APORTE CIENTÍFICO DE LA INVESTIGACIÓN.

If the influence of the group velocity dispersion in the SHG crystal is neglected and by assuming that the pattern of the spatial distribution of the field is time-independent (Mindl et al 1983), the detected second harmonic signal from a SHG autocorrelator can be expressed as

I2(B(t)~Gi = Jl2(t,-c)dt (4.9)

-oo

where I(t,x) is given by

I(t,x) = [V(t) + V(t+T)] [V(t) + V(t+T)]* (4.10)

The time integral in Eq. (4.9) represents the accumulating effect caused by the limited response time of the detector. Provided that the measured pulses are coherent, i. e.

V(t) = ACOeiWiox (4.11)

where <j)0 is a constant, substitution of Eq. (4.11) into Eq, (4.10) and then bringing the resultant expression for I(t,x) into Eq. (4.9), it can be deduced that: .

Gi(t,co) = G2bCt) + Gf(CO,T) (4.12a) where +OO G2b« = J[A4(t)+A4(t+r) + 4A2(t)A2(t+T)]dt (4.12b) -oo +OO

Gf(co,t) = J {2cos(2coT)A2(t)A2(t+ t) + 4cos(Gn)[A3(t)A(t+T) + A3(t+T)A(t)]}dt

-oo

(4.12c) Eq. (4.12a) is the general expression of the so-called interferometric autocorrelation, which contains two components. The first one expressed by Eq. (4.12b) is solely a function of time delay, while the second given by Eq. (4.12c) relies on both the time delay and the carrier frequency. If the detector system cannot follow the oscillations characterised by cos(cot), the second component will be averaged out, i. e. (Gf)avg = 0, and so only the slowly changing portion, G2b00> is recorded. G2bC0 is widely referred to

as the intensity autocorrelation (the non-background-free type)6, which can be rewritten as 2jl(t)I(t + T)dt G2BW = 2jl2(t)dt = C[l+2g2(x)] (4.13) 1 +

where I(t) = A2(t) and the indices for integral range (from to +o°) are omitted. Since g2(0) = 1 and g2(x »2At) = 0, G2b(^) has a peak-to-background ratio of 3:1. In some experimental arrangements only g2(x) is recorded. This was achieved by either polarizing the two replica pulses orthogonally (Weber 1967) or making them have a noncollinear incident angle (Ippen and Shank 1977). To differ from G2B in names, g2(x) is termed as background-free intensity autocorrelation. In a form similar to Eq. (4.13), Eq. (4.12a) can be reexpressed as

„ „ L r„ „ „ , ' „ , /2jA3(t)A(t+x)dt 2jA3(t+r)A(t)dt\

Gi = C< 1+ [2 + cos(2cox)]g9(x) + 2cos(cox)--- ---

+ ——p——— I

I jA4(t)dt jA4(t)dt J)

(4.14) It is straightforward to derive that the interferometric autocorrelation function Gi has a peak-to-background ratio [ Gi(0)/Gi(x »2At)] of 8:1. The fringes have maxima when cox

= 2u7t and minimas at cox = (2u+1)k. For the first minimum value (n = 0), cox = rc and G ~ 0. As I x I increases the maxima decrease while the minima increase. They finally

merge at the background level. The period of the fringes displayed on the time delay axis is given by Tf = 27t/co, which corresponds to one optical cycle of the carrier. This feature

leads to a special use of interferometric autocorrelation which is that the timescale on the screen can be conveniently calibrated by counting the number of the fringes in each division.

If the pulses to be measured are frequency chirped, it can be shown that Eqs. (4.12) are still valid except that in these cases co is a function of both t and x, and consequently,

d dd> d2<b

co = co(t, x) = co0 [0(t + x) - 0(t)] = w0 + + dt“X + - (4.15)

It should be pointed out that the designations of interferometric autocorrelation and intensity autocorrelation sometimes give rise to some confusion. They are both autocorrelations of optical intensity, the only difference between them is simply that one has fringes resolved with better time resolution whereas the other is fringe-averaged and therefore only sensitive to the pulse profiles.

The oscillatory terms cos(2cox), cos(cor) thus cannot be taken out of the time integral as in Eq. (4.14). An interesting feature implied by Eq. (4.15) is that for chirped pulses the fringes within the interferometric autocorrelation trace are no longer uniform. Instead, the variation of the fringe frequency with respect to the delay time follows the same pattern as the carrier chirp in the time domain. This is the basic reason why Diels et al (1985) concluded that the characteristics of the pulse phase or the carrier can be included in the interferometric autocorrelation.

Example II. Interferometric autocorrelations for chirped and unchirped pulses For a linear chirped Gaussian pulse, suppose

V(t) = e-at2ei<Kt)e-iwot (4.16)

where a = 21n2/(At)2, <X0 = bt2 (b is the chirp parameter), and so

to(t, t) = (Oq + 2bt + 2bx (4.17)

Bringing Eq. (4.16) into Eq. (4.12) produces (Saia et al 1980):

gj = = 1 + 2e‘aT2+ cos(2a>oX)e‘(l+^2/a2)aT2+ 4cos((o0x)cos(b2r2/4a)e'(3+b2/a2)aT2/4 (4 jg) The corresponding intensity autocorrelation is given by

g2B = 52EW = , + 2e-at2 (4,19)

Note that the intensity autocorrelation does not indicate whether the pulse is chirped or not. In Eq. (4.18), let b = 0, the interferometric autocorrelation for a chirp-free Gaussian pulse can be obtained, which is

gj = 1 + e_aT2[2 + cos(2cqot:)] + 4cos((doT)e"^at2/4 (4.20) Two calculated results following Eq. (4.18) and Eq. (4.20) for At = 75 fs, b/a = 20, to0 = 1-25 X 1015 Hz (for X = 1.5 pm) are shown in Fig. 4.8(a), (b) respectively, where corresponding intensity autocorrelation traces are also shown. As it is expected, for both the chirped and unchirped pulses, whose electric fields have been shown in Fig. 2.2, the intensity autocorrelations are the same while the interferometric counterparts are dramatically different. For the chirped case, because a large value has been assumed for the chirp parameter, the interferometric fringes are only visible in the central part of the traces. If we reassume b/a = 2 the corresponding interferometric autocorrelation trace will be the one reproduced in Fig. 4.8(c). Two experimentally recorded autocorrelation traces, one for unchirped pulses and one for chirped pulses, are presented in Fig. 4.9(a) and (b) respectively. A qualititive comparision of these traces with the calculated data in Fig. 4.8(a), (c) shows satisfactory agreement

Fig. 4.8. Calculated intensity and interferometric autocorrelation traces for 75 fsec Gaussian pulses, (a) chirp-free, (b), (c) chirped, where the chirp parameter b = 20a, 2a respectively.

Fig. 4.9. Experimentally recorded autocorrelation traces for the coupled-cavity mode-locked KC1:T1 colour centre laser pulses, (a) for main output pulses, (b) for pulses out of the control fibre.

The upper and lower envelopes of the interferometric autocorrelation traces can be obtained simply by setting cor in Eq. (4.12c) equal to 2utc and (2u+l)rc respectively. For the traces shown in Fig. 4.8, we have

(gl)extemal-envelope = 1 + 2e^2+ e-(l+b2/a2)a'C2+ (4.21a)

(g,)intemal-envelope = 1 + 2e™2+ e-d+b2/a2)ax2.4e-(3+b2/a2)at2/4 (4.21b) Thus, the width of the interferometric autocorrelation can be estimated from Eq. (4.21a). With the left-hand-side of Eq. (4.21a) equal to 4, we obtain

3 = 2e'a(T2c/2)2 + e-a(l+b2/a2)(X2c/2)2 + 4€-a(3+b2/a2)(T2c/2)2/4 (4.22) where T2c is the FWHM of the interferometric autocorrelation trace. From Eq. (4.22), it can be seen that T2c is frequency-chirp dependent, and for a given pulse duration a fixed relationship between X2c and the chirp parameter b can be established. In general, as demonstrated in Fig. 4.10, the larger the frequency chirp the smaller is the FWHM duration of the interferometric autocorrelation. (This can be understood by the fact that as the pulses become heavily chirped in a linear form, the leading part and the trailing part are no longer coherent, which leads to a wash-out of the fringe visibility in the wings of the interferometric autocorrelation trace and so a decrease of X2c)- For a known pulse shape, such a feature can be used to evaluate the magnitude of the chirp in the pulses, when a linear chirp is assumed. From Eq. 4.21(a), if b = 0, we have X2c = 1.696At (For chirp-free sech2 pulse shape, T2c - 1.897At), otherwise X2c <1.696At.

Fig. 4.10. Envelopes of interferometric autocorrelation traces for the Gaussian pulses which have the chirp parameters b = 0,2a, 20a.

As a measure of interference, T2c can be taken as a second-order coherence time, which may be compared with the parameter Tc described in Section 2.1. A common property of t2c and Tc is that they are both frequency-chirp related, except that T2c is expected to be more sensitive owing to the fact that it describes a higher coherence order. (As far as the statistical feature of the field is concerned, it is possible that for two sequence of pulses they may have the same Tc but their X2c's may be quite different). A similar argument applies for yet higher-order coherence times, which may be derived from the Uth order autocorrelation (tv > 2) (Saia et al 1980). However, in practice, the high order autocorrelations are often difficult to achieve because of the requirement on higher pulse power and more complicated optical arrangements.

Finally, it needs to be emphasised that because T2c is dependent upon both frequency- chirp and pulse phase, the actual duration of the measured pulses should be always deduced from the FWHM width of the intensity autocorrelation rather than the envelope of the interferometric one. In the cases where the pulse coherence is wanted to be checked, a comparison of At’s deduced respectively from intensity and the interferometric autocorrelations can be made. If the two values are the same the pulses are coherent, otherwise, (the one derived from i2c is smaller than that from Tj), implying that the pulses are not perfectly coherent.