1. INTRODUCTIONSecond-order cascaded effects are receiving increasing at-tention for use in the development of all-optical switchingdevices and passive optical modulators for laser modelocking. These processes can induce nonlinear phase oramplitude modulation orders of magnitude stronger thanthe Kerr effect, and they have the further advantage ofadjustable magnitude and sign.1–5
The second-order cascaded interaction of ultrashortlight pulses is strongly affected by the group delay amongthe different interacting frequencies and polarizationsowing to their group-velocity mismatch (GVM) (nonsta-tionary interaction). As an example, for the commonlyemployed nonlinear crystals the nonstationary conditionis approached when the pulse duration is hundreds offemtoseconds over an interaction length of few millime-ters. Until now only a few experimental investigations ofthe dynamics of the phase- and amplitude-modulationprocesses in a nonstationary regime have been discussedin the literature.4,6–8 Analytical expressions for the tem-poral solitary waves propagated in a second-order nonlin-ear medium in the presence of GVM and group-velocitydispersion have been reported.9 Several authors1,7,10
have investigated the nonlinear self- and cross modula-tions (in phase and amplitude) experienced by interactingpulses of arbitrary input envelopes by means of the nu-merical integration of the propagation equations. Thismethod, however, results in some loss of informationabout the influence of the various parameters on the pro-cess that is being tested.
In the past few years, second-order cascaded processeswere generally investigated for the passive optical-pulsemodulation to realize devices that exhibit a saturable ab-sorber action for laser mode locking and for all-opticalswitching (both in bulk materials and in waveguide struc-tures). Until now the passive mode locking with second-
order cascaded nonlinearities has been experimentallyachieved by means of two different approaches based onthe self-amplitude modulation (nonlinear mirror11,12) andon the self-phase modulation of the fundamental field (af-fecting the resonator mode structure, such as for the cas-caded second-order mode locking13,14). Theoretical15,16,17
and experimental14 results suggest that the GVM be-tween the fundamental and second harmonic in the non-linear crystal plays a key role on the dynamic of the lasersource and therefore contributes to determining theshortest achievable pulse duration.
The aim of this paper is to provide an analysis of thetype I [a single fundamental (F) field at the frequency v1and a second-harmonic (SH) field at the frequency v25 2v1] cascaded second-order interaction between planewaves. This analysis uses a perturbative approach tocalculate the effects of the group delay on the process dy-namic, and it provides analytical expressions that outlinethe nonlinear phase and amplitude modulation of the Fpulse in both single- and double-pass geometries.
This analysis clarifies the dynamic behavior of passiveoptical devices for laser mode locking. As an example,we found that, in a double-pass configuration, a properchoice of the propagation parameters (e.g., the crystallength, the phase-velocity mismatch, and the pulses’ de-lay adjustment) allows us to obtain an efficient modula-tion of the F pulse even in nonstationary interaction.
In this paper we ignored the effects related to thepropagation of finite aperture beams, such as diffractionand beam lateral walk-off. Diffraction plays a key role inthe interaction of focused beams, but it represents a veryheavy complication even with a numerical calculation.On the other hand the plane-wave picture gives at least atemporally correct picture of the process evolution. Ow-ing to similar considerations, we also ignored the beamlateral walk-off that is due to the crystal birefringence.
1998 Optical Society of America
104 J. Opt. Soc. Am. B/Vol. 15, No. 1 /January 1998 Toci et al.
In the conclusions we will report a more detailed discus-sion related to these approximations.
2. ANALYTICAL PERTURBATIVESOLUTIONUnder the plane-wave picture, we assume two fields oscil-lating at v1 , v2 5 2v1 with wave numbers k1 , k2 of theform E1,2(z, t) 5 E1,2(z, t)exp@i(k1,2 z 2 v1,2t)#, whereE1,2(z, t) are the pulses’ envelopes, propagating along thez direction in the nonlinear medium of length L. In theslowly varying envelope approximation their time-dependent propagation equations are given by18,19
S ]
]z1
1
n1
]
]t DE1 5 i4pv1
2
k1c2 cos2 b1xeff
~2 !~v1!E1* E2
3 exp@2i~2k1 2 k2!z#, (1a)
S ]
]z1
1
n2
]
]t DE2 5 i8pv1
2
k2c2 cos2 b2xeff
~2 !~v2!E12
3 exp@i~2k1 2 k2!z#, (1b)
where xeff(2) is the effective second-order susceptibility,
n1,25 (]k1,2 /]v1,2)21 are the group velocities, and b1,2 are
the birefringence angles inside the crystal. In Eqs. (1)the higher-order effects of the dispersion (i.e., the group-velocity dispersion at v1 , v2) have been neglected. Thisapproximation is well verified for pulse durations longerthan a limit that is determined by the length and the dis-persion properties of the crystal sample under the test, aswill be discussed later on. For the commonly employednonlinear materials and crystal sizes, this limit is usuallymuch lower than the pulse duration that determines theonset of a nonstationary interaction. The pulses enve-lopes E1,0(t) and E2,0(t) at the input of the crystal (z5 0) are assumed known.
Using the following normalized quantities (with uE1,0upkthe peak amplitude of the F pulse at the input of the crys-tal),
where h0 has the physical meaning of the conversion effi-ciency for the field amplitudes, which would result at thepulse peak in stationary conditions, in phase-matchedpropagation and by neglecting the F-field depletion. Thetwo temporal coordinates u1,2 are related by the expres-sions u1 5 u2 1 jL(n2
21 2 n121).
Expanding the field envelopes in a series of h0 ,
r i~j, t ! 5 (n50
`
h0nr i,n~j, t ! ~i 5 1, 2 !, (3)
and substituting in the coupled equations (2a) and (2b), itis possible to separate the terms of the same order in h0and obtain the following hierarchy of integral equations:
r1,n~j, u1! 5 iE0
j
(i1j5n21
r1i* ~x, u1!r2,j~x, u1!
3 exp~2idx !dx ~i, j > 0 !, (4)
r2,n~j, u2! 5 iE0
j
(i1j5n21
r1i~x, u2!r1,j~x, u2!
3 exp~idx !dx, ~i, j > 0 !, (5)
where r i,0(ui) 5 r i,in(ui) and r i,in(t) are the pulse shapesinjected in the crystal at j 5 0 (i 5 1, 2). Given the in-jected envelopes r i,in(t), it is possible, in principle, to cal-culate the series terms given by Eqs. (4) and (5), since theterm of order nth depends only on the terms up to the(n 2 1)th.
In the following sections we will apply this model to thesecond-order cascaded processes in single- and double-pass geometry.
3. SINGLE-PASS INTERACTIONAssuming that, at the input of the nonlinear medium,there is no SH-signal injection, the series developmentsfor the F and the SH pulses during the propagation con-tain only even and odd terms in h0 , respectively.
We will consider the configurations in single- anddouble-pass interactions, which in stationary conditionshave minimum losses for the F field and negligible SHoutput.
The values of d, yielding a zero SH field at the output ofthe crystal, calculated for cw fields employing the expres-sion approximated up to the fifth order in h0 ,
ur2~j!u 5 ur2,1~j!h0 1 r2,3~j!h03 1 r2,5~j!h0
5u, (6)
move slightly from the values found in the regime of anundepleted v1 field (i.e., in the first-order approximationfor the SH amplitude), as already reported.20 Takinginto account the higher-order terms in approximation (6),we find the following approximate expressions for the ze-roes of SH generation:
Toci et al. Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. B 105
d 5 npU1 2 4S h0
np D 2
1 4S h0
np D 4
1 352S h0
np D 6
2 16~11n2p2 2 362!S h0
np D 8
2 8~133n2p2 2 3728!S h0
np D 10U, (7)
n 5 62,64, ...We assume in this paper a F input pulse shape of the
form sech(t/t) with negligible chirping effects. Neverthe-less, the previously discussed perturbative model holdsfor an arbitrary frequency-modulated pulse shape.
As long as we deal with transform-limited F pulses, itis useful to introduce the following steadiness parameter:
s 5 F 1v2
21v1
G L2t
, (8)
which is proportional to the ratio between the propaga-tion delay across the crystal length and the F-pulse dura-tion or, equivalently, inversely proportional to the ratiobetween the pulse bandwidth and the spectral acceptanceof the crystal. This ratio becomes the relevant param-eter when dealing with non-transform-limited pulses.21,22
In the following analysis we will assume s . 0, as it re-sults for the most common frequency-doubling crystals inthe visible. The extension to the case s , 0 is straight-forward.
We can separate the phase- and amplitude-modulationeffects on the F-pulse shape at crystal output (j 5 1) us-ing the form
r1~u1! 5 r1,0~u1!exp@ 2 A~u1! 1 iF~u1!#. (9)
The exponential form in Eq. (9) gives an accurate rep-resentation of the series development (5) up to the fourthorder in h0 if the exponents are defined as follows:
A~u1! 5 a2~u1!h02 1 a4~u1!h0
4
5 2Fh02
Re@r1,2~u1!#
r1,0~u1!
1 h04X$Im@r1,2~u1!#2 2 Re@r1,2~u1!#2%
2r1,02 ~u1!
1Re@r1,4~u1!#
r1,0~u1!CG, (10a)
F~u1! 5 f2~u1!h02 1 f4~u1!h0
4
5 Fh02
Im@r1,2~u1!#
r1,0~u1!
1 h04X2
$Re@r1,2~u1!#Im@r1,2~u1!#%
r1,02 ~u1!
1Im@r1,4~u1!#
r1,0~u1!CG (10b)
It appears from Eqs. (10) that the coefficients a2 , a4 ,f2 , and f4 are independent from h0 .
The terms up to the second order are proportional tothe peak power density of the incident F pulse (seeabove). In particular in Eq. (10b), in the limit of station-ary interaction, the term f2(u1)h0
2 is the well-knownKerr-like self-phase-modulation effect. The terma2(u1)h0
2 in stationary conditions shows some analogywith a two-photon-absorption effect, where, with the cho-sen signs, a positive value of a2 leads to an attenuation ofthe fundamental pulse.
In stationary conditions (s 5 0) integrals (4) and (5)can be calculated explicitly for every order. In phase-matched conditions (d 5 0) analytical expressions existat least for the first terms in the series development (asreported in Appendix B).
In phase-mismatched, nonstationary conditions (dÞ 0,s Þ 0) and with the assumed pulse shape, the integralexpressions (4) and (5) must be numerically calculated.We employed the numerical analysis software packageMATHCAD 5.0.23 Even in this case, the resulting functionsdepend only on d and s, whereas the dependence on h0becomes explicit from the series-development scheme.
We limit our analysis to the case d . 0 because, withthe hypothesis of unchirped F input pulse and no SHseeding, the following symmetry properties hold for bothstationary and nonstationary interaction:
r1,n~2d! 5 r1,n* ~d!, (11a)
r2,n~2d! 5 2r2,n* ~d!. (11b)
Therefore [see Eqs. (10)], A and F are, respectively,even and odd functions of d.
The second-order amplitude and phase modulationa2(u1), f2(u1) can be expressed in a form that will beuseful in the following section when we deal with thedouble-pass interaction schemes. a2(u1), f2(u1) are, re-spectively, the real and imaginary parts of the functionr1,2(u1)/r1,0(u1) which, through the definitions (4) and(5), results in
Because the function r1,0* (u1) does not depend explicitlyon x, it can move out of the integrals; furthermore, aslong as we deal with transform-limited pulses, we can as-sume r1,0* (u1 , x) 5 r1,0(u1 , x), and we can cancel out thedenominator in Eq. (12). a2(u1), f2(u1) are thus thereal and imaginary parts of the function
g1~u1 , s, d! 5 2E0
j
r2,1~u1 2 2stx !exp~2idx !dx,
(13)
which depends not only on the instantaneous pulse am-plitude, but also on the shape and temporal duration ofthe pulse.
Figure 1 shows the behavior of the phase-modulationcoefficient f2(u1), which determines the h0
2 (Kerr-like)term in the phase modulation. We recall that f2(u1) is
106 J. Opt. Soc. Am. B/Vol. 15, No. 1 /January 1998 Toci et al.
Fig. 1. Phase-modulation coefficient f2 [Eq. (10b)] for thesingle-pass configuration at the output of the nonlinear medium(j 5 1) as a function of the local time u1 5 t 2 L/v1 [see Eq. (2)]varying the steadiness parameter s for the specified values of thephase-mismatch parameter d : (a), d 5 2p; (b), d 5 4p; (c), d5 6p. Mesh spacing is Ds 5 0.2, Du1 /t 5 0.1. Gray-scalelevels in the three-dimensional mesh surfaces and in the upper-most contour maps correspond to the vertical-axis main-divisionvalues.
independent from h0 [see Eq. (10b)], and the nonlinearphase modulation for a given value of h0 can be calculatedfrom Eq. (10b).
For a given crystal length, when the F-pulse durationdecreases [and the value of the steadiness parameter s in-creases; see Eq. (8)], the main peak of f2(u1) increasinglylags from the peak of the v1 pulse. This happens be-cause during the propagation the SH pulse retards withrespect to the F pulse, and therefore it modulates mainlythe F-pulse tail, whereas the F-pulse head undergoes alesser perturbation. This effect becomes more evidentwhen the F pulse shortens. Furthermore, the trailingedge of f2(u1) develops a structure of secondary maximaand minima, swinging more and more as d increases ow-ing to the quicker change of the relative phase of the Fand SH fields along the propagation. For the highest val-ues of d, the development of such a substructure has twoeffects: On one hand it prevents the main peak of f2(u1)from moving toward the trailing edge of the F pulse; onthe other hand, it quenches the trailing edge of f2(u1),preventing it (for the chosen range of s) to undergo thesignificant temporal broadening, which appears for thelowest values of d [see Fig. 1]. The overall result is that,for high values of d, the behavior of the phase-modulationprocess remains qualitatively similar to its behavior instationary conditions. For increasing values of d, thesteady-state peak value of f2(u1) decreases (roughly as1/d, as pointed out by DeSalvo et al.2).
The approaching of a nonstationary condition gives ori-gin to an amplitude-modulation profile a2(u1), which isdue to the energy drain toward a growing SH pulse oth-erwise prevented by the phase-velocity mismatch.24,25 Itdetermines also a power-density transport among differ-ent points of the F-pulse envelope. This appears in Fig.2, showing the behavior of a2(u1) [see Eq. 10(a)] for dif-ferent values of d.
Again, a2(u1) becomes more structured and swingingas the phase-mismatch parameter increases, as discussedabove for f2(u1). This phenomenon makes it impossibleto obtain a pure phase-modulation process in the nonsta-tionary condition.
The accuracy achieved by cutting the series develop-ments at the second order in h0 can be tested by evaluat-ing the fourth-order term in expressions (10a) and (10b).This requires the evaluation of the further termsr2,3(u2 , j) and r1,4(u1 , j) in the series development (3).A further discussion about the accuracy of the methodbased on the comparison with the numerical integrationof the propagation equations is reported in Appendix C.
We have evaluated these terms for the same parameterranges considered above. Instead of reporting the realand imaginary parts of r1,4(u1 , j), we found it useful toreport the pure h0
4 correction to the phase and to the am-plitude modulations, i.e., the functions f4(u1), a4(u1) in-troduced in Eqs. (10). These results are shown in Figs. 3and 4, respectively.
In stationary condition (s 5 0), the fourth-order termintroduces a pure phase modulation without furtherlosses @a4(u1 , s 5 0) 5 0#. This is understandable be-cause in the stationary condition, in a lossless mediumthe total instantaneous intensity of the two fields must beconserved, i.e., ur1u2 1 ur2u2 5 1 (the Manley–Rowe
Toci et al. Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. B 107
Fig. 2. Amplitude-modulation coefficient a2 [Eq. (10a)] for thesingle-pass configuration at the output of the nonlinear medium(j 5 1) as a function of the local time u1 5 t 2 L/v1 [see Eq.(2)], varying the steadiness parameter s for the specified valuesof the phase-mismatch parameter d : (a), d 5 2p; (b), d 5 4p;(c), d 5 6p. Mesh spacing is Ds 5 0.2, Du1 /t 5 0.1. Gray-scale levels in the three-dimensional mesh surfaces and in theuppermost contour maps correspond to the vertical-axis main-division values.
Fig. 3. Fourth-order coefficient f4 of the phase modulation [Eq.(10b)] for the single-pass configuration at the output of the non-linear medium (j 5 1) as a function of the local time u1 5 t2 L/v1 [see Eq. (2)], varying the steadiness parameter s for thespecified values of the phase-mismatch parameter d : (a), d5 2p; (b), d 5 4p; (c), d 5 6p. Mesh spacing is Ds 5 0.2,Du1 /t 5 0.1. Gray-scale levels in the three-dimensional meshsurfaces and in the uppermost contour maps correspond to thevertical-axis main-division values.
108 J. Opt. Soc. Am. B/Vol. 15, No. 1 /January 1998 Toci et al.
Fig. 4. Fourth-order coefficient a4 of the phase modulation [Eq.(10a)] for the single-pass configuration at the output of the non-linear medium (j 5 1) as a function of the local time u1 5 t2 L/v1 [see Eq. (2)], varying the steadiness parameter s for thespecified values of the phase-mismatch parameter d: (a), d5 2p; (b), d 5 4p; (c), d 5 6p. Mesh spacing is Ds 5 0.2,Du1 /t 5 0.1. Gray-scale levels in the three-dimensional meshsurfaces and in the uppermost contour maps correspond to thevertical-axis main-division values.
condition18). For d 5 2np, the first nonzero term in r2 isof the third order, and in consequence the Manley–Roweequation requires that the first depletion term for the v1field is proportional to h0
6.It is useful to estimate the relative error introduced by
neglecting the fourth-order term with respect to thesecond-order one (for given s, d, h0) at the crystal output(j 5 1) with the expression
«1,4~h0! 5 h04ur1,4~u1 , 1!upk /h0
2ur1,2~u1 , 1!upk (14)
which is the ratio between the peak value of the modulusof the fourth- and second-order terms. We found that inthe range of parameters 0 < s < 3 the relative error « isrelatively independent (within about 620 %) from s, andfor a given value of h0 it decreases when udu increases.The following inequalities hold:
«1,4 , 0.24h02 ~d 5 2p!, (15a)
«1,4 , 0.093h02 ~d 5 4p!, (15b)
«1,4 , 0.056h02 ~d 5 6p!. (15c)
We found that these inequalities provide a useful testof the accuracy reached by cutting the series developmentat the second order. The problem of the overall accuracyof the perturbative method is discussed with a differentapproach in Appendix C.
4. DOUBLE-PASS INTERACTIONWe will now consider the case of double-pass interaction,where the F pulse generates a SH pulse during a firstpass in the nonlinear crystal and both pulses are then re-injected in the crystal by means of a suitable opticalsetup. This situation has great practical importance forrealizing passive modulators for laser mode locking lyingboth on the amplitude (nonlinear mirror11,12,15–17) and onthe phase (cascaded second-order mode locking13,14) non-linear modulation effects. In comparison to the single-pass geometry, the double-pass one generates more sym-metric amplitude- and phase-modulation profiles usefulin the development of optical modulators.
The same assumptions introduced in the single-pass in-teraction hold. Moreover, the wave vectors are assumedto have counterpropagating directions with respect to thefirst pass; the nonlinear interaction of the pulses takesplace from z 5 L to z 5 0; this geometry is describedwith the same equations of the first pass, provided the v2pulse envelope is corrected with a further phase-shiftterm @exp(id)# as shown in Appendix A.
We assume control of the optical path length betweenthe first pass and the second pass independently for boththe spectral components so that the pulse envelopes canbe shifted in phase and in time with respect to each other,for example, by means of an interferometric arrangementor with suitable dispersing optical components.14,16
The pulse envelopes are also assumed to undergo anegligible distortion between the first and the secondpasses (neglecting, for example, the group-velocity-dispersion effects of the reinjection optics) except for lin-ear losses (for example, that are due to the mirror reflec-tivity). It is also useful to normalize the F- and the SH-
Toci et al. Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. B 109
field envelopes with respect to uE1upk at the first-passinput, simplifying the calculation of the overall phase andamplitude modulations and directly taking into accountthe effect of the finite mirror reflectivity.
The v1 and the v2 pulses at the input of the secondpass are thus given by
r1,in9 ~j 5 0, t ! 5 r1 exp~iu1!@r1,08 ~j 5 1, t !
1 h02r1,28 ~j 5 1, t ! 1 ...#, (16a)
r2,in9 ~j 5 0, t ! 5 r2 exp@i~d 1 u2!#@h0r2,18 ~j 5 1, t 1 Dt !
1 h03r2,38 ~j 5 1, t 1 Dt ! 1 ...#, (16b)
where the single and the double apexes denote the func-tions for the first and the second pass, respectively; r1,2are the mirror reflectivities at v1 and v2 ; u1,2 are theoverall phases at the beginning of the second pass thatare due to the optical path of the v1 and v2 pulses; and Dtis the relative time delay of the v2 pulse with respect tothe fundamental one.
The double-pass interaction scheme is shown in Fig. 5.The series (3) for the F field at the output of the second
pass now contains terms with an odd index, resultingfrom the interaction between the F and the injected SHsignal, which in turn contains only odd powers of h0 start-ing with a linear term. In consequence also the termswith the odd index in the F-pulse envelope introduce onlyeven powers in h0 at the output of the second pass. Ananalogous result holds also for the outgoing SH pulse,which contains only odd powers of h0 .
Similarly to the single pass, we want to calculate thephase and amplitude modulations on the fundamentalproportional to h0
2 (i.e., the Kerr-like second-order effect).From the previous discussion the first nonzero correctionto Eq. (16a) is of the order of h0
4.The v1 pulse envelope at the output of the second pass,
up to the second order in h0 , calculated with the sameprocedure employed to obtain Eqs. (4) and (5), is given by
Fig. 5. Ideal experimental layout describing the second-ordercascaded effect in the double-pass configuration. Open and solidblack shapes represent the v1 and v2 pulse envelopes, respec-tively; Dt is the additional time delay between the F and SHpulses before the second pass.
r19~u1 , j! 5 r1 exp~iu1!r1,08 ~u1! 1 ih02r1r2
3 exp@i~u2 1 d 2 u1!#E0
j
r1,08* ~u1 , x !
3 r2,18 ~u1 2 2stx 1 Dt, 1!exp~2idx !dx
1 ih02r1
3 exp~iu1!E0
j
r1,08* ~u1 , x !
3 r2,19 ~u1 2 2stx, x !exp~2idx !dx
1 h02r1 exp~iu1!r1,28 ~u1 , 1!. (17)
As for the single pass arrangement, we can separatethe nonlinear phase- and amplitude-modulation effects atthe output of the crystal through the exponential form[see Eq. (9)]
r19~u1 , j 5 1 ! 5 r1 exp~iu1!r1,08 ~u1 , j 5 0 !
3 exp@2A~u1! 1 iF~u1!#
5 r1 exp~iu1!
3 r1,08 ~u1 , j 5 0 !
3 exp$@2a2~u1! 1 if2~u1!#h02%,
(18)
where
a2~u1! 5 21
h02 ReH r19~u1 , j 5 1 !
r1 exp~iu1!r1,08 ~u1 , j 5 0 !2 1J ,
(19a)
f2~u1! 51
h02 ImH r19~u1 , j 5 1 !
r1 exp~iu1!r1,08 ~u1 , j 5 0 !J , (19b)
and where r19(u1 , j 5 1) is given by Eq. (17) (a positivevalue of a2 leads to attenuation). As it was already dis-cussed for Eq. (10), the coefficients a2 , f2 are indepen-dent of h0 .
We can find an alternative expression for theamplitude- and phase-modulation coefficients by substi-tuting into Eq. (19) the series development (17). As thefunction r1,08* (u1 , x) does not depend explicitly on x, it canmove out of the integrals; furthermore, as long as we dealwith transform limited pulses, we can assumer1,08* (u1 , x) 5 r1,08 (u1 , x) and we can cancel out the de-nominators in Eq. (19). We obtain the following expres-sions:
110 J. Opt. Soc. Am. B/Vol. 15, No. 1 /January 1998 Toci et al.
g2~u1 1 Dt, s, d! 5 iE0
j
r2,18 ~u1 2 2stx 1 Dt, 1!
3 exp~2idx !dx, (21)
and the function g1 was introduced in Section 3 [see Eq.(13)].
The double-pass modulation described by Eqs. (17) and(19) results from the superimposition of several effects.The second integral term in Eq. (15), which contains r2,18and gives origin to the term containing g2 in Eq. (20),arises from the cross interaction during the second pass ofthe two fields coming out from the first pass (at the lowestorder of approximation in h0
2); in this term the phasorexp@i(u2 1 d 2 2u1)# determines how g2 affects the phaseor amplitude modulation of the F pulse via Eq. (20).
The term containing r1,28 is the lowest-order perturba-tion on the F field arising from the first pass [see Eq. (4),with n 5 2, j 5 1], and the integral term containing r2,19describes the modulation of the F pulse arising from itsinteraction with the SH pulse generated in the secondpass. Both these terms have the same dependence onu1 , s, d, described by the function g1 in Eq. (20), and theydiffer only in magnitude since the zeroth-order terms inthe F development for the first and the second pass differonly in their magnitude by a factor r1
2 , owing to the linearlosses of the reflecting mirror.
The expressions (17) and (20), show explicitly the effectof the parameters r1 , r2 , h0 , u1 , u2 . They take also ex-plicitly into account the influence of the temporal delayDt, since the function g2 depends only on (u1 1 Dt), s, d,so that for given s, d a change in the delay parameterDt determines just a shift with respect to u1 , and we donot need to recalculate it. The parameters s, d affect theoverall result via the integral expressions g1 ,g2 .
Up to the second order in h0 , the v2 pulse envelopeduring the second pass is given by
which is the linear combination (with the proper lossesand phases) of the envelopes generated during bothpasses.
We are mainly interested in the configurations, whichin a stationary condition minimize the nonlinear lossesfor the v1 field, yielding the minimum output at v2 . As-suming lossless optics (r1 5 r2 5 1) at the second orderin h0 , this is obtained with d 1 Q 5 2np (for unchirpedinput fundamental pulses), where Q 5 u2 2 2u1 . To en-sure the maximum interaction effect of the two pulses,the SH pulse must be shifted in time with respect to the Fpulse of an amount Dt 5 @1/v2 2 1/v1#L 5 2st.
Figure 6 shows the nonlinear phase-modulation coeffi-cient f2(u1) as given by Eq. (20b) when s and d increase.The comparison with Fig. 1 obtained for a single pass in-teraction shows that the two geometries have quite differ-ent behaviors when moving toward the nonstationary in-teraction regime. The single-pass phase-modulation
Fig. 6. Phase-modulation coefficient f2 [Eqs. (19b) and (20b)]for the double-pass configuration at the output of the nonlinearmedium (j 5 1) as a function of the local time u1 5 t 2 L/v1[see Eq. (2)], varying the steadiness parameter s for the specifiedvalues of the phase-mismatch parameter d: (a), d 5 2p; (b), d5 4p; (c), d 5 6p, with r1 5 r2 5 1, Q 5 0, Dt 5 (1/v22 1/v1)L. Mesh spacing is Ds 5 0.2, Du1 /t 5 0.1. Gray-scale levels in the three-dimensional mesh surfaces and in theuppermost contour maps correspond to the vertical-axis main-division values.
Toci et al. Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. B 111
shows a main peak that increasingly lags from the exci-tation pulse peak and develops a substructure of second-ary maxima and minima on its trailing edge (see above).The double-pass phase modulation instead is an evenfunction of the reduced time u1 , which remains symmet-ric around the excitation pulse peak even for high valuesof s, and whose peak value is as high as approximatelytwice the value achieved in the single-pass geometry.This different behavior results because the overalldouble-pass phase modulation rises from the superimpo-sition of three contributions, as was already discussed.
The first two terms are the single-pass modulations ris-ing in the first and second transits in the crystal, which atthe second order of approximation are identical [term g1in Eq. (20)]; these two terms do not have a definite parityin u1, and within the second-order approximation theircontribution to the overall phase modulation is twice thatof the single-pass modulation.
The term resulting from interaction during the secondpass of the F pulse with the SH pulse reinjected from thefirst pass [term g2 in Eq. (20)] is antisymmetric with re-spect to u1 , and therefore it almost does not change thephase peak, but affects mainly the F-pulse wings.
The superimposition of these terms is even in u1 , pro-vided that the mirror reflectivities and the phase andtemporal delays between the two passes are properly set.
When d increases, the phase modulation reduces itssteady-state peak value, as was already noticed in thesingle pass. Thus the configurations with low values of dare more convenient for an efficient self modulation oflong pulses (low values of s). When the pulse shortensand s increases, we notice that the configuration with lowvalues of d evolves more rapidly toward a nonstationarybehavior, as we can see from the comparison in Fig. 6.For the lower value of d, the main phase-modulation peakbroadens rapidly, and around s 5 2 it splits into two sub-sidiary peaks affecting the pulse wings. When d in-creases, this same behavior appears to occur for corre-spondingly higher values of s, i.e., in the range covered inFig. 6, for d 5 4p it barely starts to appear at s 5 3 as aslight broadening and lowering of the main peak, and ithas yet to become visible for the case d 5 6p. In thislast case, the phase-modulation effect near the pulse peakremains similar to the stationary case over a broad inter-val of s. The increase in the value of s leads to thegrowth of a system of small subsidiary maxima andminima affecting the pulse wings.
Figure 7 shows the amplitude modulation that resultsfrom the double-pass interaction in the same propagationcondition described above. The comparison with the cor-responding single-pass amplitude modulation (Fig. 2)shows some relevant differences. When we approach anonstationary interaction, the growth of a single-pass SHpulse determines the rise of an amplitude-modulationprofile, which appears to be an antisymmetric function ofthe reduced time u1 , although it gets more structuredand more swinging as the values of d and s increase.The single-pass case instead does not exhibit a definitesymmetry with respect to the time. Therefore, at this or-der of approximation, the F pulse outgoing from thedouble-pass process did not suffer a net energy drain to-ward the SH pulse. Indeed, Eq. (22) shows that the v2
Fig. 7. Amplitude-modulation coefficient a2 [Eqs. (19a) and(20a)] for the double-pass configuration at the output of the non-linear medium (j 5 1) as a function of the local time u1 5 t2 L/v1 [see Eqs. (2)], varying the steadiness parameter s for thespecified values of the phase-mismatch parameter d : (a), d5 2p; (b), d 5 4p; (c), d 5 6p, with r1 5 r2 5 1, Q 5 0, Dt5 (1/v2 2 1/v1)L. Mesh spacing is Ds 5 0.2, Du1 /t 5 0.1.Gray-scale levels in the three-dimensional mesh surfaces and inthe uppermost contour maps correspond to the vertical-axismain-division values.
112 J. Opt. Soc. Am. B/Vol. 15, No. 1 /January 1998 Toci et al.
output results from the coherent superimposition of thetwo single-pass SH pulses, which cancel out completely,with the chosen temporal and phase delay and assuminga lossless reinjection; in other words, during the secondpass the SH pulse generated during the first pass fully re-pumps the F pulse, determining an energy transfer be-tween its various portions without an overall energy loss.
The absolute peak value of the double-pass amplitudemodulation is not significantly stronger than in thesingle-pass case. Because of its antisymmetric behavior,the double-pass amplitude modulation mainly affects thepulse wings. This is different from the single-pass am-plitude modulation, which always shows a strong deple-tion peak at the pulse center that is due to the powerdrain toward the v2 pulse.
5. DISCUSSIONThe previous sections show that the perturbative methodoutlined in this paper can be usefully employed to studythe second-order cascaded process when the interaction isdominated by the group-velocity mismatch (GVM) be-tween the interacting fields. In particular this method isuseful to describe the dynamics of the devices based onthe cascaded second-order nonlinearities, which exhibit asaturable-absorber action suitable for laser mode locking.
Until now, the cw laser sources mode locked by secondorder nonlinearities (Nd:YLF12 and Nd:YAG13,14 withlithium borate as the nonlinear crystal) exhibit a stableoperation characterized by fairly low values of the single-pass conversion efficiency (h0
2 ' 0.112). The occurrenceof this low value of h0
2 allows us to obtain a rather accu-rate description of the modulation dynamic just truncat-ing the series development at the second order. A suit-able group-delay compensation14 determined a pulseduration of 5.1 ps (for the nonlinear mirror configuration)and 5.9 ps (for the cascaded-mode-locking scheme)FWHM, with a hyperbolic-secant-squared pulse-intensityprofile. A reported group delay of 825 fs along the wholecrystal length determines for these experiments a value ofthe steadiness parameter of about s 5 0.2, setting an al-most nonstationary interaction.
From the above discussion, the double-pass geometriesappear to be more efficient than the single-pass ones forthe development of passive optical modulators under sev-eral aspects:
• higher self-phase modulation obtained for a givenintensity level and crystal length;
• complete repumping of the SH field to the F oneachieved even in nonstationary conditions, allowing amodulation without energy losses for the F field itself, al-though a self-amplitude-modulation effect is unavoidablewhen the process is in nonstationary interaction condi-tions;
• temporally symmetric phase- and amplitude-modulation profiles attained with proper choice of thephase and temporal delay before the second pass.
The previous results have also evidenced the strong de-pendence of the modulation process in nonstationary con-ditions from the value of the phase-mismatch parameterd. For a given crystal length high values of d appear to
be more favorable for the self phase modulation of shortpulses despite the weaker modulation amplitude becauseof the more stable temporal dependence, which remainssimilar to the steady state over a range of values of sbroadening when d increases.
It is useful to discuss the scaling of the nonlinear phaseand amplitude modulations with respect to the param-eters of the cascaded process. From the previous analy-sis, it appears that in phase-mismatched conditions, thephase-modulation profile remains almost stable with re-spect to s, up to a value spk where f2(u1 5 0) shows abroad peak with respect to s (see Fig. 6). Above thisvalue of s, the phase-modulation profile lowers andbroadens, and it develops the features typical of nonsta-tionary interactions. Furthermore, below this limit theamplitude-modulation coefficient a2 remains quite mod-est. This value spk can be considered the threshold de-termining the onset of a nonstationary interaction. Wehave found that in the double-pass configuration with op-timal group-delay compensation the steadiness thresholdspk increases almost steadily with d, according to the for-mula
spk~d! ' 0.5udu/p, (23)
whereas, as already noticed, the peak value of the phasemodulation decreases as approximately
Fpk ' 0.7h02p/udu. (24)
Recalling the definitions of d, s [see Eq. (8)] Eq. (23) canbe interpreted as follows: For a given pulse duration anda given crystal length the cascaded process is forced to aquasi-stationary interaction [that is, s remains under thethreshold set by Eq. (23)], if the phase mismatch Dk islarger than
uDku 5 p~1/v2 2 1/v1!/t. (25)
Not so surprisingly, this value of Dk sets a coherencelength close to the longitudinal walkoff length of the Fand SH pulses. If the pulse duration, the crystal length,and the crystal GVM are such that s is significantlygreater than 0, Eq. (25) sets the value of Dk, which yieldsthe maximum phase modulation: Lower values of Dkwould induce a completely nonstationary interaction,higher values of Dk would reduce the phase modulationaccording to Eq. (24). From Eq. (24), with the condition(25), for a given pulse duration and peak power the maxi-mum achievable peak phase modulation (to the second or-der in h0) results in
Fpk 5 0.7S 4v12
c2 cos2 b1 cos2 b2D
3 F ~xeff~2 !!2
n1n2~1/v2 2 1/v1!G uE1upk
2 tL. (26)
We can make some conclusions from Eq. (26). First,provided that the phase mismatch is chosen according toEq. (25), a strong phase modulation can be obtained de-spite the nonstationary interaction by increasing the crys-tal length (even though this is done at the price of somenonlinear amplitude modulation and some ringing in thephase modulation; see Section 4). In this case the phase
Toci et al. Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. B 113
modulation increases linearly with the crystal length,whereas in stationary conditions it is proportional to L2,as has been widely reported.1,2
Second, the factor in the square bracket depends onlyon the material properties and the frequency-doublingprocess parameters, both linear (indices of refraction,GVM, phase-matching angle for the type I second-orderinteraction) and nonlinear (second-order susceptibilitytensor). Defined as
M 5 F ~xeff~2 !!2
n1n2~1/v2 2 1/v1!G , (27)
this can be therefore regarded as a figure of merit for thenonlinear material for the cascaded process in nonstation-ary conditions, which accounts not only for the materialnonlinear properties, but also for its dispersion character-istics and their effects on the self-phase-modulation pro-cess.
We have calculated the figure of merit M for the type Isecond-order cascaded interaction at l 5 1064 nm forseveral commonly employed nonlinear materials; for eachmaterial the value of the various parameters was calcu-lated using the proper phase-matching angles and polar-ization directions. The results are summarized in Table1.
We can see that M undergoes a quite large variationover the considered materials. In particular, althoughKDP and its isomorphs (ADP, KD*P) exhibit a very lowGVM, they are penalized by the very low value of theirnonlinear coefficient; BBO provides the better perfor-mance among the considered materials despite its quitehigh GVM, owing to its large nonlinear coefficient, whichallows the use of short samples.
The evaluation of the figure of merit introduced abovecan guide the choice of the proper nonlinear material forthe design of experiments or devices. Nevertheless,there are other material parameters, which become im-portant when dealing with finite-aperture or focusedbeams, such as angular acceptance and lateral beamwalk-off that is due to the birefringence. In this sense,BBO (and even more, LiIO3) exhibits both a rather lowangular acceptance and a rather high walk-off, whereasthe large acceptance angle and the null walk-off achiev-able in LBO by means of temperature-noncritical phasematching are attractive characteristics which were use-fully exploited in cascaded mode locking and nonlinearmirror mode locking.12–14
We should discuss now the approximation that resultsfrom neglecting the group-velocity-dispersion terms in
Eq. (1). The group-velocity-dispersion coefficient k1,295 (]2k/]v2)uv5v1 ,v2
determines, for a given (unchirped)pulse duration t, a characteristic propagation length LD
5 t 2/k9, and the temporal-diffraction effects can be ne-glected as long as the crystal thickness is L ! LD . Thiscondition can be expressed as s ! @1/v2 2 1/v1#3 L/(4Lk1,29 )1/2. Considering the experiments of cas-caded mode locking and of nonlinear mirror mode lockingrecalled before,12–14 in which a 15-mm-long LBO crystaloperating at a fundamental wavelength of 1064 nm wasused, we obtain (from the Sellmeier coefficients reportedin Ref. 29) k19 ' 18 fs2/mm, k29 ' 85 fs2/mm, and it re-sults in s ! 12 (determined by the dispersion of the har-monic), much higher than the values in the experimentsand those considered in this discussion.
The method presented in this paper deals only withplane waves, but when it is possible to neglect the diffrac-tion effects and the eventual lateral walk-off over the non-linear medium length, the analysis can be extended to de-scribe the effects on the propagation of finite aperturebeams (as required, for example, for the analysis of cas-caded mode locking). This is achieved by considering thefield amplitudes and thus the coupling coefficient h0 asfunctions of the distance from the beam axis. Within thisassumption, the temporal dynamics of the pulse and thespatial evolution of the beam are decoupled, as the time-dependent coefficients a2 , a4 , f2 , f4 introduced in Eqs.(10) and (20) remain independent from the beam-axis dis-tance. This allows a combination with perturbativebeam-propagation techniques such as Gaussiandecomposition.32,33 The lateral beam walk-off must bekept at its minimum value by either a proper choice of thecrystal cut or adopting, when possible, noncritical phasematching schemes, i.e., temperature tuning. Actually,the lateral walk-off is likely to perturb the resonatormode because it breaks the cylindrical geometry of thecascaded interaction.
6. CONCLUSIONSWe have presented a perturbative method to describe thesecond-order cascaded processes with type I phase-matching for plane waves. The approach considers theeffects of the group-velocity mismatch between the inter-acting fields at v1 and 2v1 . It is based on a series de-velopment of the fundamental- and second-harmonic-pulse envelopes in respect to the coupling parameter h0as defined in Section 2. The coefficients of the series turnout to be functions of time and the longitudinal coordinate
Table 1. Optical Parameters and Figure of Merit M of Several Nonlinear Materialsfor the Type I Second-Order Cascaded Process at 1064 nm
114 J. Opt. Soc. Am. B/Vol. 15, No. 1 /January 1998 Toci et al.
along the crystal. The perturbative approach presentedabove can also be used to study three-wave-mixing pro-cesses such as type II SH generation (and the relatedcascaded-interaction schemes) and parametric oscillationand amplification.
In this paper we have employed the perturbativemethod to study the effect of self-phase and self-amplitude modulation experienced by the fundamentalfield in a cascaded second-order interaction with the non-linear medium in a nonstationary condition. This ap-proach represents a useful alternative to the numericalintegration of the propagation equations (1) and (2) em-ployed by several authors for the description of this classof phenomena,7,15–17 because it allows us to obtain semi-analytical (or, in particular cases, completely analytical,as shown in Appendix B) expressions for the pulse enve-lopes during the propagation and the interaction. Fur-thermore, it permits us to separate the different-orderself-modulation terms, i.e., the contribution of Kerr-likeand two-photon-like modulation effects (which scale lin-early with the intensity) and the higher-order modulationeffects. Furthermore, the series-development scheme ofthis method allows a combination with perturbativebeam-propagation techniques such as Gaussian decom-position.32,33 Our activity is in progress on this subject.
Our results at the present state of the art can be usedas a guideline for defining the temporal region of the ex-periments, for example, whether a mode-locked laser isachievable for a given crystal and laser material. Stablemode-locking operation with cascaded second-order non-linearities with broadband laser media in the region oftens or hundreds of femtoseconds requires a suitable cav-ity design to obtain both a proper resonator mode struc-ture (to be modulated by the self-phase-modulation effect)and a stable laser oscillation against the wavelength wan-dering outside the acceptance bandwidth of the nonlinearcrystal. Furthermore, a careful study is required to finda proper balance between the temporal-broadening effects(which are due to the linear dispersion and the laser me-dium finite bandwidth) and the pulse-shortening effect(which is due to the active presence of the nonlinear ma-terial). We already reported34 an analysis applied to thenonlinear mirror (that is, in phase-matched conditions),which provides a satisfactory comparison with theexperiments.12,14
APPENDIX A: EQUATIONS FOR THEBACKWARD PROPAGATIONFor the backward propagation from z 5 L to z 5 0, in thetraveling wave form introduced in Section 2 we mustchange k1,2 to 2k1,2 (and therefore v1,2 to 2v1,2). Settingz8 5 L 2 z, we thus obtain for the backward (pedix b)equations
S ]
]z81
1
v1
]
]t DE1b 5 i4pv1
2
k1c2 cos2 b1xeff
~2 !~v1!E1b* E2b
3 exp@i~2k1 2 k2!L#
3 exp@2i~2k1 2 k2!z8#, (A1)
S ]
]z81
1
v2
]
]t DE2b 5 i8pv1
2
k2c2 cos2 b2xeff
~2 !~v2!
3 E1b2 exp@2i~2k1 2 k2!L#
3 exp@i~2k1 2 k2!z8#. (A2)
The fixed phase factor exp@2i(2k1 2 k2)L# can be en-closed in E2b setting E29 5 E2b exp@i(2k1 2 k2)L#. We ob-tain a couple of equations of the same form of Eq. (1) tointegrate between z8 5 0 and z8 5 L with the same re-sults shown above.
APPENDIX B: ANALYTICAL EXPRESSIONSWe report again for completeness some analytical expres-sions that result from interaction with matched-phase ve-locities (d 5 0).
1. Single-Pass InteractionAs discussed above, we assume, for the input fundamen-tal pulse,
r1,in~t ! 5 sech~t/t!, (B1)
and no input second-harmonic signal; we obtain duringthe propagation the following forms: for the F pulse,35
r1,0~u1! 5 sech~u1 /t!, (B2)
r1,2~u1 , j! 5 2sechS u1
t D H j
2stanhS u1
t D1
14s2 lnFcosh~u1 /t 2 2sj!
cosh~u1 /t! G J , (B3)
and for the SH pulse,22,24
r2,1~u2 , j! 5i
2s F tanhS u2
t1 2sj D 2 tanhS u2
t D G .(B4)
2. Double-Pass InteractionIn this case it is more useful to collect the terms of thesame order in h0 , rather than to report the developmentcoefficients as in the case of the single pass. At the out-put of the second pass we obtain, for the fundamental upto the third order in h0 ,
r19~u1 , j 5 1 !
5 r1 exp~iu1!XsechS u1
tD 1 r2 exp~iQ!
h02
4s2 sechS u1
tD
3 ln5 coshF ~u1 1 Dt !
t2 2sG2
coshF ~u1 1 Dt !
tGcoshF ~u1 1 Dt !
t2 4sG 6
1 ~1 1 r12!h0
2r12~u1 , j 5 1 !C. (B5)
Toci et al. Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. B 115
For the second harmonic, up to the second order in h0 ,Eq. (22) becomes
Fig. 8. Time dependence of the phase-modulation coefficient f2and normalized phase modulation fnum (calculated from the nu-merical integration) evaluated at s 5 0.6, (a) d 5 2p, (b) d5 4p, (c) d 5 6p, Q 5 0, Dt 5 2st, r1 5 r2 5 1 for increasingvalues of h0 . Lower frame: residuals (fnum 2 f2).
where r2,1 results from Eq. (B4).
APPENDIX C: COMPARISON WITH THENUMERICAL INTEGRATION RESULTSTo carefully check the accuracy of the series development,we compared the results of the method with the output of
Fig. 9. Time dependence of the phase-modulation coefficient a2and normalized phase modulation anum (calculated from the nu-merical integration) evaluated at s 5 0.6, (a) d 5 2p, (b) d5 4p, (c) d 5 6p, Q 5 0, Dt 5 2st, r1 5 r2 5 1 for increasingvalues of h0 . Lower frame: residuals (anum 2 a2).
116 J. Opt. Soc. Am. B/Vol. 15, No. 1 /January 1998 Toci et al.
a numerical integration of the coupled equations (2). Wedeveloped a computer code for the analysis of cascadedsecond-order optical processes,10 which is based on a fi-nite difference-integration scheme (the two-step Lax–Wendroff method36). This program provides, for a givenF-pulse shape at the input of the crystal, the pulse enve-lopes for the F and SH pulses at the output of the first andsecond passes through the nonlinear crystal.
We verified in particular the accuracy of the expres-sions (19) for the nonlinear phase and amplitude modula-tion experienced by a hyperbolic-secant pulse in a double-pass interaction. Setting the values of r1 , r2 , h0 , d, sfrom the pulse envelopes obtained through the numericalsimulation, we have calculated the following expressions:
fnum 5 arcsin@Im~r19 !/ur19u#/h02, (C1a)
anum 5 2ln~ ur19u/ur10u!/h02. (C1b)
These are the exponential phase and amplitude modu-lations resulting from the interaction, normalized in re-spect to h0
2. This normalization allows an immediatecomparison with the second-order-modulation coefficientscalculated with the analytical procedure. As long as h0is small, the expressions (C1) coincide with the second-order coefficients (19). When h0 increases, the results ofthe numerical simulation separate increasingly from thesecond-order truncation of the series development.
Figures 8 and 9 show the temporal dependence of thesecond-order phase-modulation coefficient f2 and of thesecond-order amplitude-modulation coefficient a2 givenby the expressions (C1) in comparison with the normal-ized phase and amplitude modulation fnum and anumevaluated from the numerical results. These are evalu-ated for s 5 0.6, d 5 2p [Figs. 8(a) and 9(a)], d 5 4p[Figs. 8(b) and 9(b)], d 5 6p [Figs. 8(c) and 9(c)], Q 5 0,Dt 5 2st, for increasing values of h0 . The residuals(fnum 2 f2) and (anum 2 a2) enlighten the truncation er-ror. When h0 increases, the accuracy of the second-orderapproximation given by f2 worsens. Nevertheless, thereis a wide range of values of h0 for which the accuracy ofthe second-order approximation is quite good. Thisrange increases with increasing values for d (for instance,a maximum error in f2 , equal to 10% of the peak value, isobtained at d 5 2p for h0 ' 0.6, but this limit increasesto h0 ' 1.4 for d 5 4p and to h0 ' 1.8 for d 5 6p), show-ing that, for a given accuracy, this method models largerand larger values of the phase and amplitude modula-tions when d increases. Anyway, these limits in h0 aremuch higher than those reported until now in the experi-ments of cw laser mode locking by second-order cascadednonlinearities (h0
2 ' 0.1 in Refs. 12–14).
ACKNOWLEDGMENTThe authors thank Peter White for his critical and patientrevision of the English of this manuscript.
REFERENCES1. H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. La-
gendijk, ‘‘Phase modulation in second-order non-linear-optical processes,’’ Phys. Rev. A 42, 4085 (1990).
2. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E.W. Van Stryland, and H. Varhenzeele, ‘‘Self-focusing andself-defocusing by cascaded second-order effects in KTP,’’Opt. Lett. 17, 28 (1992).
3. D. Pierrotet, B. Berman, M. Vannini, and D. McGraw,‘‘Parametric lens,’’ Opt. Lett. 18, 263 (1993).
4. M. L. Sunderheimer, Ch. Bossard, E. W. Van Stryland, G. I.Stegeman, and J. D. Bierlein, ‘‘Large nonlinear phasemodulation in quasi-phase-matched KTP waveguides as aresult of cascaded second-order processes,’’ Opt. Lett. 18,1397 (1993).
5. G. I. Stegeman, M. Sheik-Bahae, E. V. Van Stryland, andG. Assanto, ‘‘Large nonlinear phase shifts in second-ordernonlinear-optical processes,’’ Opt. Lett. 18, 13 (1993).
6. G. Toci, D. McGraw, R. Pini, R. Salimbeni, and M. Vannini,‘‘Time-dependent analysis of a parametric lens detectedwith a 100-fs Ti:sapphire laser,’’ Opt. Lett. 20, 1547 (1995).
7. F. Hache, A. Zeboulon, G. Gallot, and G. M. Gale, ‘‘Cas-caded second-order effects in the femtosecond regime in b-barium borate: self-compression in a visible femtosecondoptical parametric oscillator,’’ Opt. Lett. 20, 1556 (1995).
8. R. Danielius, A. Dubietis, and A. Piskarkas, ‘‘Linear trans-formation of pulse chirp through a cascaded optical second-order process,’’ Opt. Lett. 20, 1521 (1995).
9. C. R. Menjuk, R. Schieck, and L. Torner, ‘‘Solitary wavesdue to x (2):x (2) cascading,’’ J. Opt. Soc. Am. B 11, 2434(1994).
10. G. Toci, R. Pini, R. Salimbeni, S. Siano, and M. Vannini,‘‘Optical modulators based on second order non-linear pro-cesses in non-stationary conditions,’’ Proceedings of the In-ternational Conference on Lasers, 1995 (STS, McLean, Va.,1996), p. 761.
11. K. A. Stankov and J. Jethwa, ‘‘A new mode-locking tech-nique using a nonlinear mirror,’’ Opt. Commun. 66, 41(1988).
12. M. B. Danailov, G. Cerullo, V. Magni, D. Segala, and S. DeSilvestri, ‘‘Nonlinear mirror mode locking of a cw Nd:YLFlaser,’’ Opt. Lett. 19, 792 (1994).
13. G. Cerullo, S. De Silvestri, A. Monguzzi, D. Segala, and V.Magni, ‘‘Self-starting mode locking of a cw Nd:YAG laserusing cascaded second-order nonlinearities,’’ Opt. Lett. 20,746 (1995).
14. G. Cerullo, V. Magni, and A. Monguzzi, ‘‘Group-velocitymismatch compensation in continuous-wave lasers modelocked by second-order nonlinearities,’’ Opt. Lett. 20, 1785(1995).
15. K. A. Stankov, V. P. Tzolov, and M. G. Mirkov, ‘‘Frequency-doubling mode locker: the influence of group-velocity mis-match,’’ Opt. Lett. 16, 1119 (1991).
16. K. A. Stankov, V. P. Tzolov, and M. G. Mirkov, ‘‘Compensa-tion of group-velocity mismatch in the frequency-doublingmodelocker,’’ Appl. Phys. B 54, 303 (1992).
17. I. Buchvarov, G. Christov, and S. Saltiel, ‘‘Transient behav-ior of frequency doubling mode-locker. Numerical analy-sis,’’ Opt. Commun. 107, 281 (1994).
18. Y. R. Shen, The Principles of the Nonlinear Optics (Wiley,New York, 1984).
19. N. Bloembergen, Nonlinear Optics (Addison-Wesley, Red-wood City, Calif., 1992), and references therein.
20. R. C. Eckardt and J. Reintjes, ‘‘Phase-matching limitationof high efficiency second harmonic generation,’’ IEEE J.Quantum Electron. 20, 1178 (1984).
21. E. Sidick, A. Knoesnen, and A. Dienes, ‘‘Ultrashort-pulsesecond-harmonic generation. I. Transform-limited fun-damental pulses,’’ J. Opt. Soc. Am. B 12, 1704 (1995).
22. E. Sidick, A. Knoesnen, and A. Dienes, ‘‘Ultrashort-pulsesecond-harmonic generation. II. Non-transform-limitedfundamental pulses,’’ J. Opt. Soc. Am. B 12, 1713 (1995).
23. MATHCAD 5.0 Plus, MathSoft, Inc. (Cambridge, Mass., 1995).24. J. T. Manassah, ‘‘Effects of velocity dispersion on a gener-
ated second harmonic signal,’’ Appl. Opt. 27, 4365 (1988).25. H. J. Bakker, P. C. M. Planken, and H. G. Muller, ‘‘Numeri-
cal calculation of optical frequency-conversion processes: anew approach,’’ J. Opt. Soc. Am. B 6, 1665 (1989).
26. K. Kato, ‘‘Second harmonic generation to 2048 Å in
Toci et al. Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. B 117
b-BaB2O4,’’ IEEE J. Quantum Electron. QE-22, 1013(1986).
27. R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, ‘‘Ab-solute and relative nonlinear optical coefficients of KDP,KD*P, BaB2O4, LiIO3, MgO:LiNbO3 and KTP measured byphase-matched second harmonic generation,’’ IEEE J.Quantum Electron. QE-26, 922 (1990).
28. S. P. Velsko, M. Webb, L. Davis, and C. Huang, ‘‘Phase-matched harmonic generation in lithium triborate,’’ IEEEJ. Quantum Electron. 27, 2182 (1991).
29. M. J. Weber, CRC Handbook of Laser Science and Technol-ogy, Vol. 3 (CRC Press, Boca Raton, Fla., 1986), p. 108.
30. G. C. Ghosh and G. C. Bhar, ‘‘Temperature dispersion inADP, KDP, KD*P for nonlinear devices,’’ IEEE J. QuantumElectron. QE-18, 143 (1982).
31. R. L. Sutherland, Handbook of Nonlinear Optics (MarcelDekker, New York, 1996).
32. D. Wearie, B. S. Wherret, D. A. B. Miller, and S. D. Smith,
‘‘Effect of low-power nonlinear refraction on laser beampropagation in InSb,’’ Opt. Lett. 4, 331 (1974).
33. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E.W. Van Stryland, ‘‘Sensitive measurement of optical nonlin-earities using a single beam,’’ IEEE J. Quantum Electron.26, 760 (1990).
34. G. Toci, M. Vannini, and R. Salimbeni, ‘‘Temporal dynamicof the non-linear mirror: an analytical description,’’ Opt.Commun. (to be published).
35. G. Toci, R. Pini, R. Salimbeni, M. Vannini, ‘‘Group velocitymismatch effects in ultrafast optical modulators based oncascaded second order nonlinearities,’’ in Ultrafast Pro-cesses in Spectroscopy, O. Svelto, S. De Silvestri, and G. De-nardo, eds. (Plenum, New York, 1996).
36. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T.Wetterling, Numerical Recipes (Cambridge University,Cambridge, England, 1989).
