1704 OPTICS LETTERS / Vol. 16, No. 21 / November 1, 1991

General compressor for ultrashort pulses with nonlinear chirp

Juan M. Simon, Silvia A. Ledesma, and Claudio C. Iemmi

Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,Ciudad Universitaria, Buenos Aires, Argentina

Oscar Eduardo Martinez

Departamento de Fisica, Comision Nacional de Energia Atomica, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina

Received June 27, 1991

A new type of grating compressor capable of compensating, in principle, for any arbitrary group-delay disper-sion is described. The system relies on a specially designed grating with a variable groove spacing along itssurface as well as a variable angle of incidence. In this manner each wavelength actually sees a different grat-ing, always at Littrow incidence, that gives rise to distinct group delays. The design equations are derived, andthe particular case of a cubic compressor is discussed. It is shown that, within practical limits, both quadraticand cubic terms can be simultaneously compensated for in typical optical fibers.

Grating pulse compressors' are routine tools forchirp compensation of ultrashort light pulses. Typ-ical applications that require large compression fac-tors are optical-fiber dispersion compensation'-4 andchirped pulse amplification.5 6 When large com-pression factors are needed it has long been estab-lished that, besides the linear chirp (quadratic termin the frequency dependence of the phase), higher-order terms are present in the pulse as a result offiber dispersion6'7 or appear because of the compres-sor.6 For short pulses such higher-order terms canbe compensated for by prism compressors 8 or combi-nations of prisms and gratings.9

As Frenkel et al.4 showed, once the quadratic termhas been compensated for, the higher-order termsstill set a limit on system performance. In this Let-ter we describe a new type of grating compressorthat can be designed to compensate in principle forany arbitrary functional dependence of the phasewith frequency. The method relies on the use ofspecially designed nonflat gratings with unhomo-geneous groove spacing.

The system proposed is shown schematically inFig. 1. The first grating, G1, introduces angulardispersion; the parabolic mirror is located such thatthe beams corresponding to the different frequen-cies are reflected parallel to one another and arefocused (to first order) onto the retroreflecting grat-ing, G2. In this configuration the optical paths forthe different frequencies before grating G2 are iden-tical over a plane perpendicular to the beams, andhence it is the second grating shape that is used tointroduce the desired frequency dependence of thegroup delay. As we show, the groove spacing mustbe adjusted to meet the retroreflection condition.

To derive the grating design, we follow Treacy'sformulation' for the grating phase shift. Once thefirst grating and the mirror are set, there will be aunique relation between the transverse coordinate xand the beam center for the wavelength A. The

shape of grating G2 can then be characterized by thesurface g(x). There is a one-to-one correspondencebetween the frequency w and the coordinate x atwhich the center of a beam of frequency w hits thesurface of the grating. Hence the grating surfacecan be written in terms of the frequency as k(w)or of the wavelength as g(A). In general, given anyfunction f(x), the correspqnding functions of wand A are denoted f(w) and f(A), respectively.

The diffracted wave at order m after a grating isobtained by assuming that there is a phase shift217rm between consecutive grooves. That is, an ad-ditional phase shift 2IrrmN, where N is the numberof grooves from a reference point chosen on thegrating, is assigned. The phase shift after an opti-cal path 1 and reflection on the grating is

4:(A) = 27rmN + 2wrl/A. (1)

Optical path 1 can be expressed in terms of thesurface equation g(A), and Eq. (1) becomes

¢$(A) =- A [lo - 2g(A)] + 27rN(A), (2)

where lo is the optical path to the plane R(A) = 0.N(A) can be obtained in terms of g(A) from theretroreflection condition. This condition is thatthe optical path difference 2Ag(A) between AN suc-cessive grooves must compensate for the additionalphase shift from the grating, that is, that

- 2 Ag(A) (3)

m A

so that

dN 2 dg(A) (4)dA mA dA

or, integrating,

N(A) [ J mt2 dg(X) dR(A= f~r~ dX dA' + N(Ao). (5)

0146-9592/91/211704-03$5.00/0 © 1991 Optical Society of America

November 1, 1991 / Vol. 16, No. 21 / OPTICS LETTERS 1705

.xo

Fig. 1. Schematic drawing of the proposed setup. Thebeam is dispersed by the first grating, G1, at the focus ofmirror M. The mirror sends the different frequenciesparallel to one another and focused onto grating G2. G2has a groove spacing designed in such a manner that itreflects each color back over its path (Littrow incidence)with a time delay given by the shape of the grating sur-face. The dashed lines (R's) show the path of the centralfrequency wo, which hits the grating at coordinate x0 .

We define

(6a)

mirror, which can be selected beforehand. In whatfollows we present a particular design for a cubiccompressor.

Let us assume that we wish to compensate for thedispersion from a single-mode fiber of length z witha phase shift per unit length ,8(w) and that thegroup delay is to be compensated for to second orderin powers of (w - wo), where w0 is the central fre-quency of the pulse. The group delay introduced bythe fiber is approximated by

T = dDf/dw = z/o' + zf3'"(w - wo)

+ 2 z3o"'(w - w) 2 , (11)

where Of is the phase at the fiber exit.Hence, from Eq. (8), to compensate for the fiber

dispersion g(w) must satisfy

g(w) = -1 + - czGo' + 2cz,30"(w - wo)2 2

+ +cz,3o0"(w - wo). (12)

The grating surface can be expressed in terms ofthe coordinate x as g(x) by means of the Taylor seriesexpansion of w(x) around x = xO, with

9(w) = g(27rc/w), (6b)

N(w) = N(2irc/w). (6c)

From Eqs. (2) and (5):

1)(w) = [lo - 2g(w)]

+ - w drw dw' + 27rmNI(wo). (7)C d

Equation (7) gives the phase term introduced bythe compensator. It can be used to obtain the de-sign of the grating needed to compensate for a givenphase term. As the constant phase term is irrele-vant, one actually needs to compensate for the groupdelay, defined as the derivative of the phase with re-spect to the frequency. The group delay X intro-duced by the system is

. = d=[ 1'r= -= -[lo - 2R(w)],dw c

xo = RAo/d, (13)

where R is the focal length of the mirror and d is thegroove spacing of the first grating:

w(x) = Wo [1-( xo) + (x - Xo)2]xO xO(

Using Eq.. (14) in Eq. (12) and keeping terms up tosecond order in powers of (x - xo), we obtain

g(x) = go + a(x - xo) + b(x - xo)',

where

a=-czfpo w02xo

czwo 0(3...wo2X02 2 + 3off).

1 1,go= - -10 +- -czf0

2 2(8)

which is the optical path divided by the speed oflight. The same result could have been obtainedby following the derivation presented by Brorsonand Haus.10

If the incoming pulse has a group delay r(w), thegrating design R(w) is obtained from Eq. (8) and is

d&(w) _ 2 dg(w)dw c dw '

the groove spacing, from Eqs. (4) and (9), is

dN(w)dw

w d&(w)

27rm dw

The design is completed after we set the rtion w(x) that will depend on grating G1 and

(15)

(16)

(17)

(18)

is a constant time delay, for which it is not necessaryto compensate to recover the pulse shape. The tworelevant parameters are a and b.

The grating groove spacing can then be obtainedfrom Eqs. (8), (10), and (15), which yields

dN = 2 dg = 2 [a + b(x - xo)]. (19)

(9) We now give several examples to get an insightinto the order of magnitude of the parameters in-volved. For the fiber we use the dispersion curveused by Marcuse,6 characterized by the dimension-

(10) less parameters

ffi-e_ D = 03o"z/(2T2),

.1

(D(w) = (D(27rc/w),

(14)

.Ulu-

Lthe B = Bo ... zl(6T'),

1706 OPTICS LETTERS / Vol. 16, No. 21 / November 1, 1991

Table 1. Values of the Parameters of Grating G2for a Cubic Compressor at Three Different Wavelengths'

f (I/mm) ro (cm)A0(,m) D B a b (i/m) a (tan a = a) (dN/dx/cos a) [1/(2b cos3 a)]

1.2 15 0.08 -2.45 24.8 -67.8° 1500 37.31.275 0 0.1 0 21 0 0 2.41.4 -23 0.14 2.76 11.3 700 1350 11.2

aThe compressor is designed to compensate for 1 km of a typical single-mode fiber.6 D and Bare the fiber parameters, a and b define the grating surface [Eq. (15)], a is the angle of incidenceat the central frequency (Fig. 1), f is the groove frequency measured along the surface, and ro is thegrating's radius of curvature.

where 2T is the pulse width. We use z = 1 km and2T = 1 ps for our examples, and we analyze threedifferent cases: with the central wavelength belowzero dispersion (A = 1.2 gtm), at the zero-dispersionpoint (A = 1.275 Am), and above that point (A =1.4 Am). The results are shown in Table 1, where Dand B are obtained from Ref. 6, a and b are fromEqs. (16) and (17), ro is the curvature of gratingG2, a is the angle of incidence, and f is the groovefrequency measured along the grating surface, allevaluated at the grating center x0.

As can be seen from the examples presented, b isnot strongly dependent on the wavelength, but theradius of curvature of the grating is sensitive to theangle a, and larger second-order terms (larger valuesof b) yield larger angles and hence larger radii ofcurvature. Hence it is easier to compensate forthird-order terms with this design if second-orderterms are also to be compensated for. The practicallimits are set by the small radius of curvature whenthe wavelength is set too close to the zero-dispersionpoint on one side and by the angle too close to graz-ing incidence that is required when the wavelengthis set too far from the mentioned point. For ex-ample, if a practical limit of a = 800 is set, we cancompensate for 2 km of fiber at A = 1.4 Aum, 10 kmat A = 1.3 ,m, and 50 km at A = 1.28 ,m. In theexamples given the gratings can be constructedholographically. We can include computer-generatedholograms in the setup to obtain the required groovespacing distribution.

We have described a new type of grating compres-sor capable of compensating, in principle, for any ar-bitrary group-delay dispersion. The system relieson the use of a specially designed grating with a vari-able groove spacing along its surface as well as avariable angle of incidence. In this manner eachwavelength actually sees a different grating, alwaysat Littrow incidence, that gives rise to distinct group

delays. The design equations were derived, and theparticular case of a cubic compressor was discussed.It was shown that, within practical limits, bothquadratic and cubic terms can be simultaneouslycompensated for in typical optical fibers. Thescheme does not provide a way to adjust the phasecontinuously once it is constructed, so that, aspresented, it can be conveniently used only withfixed systems. Other systems with fixed large non-linearities in the chirp, such as chirp pulse am-plification schemes, can be compensated for inthis manner.

The authors acknowledge partial support from theConsejo Nacional de Investigaciones Cientificas yTecnicas de la Republica Argentina through grantsPID3-127800/88 and PID-00333/88.

References

1. E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454(1969).

2. 0. E. Martinez, IEEE J. Quantum Electron. QE-23,59 (1987).

3. A. Frenkel, J. P. Heritage, and 0. E. Martinez, in Di-gest of Conference on Lasers and Electro-Optics (Op-tical Society of America, Washington, D.C., 1988),paper TUP4.

4. A. Frenkel, J. P. Heritage, and M. Stern, IEEE J.Quantum Electron. 25, 1981 (1989).

5. 0. E. Martinez. IEEE J. Quantum Electron. QE-23,1385 (1987).

6. D. Marcuse, Appl. Opt. 19, 1653 (1980).7. W J. Tomlinson, R. H. Stolen, and C. V Shank, J. Opt.

Soc. Am. B 1, 139 (1984).8. 0. E. Martinez, J. P. Gordon, and R. L. Fork, J. Opt.

Soc. Am. A 1, 1003 (1984).9. R. L. Fork, C. H. Brito Cruz, P. C. Becker, and C. V

Shank, Opt. Lett. 12, 483 (1987).10. S. D. Brorson and H. A. Haus, J. Opt. Soc. Am. B 5,

247 (1988).