see that the

first term is the chirp introduced by the geometrical phase shift, which is negative. The secondterm is the chirp introduced by the radially dependent phase term which has an x-coupling,and this is positive. The chirp introduced by the Guoy phase shift is usually far less than thefirst two terms, so for simplicity we have left it out of Eq. (16) and will neglect it for the restof our discussion.

After the pulse has passed through the second element (which is anti-parallel with the firstelement and separated by a distance L1), the angular dispersion is totally eliminated butdifferent spectral component of the pulse are still separated transversely in space, or spatial

chirp, and also thex- coupling term in the phase (wave front tilt dispersion) are present [2428]. In this interval, the relevant distances are:

2 2,z z const= + (17)

2 2 .x x const = + + (18)

where = L1is the spatial chirp present in the pulse before entering the second element.Then the corresponding first and second derivatives of z2and x2with respect to can beobtained as follows:

'

2 0,z = (19)

''

2 0,z = (20)

'

2 1,x L = (21)''

20.x

= (22)

And thus the chirp of the pulse after passing through the second element is:

2 2

2 2 2 2 2 1 1 2

2 1 1 12 2 2 2

1 2 1 2

'' ( )1( ) [1 ].

( ) ( )R

G

L d L zz k L k L k L

R d L z d L z z

+ += + =

+ + + + +(23)

Comparing Eq. (23) with the phase function2

(Eq. (7)), we can see that the first term is

the chirp introduced by the geometrical phase shift, which is negative. As the second elementremoves angular dispersion, no extra chirp will be introduced by the geometrical phase shiftafter passing through the second element. The second term represents the frequency chirpintroduced by the radially dependent phase when the pulse travels the distance z2, which ispositive. Here we can see that though the angular dispersion has been eliminated after passing

through the second element, x-couplings, namely, spatial chirp and wave-front-curvaturedispersion are still present, and these combined with a changing beam spot size introduceadditional frequency chirp as the pulse propagates through this region of the compressor.

After the pulse has passed through the third element, angular dispersion is present again.As shown in Fig. 1, the distance for any spectral component between the third and the fourthelements isL3cos3[2], and thus the relevant formulae are:




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3 3 3 3 3

cos sin ,z z x = (24)

3 3 3 3 3 3( )sin cos .x L z x = + (25)

Then the first and second derivatives of z3and x3with respect to frequency can beobtained as follows:

'

30,z

= (26)

2

'' 3

3 3 ,d

z zd

=

(27)

' 33 3 3

( ) ,d

x L zd

= (28)

2

'' 3

1 3 3 2( ) .

dx L z

d

= (29)

where the angular dispersion introduced by the third element is d3/d= -. Also, the chirp ofthe femtosecond laser pulse after passing through the third element is:

( ) ( )

( )

2 2 2 23 1 3 3 3 2 2

1 2 3

2 2 2

1 2 322 2 2

1 3 3 3 22 2 2

1 2 3

'' 1( ) ( )( / / )

.

R

G z k L k z k L zR d a L L z

d L L zk L k z k L z

d L L z z

= + + + +

+ + += +

+ + + +

(30)

Comparing the above expression with the phase function3

in Eq. (8), we see that the first

term is the chirp from the geometrical phase term acquired from propagating between the firstand the second elements, which is negative. The second term is the chirp from the geometricalphase shift due to propagation between the third and the fourth elements, which is alsonegative. Finally, the third term is the chirp introduced by the radially dependent phase shiftdue to propagation through the same distance, which is positive. If the propagation distancez3=L3, i.e. when the pulse arrives at the entrance of the fourth element, the chirp from the lastterm becomes zero.

Provided thatz3=L3, and the angular dispersion from the first and third elements are equaland opposite, after the pulse has passed through the fourth element, all of the spectralcomponents overlap transversely in space (i.e. no spatiotemporal couplings are present), andboth the angular dispersion and all other spatiotemporal couplings have been removed, giving:

4 4,z z= (31)

4 4.x x = (32)

So, after passing through the fourth element, neither the geometrical phase shift nor theradially dependent phase shifts introduce any chirp and then we get [15]:

'' 2 2 2

4 1 3 1( ) 2 .

Gz k L k L k L = = (33)

In this formula we can see that the final frequency chirp of the Gaussian beam is the sameas that of a plane wave after passing through the pulse compressor. This final expression

depends only on the geometrical phase shift and has no dependence on the radially dependentphase shift. From the above analysis of the chirp of the Gaussian beam, we can easily obtainthe corresponding chirp of the plane wave by considering only the geometrical phase term. Asthe term describing the geometrical phase shift of the Gaussian beam is equal to the phaseterm of the plane wave, the chirp introduced by this term is the corresponding chirp of theplane wave:




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'' 21 1

( ) ,P

z k z = (34)

'' 2

2 1( ) ,

Pz k L = (35)

'' 2 2

3 1 3( ) ,

Pz k L k z = (36)

'' 2

4 1( ) 2 .

P z k L = (37)

From the above analysis, we can see that, for the Gaussian beam, the chirp introduced bythe geometrical phase term is negative while that introduced by the radially dependent phaseshift is positive at any position after passing through the element; it becomes zero when no -xcoupling terms are present in the phase. Also, the radially dependent phase shift in the phasefunction of the Gaussian beam is the fundamental reason for the difference between thefrequency chirp of a Gaussian beam and plane wave when passing through the classical pulsecompressor. The frequency chirp evolution of the Gaussian beam and plane wave over thewhole propagation process are shown in Fig. 2, using the parameters = 1, = 0.1 rad/m, d=zR=L1=L2=L3= 1 m.

Fig. 2. Comparison of the chirp evolution of a Gaussian beam (green) and plane wave (black)when passing through a 4-dispersive element pulse compressor. The red arrows in the figureshow the difference in the two models.

As shown in Fig. 2, though the frequency chirp evolution of Gaussian beams is differentfrom that of plane waves, the final result is the same for both models. In the following section,we provide a detailed analysis of the physical mechanism of this phenomenon.

4. Comparison of the Gaussian beam and plane wave models for a pulse compressor

Here we compare the propagation dependent frequency chirps predicted by the two models inorder to better understand their differences. For the plane wave model, in each interval ofpropagation, the following expressions are obtained:

2

1 1

''( ) ,

Pz k z = (38)

2

''( ) 0,

Pz = (39)

'' 2

3 3( ) ,P z k z = (40)




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''4

( ) 0.P

z = (41)

where thezs are the corresponding propagation intervals as shown in Fig. 1. We can see thatthe frequency chirp of the plane waves only changes in the interval where the angulardispersion is nonzero (e.g. after the pulse has passed through the first and third elements, asshown in Fig. 2). The corresponding chirp evolution of the Gaussian beam in each interval is:

2 2

2 1 1

1 1 2 2 2

1

'' ( )( ) [1 ],( )

R

Gz d zz k z

d z z

+ = + +

(42)

2 2

2 2 2 1 2 1

2 1 2 2 2 2 2 2

1 2 1

2 2 2

1 2 2

1 2 1

'' ( ) ( )( ) [ ]

( ) ( )

1 1[ ],

( ) ( )

R R

G

d L z d Lz k L

d L z z d L z

k LR d L z R d L

+ + +

= + + + + +

= + + +

(43)

2 2 2 2 2

2 2 2 21 2 3 1 1 2

3 3 3 3 12 2 2 2 2 2 2

1 2 3 1 2

2 2 2 2 23 3

3 1 2 2 2

31 2 1 2 3

'' ( ) ( )( ) ( )

( ) ( )

1 1[ ( ) ],

( ) ( )

R R

G

d L L z L d L Lz k z k L z k L

d L L z z d L L z

L zk z k L

LR d L L R d L L z

+ + + + +

= + + + + + + + +

=

+ + + + +

(44)

4

''( ) 0.

Gz = (45)

In contrast to the plane wave model, for Gaussian beams, the chirp changes as the pulsepropagates even when the angular dispersion is zero. When the angular dispersion is zero butthe x- coupling in the phase is nonzero, the frequency chirp of the Gaussian beam stillchanges along the propagation distance (as shown in Fig. 2, after the pulse has passed throughthe second element). After the pulse has passed through the first and third elements (where theangular dispersion is not zero), the changes in the frequency chirp consist of two parts; one isintroduced by the geometrical phase shift (corresponding to the first term in Eqs. (42) and(44)) and the other is due to the radially dependent phase shift (corresponding to the secondterm in Eqs. (42) and (44)).

Calculating the difference in the chirp predicted by the two models over correspondingintervals (Eq. (42) is substracted from Eq. (38), and, etc.), we get the following results:

2 2 2

1 1 2

1

'' 1( )

( ),PG z k z

R d z

=

+ (46)

2 2 2

2 1 2 2

1 2 1

'' 1 1( ) ,

( ) ( )PG

z k LR d L z R d L

= + + +

(47)

2

2 2 2 3 3

3 1 2 2 2

31 2 1 2 3

'' 1 1( ) ,

( ) ( )PG

L zz k L

LR d L L R d L L z

=

+ + + + +

(48)

4

''( ) 0.PG z = (49)

From the above equations we can clearly see that over any propagation interval, the

fundamental reason for the difference between frequency chirp of the Gaussian beam andplane wave is due to the radially dependent, or the x- coupling term in the phase of theGaussian beam.

As shown in Fig. 2, (A) In the interval 0-L1, the chirp of the Gaussian beam increases lessrapidly than that of the plane wave because the radially dependent phase shift introduces apositive change in the chirp (see Eq. (46)). (B) In the interval L1 to L2, there is no further




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change of the chirp of the plane wave while the chirp of Gaussian beam continues to change,due to the spatiotemporal couplings in the radially dependent term. However, these changescan be either positive or negative, as determined by the parameters of the beam itself (such asthe Rayleigh lengthzR) and the propagation distances (such as d,L1,L2) (see Eq. (47)). (C) Inthe interval L2 to L3, the chirp of the Gaussian beam increases faster than that of the planewave, because in this interval, different spectral components of the pulse show a tendency toconverge, that is to say, the spatial chirp is becoming smaller, which is the opposite of what

happens between 0 toL1(where the beam is diverging and the spatial chirp is increasing), sothe chirp introduced by radially dependent phase shift becomes negative (see Eq. (48)). Whenthe propagation distance z3 = L3 (at the entrance of the fourth element), the changes in thedifference of the chirp between the Gaussian beam and the plane wave in this interval offsetstheir difference in the intervals 0 to L1 and L1 to L2 exactly, leading to the final chirp ofGaussian beam and plane wave being the same. Actually, Eqs. (46) ~(48) also reveal this rule:

1 2 3

'' '' ''( ) ( ) ( ) 0.

PG PG PGL L L + + = (50)

Thus, for both the Gaussian beam and the plane wave, the final chirps are the same. Thefundamental reason for this difference is that the Gaussian beam has an x-coupling (or aspatiotemporal coupling) in its phase which introduces some chirp, and this term is absent inthe phase of plane wave.

If the spot size of the Gaussian beam increases, or if the Rayleigh range is longer and thebeam is better collimated, the propagation dependence of the chirp predicted by the Gaussianbeam is closer to that of the plane wave, as shown in Fig. 3.

Fig. 3. The chirp evolution of a femtosecond laser pulse when passing through the pulsecompressor. The Gaussian beam (green line) compared with a plane wave (black line). Each ofthe subfigures has different Rayleigh range: from (a) to (d), they are 1m, 3m, 5m, and 10m,

respectively. The other parameters are the same as those of Fig. 2. When the Rayleigh rangeincreases, the chirp evolution of the Gaussian beam becomes closer to that of a plane wave.

This phenomenon is also shown in Eqs. (46) ~(49) where you can see that if the Gaussianbeam is well collimated, that is to say, d, L1, L2, L3


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the Gaussian beam can be either positive (see subfigures b and c) or negative (see figure a) inthe intervalL1toL2, and depends on the parameters of the Gaussian beam and the propagationdistance.

5. Discussion

Using the Kirchhoff-Fresnel diffraction integral, we can obtain the complete electric field ofthe pulse while propagating through compressor which will tell us not only the frequency

chirp, but also the spatial chirp, pulse front tilt, spot size, and etc [68,15,16]. However, forstudying the evolution of the chirp, an analysis of the whole electric field, is not necessary, asthe chirp is directly determined by the phase shift of the laser pulse. In this paper, we used amuch simpler and more straightforward method, by analyzing the phase acquired by aGaussian pulse due to propagation through a compressor, and directly calculated the secondorder derivative of the phase with respect to the spectral frequency . This gives us the chirpevolution of the pulse when passing through a pulse compressor. We found the x-couplingterm in the phase of the Gaussian beam to be the fundamental reason for its chirp evolutioninside a pulse compressor being different from that of a plane wave.

When the spot size of the Gaussian beam increases, or when it is well collimated and has alonger Rayleigh range, the chirp acquired by the Gaussian beam is much closer to that of aplane wave, as shown in Fig. 3. The reason for this is that the radially dependent phase term ofthe Gaussian beam becomes less significant (i.e. the curvature of the wave front increases). Ifthe Rayleigh range is long enough, no radially dependent phase term exists, leading toequivalent chirps for the Gaussian beam and the plane wave models. Although the spot size ofthe Gaussian beam has no influence on the final chirp after propagation through the 4thdispersive element (assuming a well aligned compressor), it could significantly influence thevalue of the chirp when the pulse is passing through a single angular dispersion element, apair of elements in a single-pass structure, or when the pulse compressor is not perfectlyaligned. This of course affects the duration of the output laser pulse.

Another phenomenon worth noting is that for Gaussian beams, as long as spatiotemporalcouplings exist (i.e.x-cross terms), even though there is no angular dispersion, an additionalfrequency chirp will be introduced as the pulse propagates. Martinez implied such aphenomenon but provided no explanation [15]. In essence, propagation of an ultrashort pulsein the presence of spatiotemporal couplings is a three-dimensional (x, and z) effect, and itcauses the spatial terms of the Gaussian to mix with the frequency terms, and phase terms aretransferred to the intensity (and vice versa). Namely, the first dispersive element introduces

angular dispersion which propagation changes into spatial chirp, wave-front-curvaturedispersion (or pulse front tilt, if viewed in the time domain), frequency chirp, and others. Andonce the angular dispersion vanishes after the second prism, there are still coupling termsremainingthe spatial chirp and the wave-front-tilt dispersionso propagation againtransfers these terms into frequency chirp. Specificially it is the radially dependent phase termin the Gaussian model that allows for this mixing, and if this term vanishes, then thefrequency chirp of Gaussian beam will be the same as that of plane wave (such as after thepulse has passed through the fourth element, which also makes this term vanish). The spatialchirp and frequency chirp are commonly regarded as independent parameters; the spatial chirpdescribes the transverse separation (perpendicular to the propagation direction) of differentspectral components while the frequency chirp describes the longitudinal delay (along thepropagation direction). Here we again illustrate that this is not the case [2628], and we haveshown that these two quantities can be coupled by propagation.

6. Conclusion

In this paper, we studied the chirp evolution of a Gaussian beam when passing through aclassical pulse compressor by directly calculating the acquired spatio-spectral phase.Compared with the chirp evolution predicted by the plane wave model, we found that aspatiotemporal coupling or an x-dependent term in the phase of the Gaussian beam is thefundamental reason for the difference in these two models predictions for pulse compressors.




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For a Gaussian beam, the existence of spatiotemporal couplings also introduces an additionalfrequency chirp even when no angular dispersion exists. If the Gaussian beam is wellcollimated, the frequency chirp evolution is closer to that of a plane wave after passingthrough the angular dispersion elements. This work provides a deeper understanding of thephysical mechanism of the frequency chirp evolution when a laser pulse passes through aclassical pulse compressor. Our analysis will also be helpful for optimization of dispersioncompensation schemes in many applications of femtosecond laser pulses.

Acknowledgments

This work was supported by the National Natural Science Foundation (NSFC) (30900331,30927001), and Program for Changjiang Scholars and Innovative Research Team inUniversity.



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