388 J. Opt. Soc. Am. B/Vol. 18, No. 3 /March 2001 T. Tomaru and H. Petek

Effect of third-order dispersion on the phases ofsolitonlike Cr41:YAG-laser pulses

characterized by the second-harmonic generationfrequency-resolved optical gating method

Tatsuya Tomaru and Hrvoje Petek*

Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama 350-0395, Japan

Received May 24, 2000; revised manuscript received October 23, 2000

The effect of third-order dispersion on solitonlike operation of the L-fold cavity Cr41:YAG laser is studied.Dispersion equations are derived to the fourth order for an L-fold cavity, and dispersion values for a Cr41:YAGlaser are calculated. Third-order dispersion is found to be comparable with the second-order dispersion in a60-fs-class Cr41:YAG laser, and fourth-order dispersion is negligible. The temporal and spectral phases of a60-fs pulse are calculated from the dispersion properties of the cavity. The spectral phase is constant, despiteconsiderable third-order dispersion. A second-harmonic generation (SHG) frequency-resolved optical gating(FROG) trace is calculated for the simulated pulse. The SHG spectrum has a tail to the short-wavelength sideat zero delay, consistent with experimental results. The calculated phase and the FROG trace should be rec-ognized as characteristic of the third-order-dispersion effect on solitonlike lasers. © 2001 Optical Society ofAmerica

OCIS codes: 320.7090, 320.7100, 320.1590, 140.3580, 190.5530.

1. INTRODUCTIONThe understanding of how high-order (third- and higher-order) dispersion affects the generation and propagationof femtosecond laser pulses has been the key to optimiz-ing the operation of Kerr-lens mode-locked lasers. Themost important concern has been the limit of pulse short-ening imposed by high-order dispersion. Consequently,generation of ,10-fs pulses has been made possible by in-vestigation of dispersion compensation involving prismsof different optical materials and chirped mirrors of vari-ous designs.1,2 Furthermore, high-order dispersion notonly restricts the pulse width but also can result in pulsesplitting, spectrum splitting, and sideband generation.3–8

In addition, the pulse phase is also affected by high-order dispersion. Haus et al. developed an analyticaltheory for evaluating the effect of high-order dispersionon phase.3 Hermann et al.9 and Kalosha et al.10 used nu-merical calculations to investigate the effect of high-orderdispersion on phase. Moreover, Penman et al. experi-mentally studied the influence of intracavity dispersionon pulse phase for the Ti:Al2O3 laser.11 However, our in-tuitive understanding of high-order-dispersion effects isnot always correct. For example, it is intuitive to antici-pate that the third-order dispersion (TOD) will appear asa third-order function of the spectral phase (see, e.g., Fig.2 of Ref. 12), as would be expected for linear propagationof a pulse through dispersive media. However, becausethe phase of a femtosecond-laser pulse is determined by adynamic balance of self-phase modulation (SPM), group-delay dispersion (GDD), and TOD, the spectral phase can-not be a simple third-order function for nonlinear propa-gation in a cavity. Although recent improvements inpulse-characterizing methods such as frequency-resolved

0740-3224/2001/030388-06$15.00 ©

optical gating12 (FROG) provide a general means of ex-tracting the pulse phase, if we do not know a typicalphase of laser pulse affected by TOD it is difficult to in-terpret what factors are responsible for the observedphase. It is our purpose in this paper, therefore, to showa typical pulse phase affected by TOD, based on thesimple perturbation theory of Haus et al.3 As an ex-ample, we show calculated results of using parameters ofan optimized L-fold cavity Cr41:YAG laser and comparethem with our recent experimental results.13 As second-harmonic-generation (SHG) FROG is the most commonlyused method of evaluating phase, we compare the calcu-lated and experimental SHG FROG traces for the L-foldcavity with TOD.

Optimization of the Cr41:YAG laser operation is of in-terest for applications in optical communications becauseof the laser’s operating wavelength of 1.5 mm. The L-foldcavity offers a simple and compact design for attainingthe high repetition rate required for such applications.However, a dispersion formula for an L-fold cavity has sofar been reported only up to second order.14 Thereforeour second purpose in this paper is to derive the disper-sion formula up to fourth order. In the 1.5-mm region,TOD limits the pulses from the Cr41:YAG laser to ;60-fsduration. TOD cannot be simply eliminated through ajudicious choice of materials, because standard opticalglasses generally have TOD of the same sign as theCr41:YAG crystal.15 By contrast, compensation of theTOD in a Ti:Al2O3 laser at 850 nm by a fused-silica prismpair makes possible the generation of pulses as short as10 fs because there is little restriction by TOD.16,17 Inwhat follows, we first derive the dispersion formula tofourth order for an L-fold cavity and then show the impor-

2001 Optical Society of America

T. Tomaru and H. Petek Vol. 18, No. 3 /March 2001/J. Opt. Soc. Am. B 389

tance of TOD in the case of a Cr41:YAG laser. We calcu-late phase and FROG traces and compare them with ex-perimental results. The experimental results are wellexplained by the effects of third-order dispersion on non-linear propagation. Because a similar solitonlike pulseformation is likely to operate in other femtosecond lasersystems, the calculated phase and the SHG FROG traceare important as primary diagnostics for the TOD effect.

2. DISPERSION EQUATIONSCalculations of dispersion equations for an L-fold cavityare given to second order in Ref. 14. Here we calculatethem to the fourth order by following the same approach.We define P(l) as the wavelength-dependent intracavityoptical path length of the propagation axes. Then thesecond-, third-, and forth-order dispersions, respectively,are described by

d2f

dv2 5l3

2pc2

d2P

dl2 , (1)

d3f

dv3 5 2l4

4p2c3 S ld3P

dl3 1 3d2P

dl2 D , (2)

d4f

dv4 5l5

8p3c4 S l2d4P

dl4 1 8ld3P

dl3 1 12d2P

dl2 D , (3)

where c is the speed of light.18,19 Figure 1(a) shows anequivalent L-fold cavity, where for clarity the folding mir-ror is replaced with a lens.14 Planes of the gain mediumand the Littrow prism are cut at the Brewster angle.When this cavity operates in a femtosecond mode-locked

Fig. 1. (a) Equivalent L-fold cavity with a folding curved mirrorreplaced by a biconvex lens. The cavity is divided into two partsfor the calculation of dispersion. (b) Magnification of Part 2 of (a).

regime, all wavelengths that contribute to the pulse haveto satisfy the resonator condition. The individual wave-lengths that compose the pulse determine the refractionangles at the Brewster planes and, therefore, the propa-gation axes in the cavity. The dashed line in Fig. 1(a)represents a propagation axis for a wavelength lc passingthrough the center of the lens. The solid line representsthe propagation axis for other wavelengths. Point O rep-resents the intersection point of the solid and dashedlines, O8 its virtual image, and C the center of the lens.The law of ray optics requires that

1/f 5 21/CO8 1 1/CO, (4)

where f is a focal length of the lens. Focal length f de-pends slightly on the position of the ray that passesthrough the lens owing to spherical aberration. How-ever, because we calculate only dispersions at l → lc inwhat follows, f can be approximated to be constant. Thisapproximation is justified by the calculated dispersions inSection 3 below, which show negligible dependence on thefocal length. Using ug and up as defined in Fig. 1(a), wecan rewrite Eq. (4) as

CO8 5 f ~tan up /tan ug 2 1 !, (4a)

CO 5 f ~1 2 tan ug /tan up!. (4b)

As liml→lcug 5 liml→lc

up 5 0, we can show that

liml→lc

up

ug5 lim

l→lc

dnup /dln

dnug /dln , (5)

liml→lc

dn~up /ug!

dln 5 0, (6)

where n is a natural number, by repeated use ofL’hospital’s theorem. Therefore

liml→lc

dn CO

dln 5 liml→lc

dnCO8

dln 5 0 (7)

is satisfied. Equation (7) signifies that O and O8 are con-stant points in the calculation of Eqs. (1)–(3) in the limitof l → lc . We therefore divide the cavity into two partsbased on points O and O8, i.e., Part 1 and Part 2 in Fig.1(a); this allows us to calculate the cavity dispersion foreach independently. Parts 1 and 2 appear to overlap be-tween O and O8, but such is not the case because P(l) isalways differentiated by l in dispersion calculation. Thegeometrical method for calculating P(l) is similar to thatof Ref. 20. First we calculate the path OD1C1 in Part 2of Fig. 1(a). When we define l and b as shown in Fig.1(b), path OD1C1 is written as

POC1 5 OD1 1 np D1C1 5 l cos b, (8)

where np is the refractive index of the Littrow prism. Asl is constant in the limit l → lc , POC1 is a function onlyof b. The derivative of POC1 , therefore, is calculatedthrough the relation

dPOC1~lc!

dl5

dPOC1~lc!

db

db~lc!

dnp

dnp~lc!

dl. (9)

390 J. Opt. Soc. Am. B/Vol. 18, No. 3 /March 2001 T. Tomaru and H. Petek

The higher derivatives are similarly calculated.20 FromEq. (8) and referring to Fig. 1(b), we obtain

dPOC1~lc!

db5 2

d3POC1~lc!

db3

5 2l sin b~lc!

5 2D0C0

cos wp~lc!

sin a, (10)

d2POC1~lc!

db2 5 2d4POC1~lc!

db 3

5 2l cos b~lc!

5 2~OD0 1 npD0C0! (11)

for l 5 lc , where a and wp are defined in Fig. 1(b) and C1and D1 are replaced by C0 and D0 because l 5 lc .Angle wp is given by Snell’s law, wp 5 sin21(np sin a). Wedefine u 5 wp 2 a, and then the deviation du is equal todwp because a is constant. Moreover, dfp is equal to 2dbbecause point O is fixed and a is constant [Fig. 1(b)].Therefore

2db

dnp5

du

dnp5 @~sin a!22 2 np

2#1/2, (12)

2d2b

dnp2 5

d2u

dnp2 5 npS du

dnpD 3

, (13)

2d3b

dnp3 5

d3u

dnp3 5 S du

dnpD 3

1 3np2S du

dnpD 5

,

(14)

2d4b

dnp4 5

d4u

dnp4 5 9npS du

dnpD 5

1 15np3S du

dnpD 7

(15)

are obtained. Refractive index np(l) and its derivativesare calculated from Sellmeier’s equation.21 Substitutionof Eqs. (10)–(15) into Eq. (9) and its derivatives yieldsgeneral equations for calculating dispersions that aresomewhat cumbersome. When the Brewster conditiontan wp 5 np is applied, the equations are simplified. Theabove equations (1)–(13) have been calculated generally

for an arbitrary angle a. The simplified equations for theBrewster condition are given for the one-way dispersionsof Part 2 (Fig. 1) by

d2POC1~lc!

dl2 5d2np

dl2 C0D0 2 S dnp

dlD 2

OD0 , (16)

d3POC1~lc!

dl3 5d3n p

dl3 C0D0

2 3FnpS dnp

dlD 3

1dnp

dl

d2np

dl2 GOD0 , (17)

d4POC1~lc!

dl4 5d4np

dl4 C0D0

2 3F ~1 1 5np2!S dnp

dlD 4

1 6npS dnp

dlD 2 d2np

dl2 1 S d2np

dl2 D 2

14

3

dnp

dl

d3np

dl3 GOD0 . (18)

The one-way dispersions of Part 1 (Fig. 1) are calculatedaccording to the same procedures and produce the sameresults as Eqs. (16)–(18) if POC1 , C0D0 , OD0 , and npare replaced by POA1 , A0B0 , O8B0 , and refractive indexng , respectively, of the gain medium. DistancesOD0 and O8B0 are calculated from Eqs. (4a) and (4b)by use of the relation liml→lc

(tan up /tan ug)5 liml→lc

(up /ug) 5 (dnp /dl)/(dng /dl).

3. RESULTSTable 1 lists the calculated cavity round-trip dispersions.The parameters are those for the optimized Cr41:YAG la-ser described in Ref. 13. The Littrow prism material islow-OH2 fused silica, for which np(l) was evaluated bySellmeir’s equation given in Ref. 21. For ng(l) we usedthe refractive indices of undoped YAG crystal,22 and weevaluated ng(l) by fitting it to the expression ng

2 5 B01 B1 /(l2 1 C1) 1 B2l2, where B0 5 3.29615, B1

Table 1. Cavity Round-Trip Dispersiona

Calculated Part

Dispersion

f9( fs2) f-( fs3) f-8( fs4)

Total cavity (sagittal) 2381 8,614 223,860Total cavity (tangential) 2381 8,611 223,852YAG 252 5,326 212,122Fused silica 2489 2,802 28,842

f9/t2 f-/t3 f-8/t4

Total cavity (sagittal) 20.339 0.229 20.0189Total cavity (tangential) 20.340 0.229 20.0189

a The cavity parameters are as follows: A0B0 5 18 mm; B0C5 10.5 mm; CD0 5 67.4 mm; D0C0 5 10 mm; f 5 18.1 and f 5 12.4 mm for sagittal andtangential planes, respectively (the radius of curvature of the folding mirror is 30 mm and the incident and reflected angles are 34°); l 5 1.52 mm;t 5 33.5 fs (corresponding to a full width at half-maximum of 59 fs for sech2 shape); the repetition rate is 1.2 GHz.

T. Tomaru and H. Petek Vol. 18, No. 3 /March 2001/J. Opt. Soc. Am. B 391

Fig. 2. (a1) Temporal phase of the soliton pulse with TOD simulated (sim.) from Eq. (20). The pulse envelope’s shape is assumed to besech2(t/t). The sign of phase is defined such that a positive second derivative with respect to time implies positive dispersion. Intensityis normalized to unity, indicated as ‘‘Norm.’’ (a2) The Fourier transform of (a1) gives a flat phase, but the spectral intensity has a tailat the short-wavelength side. (a3) The calculated FROG trace from the pulse in (a1). The spectrum at zero delay is elongated to theshort-wavelength side. (b1) Pulse envelope and phase in the time domain retrieved from experimental (Exp.) FROG data for the opti-mized L-fold-cavity Cr41:YAG laser. The qualitative shape of the phase curve is the same as in (a1). Phase behavior outside of 2150–150-fs delay has no significance because of low intensity. (b2) Fourier transform of (b1). The phase is almost flat, and the intensitycurve has a tail at the short-wavelength side as in (a2). The extra peaks at 1575 and 1610 nm are discussed by Ishida et al.23 Thephase behavior for large delays has no significance, as in (b1). (b3) The measured FROG trace to produce (b1) and (b2). The elongatedspectrum to the short-wavelength side at zero delay is similar to that in (a3).

5 0.0252, B2 5 20.0124, and C1 5 20.02311 in units ofmicrometers. Table 1 shows the round-trip dispersionsindependently for sagittal and tangential planes to indi-cate the effect of astigmatism, which apparently is not im-portant for dispersion. In other words, dispersion is es-sentially independent of the focal length of the lens. Thisresult supports the approximation in Eq. (4). For com-parison, we also give material dispersions of the YAGcrystal and the Littrow prism.

To investigate quantitatively the effect of the third- orhigher-order-dispersion on the operation of the laserwould require a numerical simulation. However, the es-sential effect of the dispersion on the phase can be ascer-tained from an analytical equation for the phase of asech2 pulse derived from the first-order perturbationtheory,3 which is convenient for comparing the effects ofvarious parameters. According to Ref. 3, the temporalphase w(t) of a pulse is given by

2f9

t2 tdw

d t1

f-

3!t 3 F1 2 6 sech2S t

tD G 1

DT

t5 0,

(19)

where f9 5 d2f/dv2, f- 5 d3f/dv3, t is the pulse-widthparameter that appears in the sech2(t/t) pulse shape, andDT is an adjustable parameter, which could be set to zeroif coordinates moved with the pulse. When we select w5 0 for t 5 0, then for DT 5 0

w 5f-

6f9t2 F t 2 6t tanhS t

tD G . (20)

Because f9 and f-, respectively, appear as f9/t 2 andf-/t 3 in Eq. (19), the relative importance of f9 and f- isestimated from f9/t 2 and f-/t 3. Similarly, f99/t 4 willbe a good measure of f99. These values also are shown inTable 1. The values of f9/t 2 and f-/t 3 are comparable,but f99/t4 is an order of magnitude less. Therefore,third-order dispersion has a significant effect on the op-eration of a 60-fs-class Cr41:YAG laser, whereas fourth-order dispersion is insignificant.

Figure 2(a1) shows pulse phase estimated from Eq. (20)with the parameters for our laser. It shows curvature ofthe opposite sign at the pulse edges, and an inflectionpoint is located at the pulse center. Its Fourier trans-

392 J. Opt. Soc. Am. B/Vol. 18, No. 3 /March 2001 T. Tomaru and H. Petek

form is plotted in Fig. 2(a2) with respect to the wave-length. The spectral phase is flat in spite of the presenceof second- and third-order dispersion. The effect of cur-vature in the temporal phase translates to an asymmetryin the spectral intensity that is characterized by a tail atthe short-wavelength side of the spectrum. The startingpoint for deriving Eq. (20) is the fundamental soliton so-lution of nonlinear Schrodinger equation, for which tem-poral and spectral phases are flat even if second-order dis-persion exists, because the second-order dispersion isexactly balanced by SPM. When third-order dispersionis included as a perturbation, the temporal phase and thespectral intensity are distorted, while the spectral phaseremains flat. The tail on the short-wavelength side ofspectrum can be qualitatively understood from the argu-ments of Rundquist et al.17 Figure 3 plots relative groupdelay ]w/]v 2 f8 5 f9Dv 1 f-Dv2/2 1 ... versuswavelength. The dashed curve shows the relative groupdelay in the presence of only GDD, whereas the solidcurve also includes the contribution from TOD. Negativedispersion pushes shorter-wavelength components to thefront of a pulse, and SPM causes a redshift in the frontpart. The effect is reversed for the longer-wavelengthcomponents. The balance between negative GDD andSPM leads to a stable soliton pulse. When TOD is in-cluded, u]w/]v 2 f8u on the short-wavelength side andthe redshift by SPM become small, whereas on the long-wavelength side the opposite effect exists and results in alarge blueshift. Because the blueshift is dominant, theresultant pulse has a tail at the short-wavelength side ofthe spectrum. The flat spectral phase is a result of thespectral energy transfer described above; in other words,dispersions including TOD are balanced by SPM.

Figure 2(a3) shows a calculated FROG trace for the cal-culated pulse in Fig. 2(a1); the trace shows a characteris-tic triangular shape. The predicted characteristics of thepulse based on the cavity dispersion analysis in Figs.2(a1)–2(a3) are all qualitatively observed for the opti-mized L-fold cavity laser, as shown in Figs. 2(b1)–2(b3).13

The configuration of FROG measurement is the same as

Fig. 3. Relative group delay ]w/]v 2 f8 5 f9Dv(1f-Dv2/2)plotted with respect to wavelength. Negative dispersion causesan advance for the short-wavelength components of a pulse and adelay for the long-wavelength components. The front part of apulse is redshifted by SPM, and the back part is blueshifted. Inthe presence of GDD only, the red and blue shifts are balanced,but the addition of TOD breaks the balance, resulting in a tail atthe short-wavelength side of the spectrum.

described in Ref. 13, and a retrieval algorithm is that ofKane et al.24 Although there is good qualitative agree-ment between the observed and the simulated pulses inFig. 2, the agreement is not perfect because the simula-tion is based on the first-order perturbation theory for thephase evolution. Other experimental characteristicsthat are not reproduced by the calculation, such as ashoulder at the pulse’s trailing end [Fig. 2(b1)] (Ref. 3)and extra spectral peaks at 1575 and 1610 nm [Fig. 2(b2)](Ref. 23), probably also originate from the third-order dis-persion. However, to reproduce these details would re-quire either numerical simulation or a perturbationtheory treatment of the amplitude.8,25

Because third-order dispersion is commonly present infemtosecond lasers with soliton pulse-shaping mecha-nisms, the present results should be of broad interest. Inparticular, as FROG measurements are commonly usedfor pulse characterization, the specific features of the cal-culated FROG trace and the phase curves in Fig. 2 shouldbe kept in mind as a typical example of distortion of apulse by the TOD. Such pulse characteristics shouldcommonly appear as a consequence of the third-order dis-persion in transmission of solitonlike pulses that are com-monly encountered in fiber optics.26

4. PROPOSAL FOR REDUCING THIRD-ORDER DISPERSIONThe Cr41:YAG laser described here has third-order dis-persion comparable with GDD. To get a narrower pulsewidth, eliminating TOD is essential. The most elegantmethod is to use a chirped mirror coating to eliminateTOD for a standard high-reflection coating.1,2 However,there are some alternatives for prism materials, whichmay lead to shorter-pulse operation of a Cr41:YAG laser.One is to use high-refractive-index material, e.g., SF10 orSF57 glass, for the Littrow prism. These glasses, how-ever, have positive material dispersion near 1.5 mm,27 andtherefore the distance between the gain medium and theLittrow prism has to be sufficiently long to yield negativedispersion. For an 18-mm-long Cr41:YAG crystal, a 0.5-GHz cavity is possible, with a round-trip GDD of 2400 fs2

[the typical GDD for stable mode locking of our Cr41:YAGlaser; see Table 1 (Ref. 13)] and a TOD of 6000 fs3 forSF10 or SF57 (compared with 8000 fs3 for fused silica).Another option is to use higher-refractive-index materialsuch as ZnSe, which would permit 1.2-GHz operationwith a round-trip GDD of 2400 fs2 and a TOD of 1800fs3.28 However, unlike fused silica, these materials arenot scalable to higher-repetition-rate operation.

5. SUMMARYWe have derived dispersion equations to the fourth orderfor an L-fold cavity and calculated the dispersions of aCr41:YAG laser. Using these values, we calculated tem-poral and spectral phases based on a perturbed nonlinearSchrodinger equation. The temporal phase of the pulsehas a characteristic feature. The spectral phase is con-stant in spite of the presence of third-order dispersion,whereas the spectrum has a tail at the short-wavelengthside. These are distinct characteristics of soliton solution

T. Tomaru and H. Petek Vol. 18, No. 3 /March 2001/J. Opt. Soc. Am. B 393

perturbed by TOD. These distinct features lead to acharacteristic SHG FROG trace. The calculated FROGtrace is in agreement with the output of the L-fold cavity60-fs-class Cr41:YAG laser. To get narrower pulsewidths, reducing TOD is essential. One can do this bycompensating for TOD with positively dispersive prismsmade from SF10, SF57, or ZnSe, at the cost of limiting theultimate repetition rate of the laser. Multi gigahertz-repetition-rate operation of the Cr41:YAG laser should bepossible with GDD compensation by fused silica and TODcompensation by chirped mirrors.

T. Tomarv’s e-mail: address is tomaru@harl.hitachi.co.jp.

*Present address, Department of Physics and As-tronomy, University of Pittsburgh, Pittsburgh, Pennsyl-vania 15260.

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28. Calculated by use of refractive indices of ZnSe for 0.54–10.6-mm wavelengths from the optics catalog of II–VI, Inc.,and evaluated by fitting of np(l) to the expression np

2

5 B0 1 B1 /(l2 1 C1) 1 B2l2, where B0 5 5.92833, B15 0.24165, B2 5 20.00139, and C1 5 20.09441 in units ofmicrometers.