Pulse distortion by dispersion

3.4 Light pulses in dispersive materials

3.4.1 Pulse distortion by dispersion

Let us now discuss the influence of the dispersive contributions in more detail. If the dispersion is independent of frequency, then we obtain the wave equation (2.12) once

3.4 Light pulses in dispersive materials 85 more, in which the velocity of light in vacuum is substituted by the material-dependent phase velocity,

00 = 27cn(vo)vo /c = 2nvo/vo.

Let us first consider the case where /3" = O. Indeed, this case occurs with glass, and one may realize qualitatively in Fig. 3.11 that somewhere between lattice absorption and electronic absorption the curvature of the refractive index must disappear. This happens at a wavelength of A = 1.3 pun, which therefore offers an important window for transmission of information by optical communication. The pulse shape after a propagation length z is obtained from

00

E(z,t) = 7-0 eir3°' foo eioi(u^-voz e(0, e^-i27(vt dv.

Substituting /3'z --> 27rtg , after some algebra this yields the form oc

E(z ,t) = To e iooz

e -i27(v o t f ûo E ( 0 , 0 e -i2Tc(v - vo)(t-to dv -i(27E,,,t- 0 0 z)

=

^{To e E (0, t -}

tg ).

The only effect of dispersion is a delay of the pulse transit time by tg = z/vg , which we interpret as a group delay time. This can be used for the definition of a group velocity vg , which can be associated with a 'group index of refraction' m g :

v 2n dv c

1 1

d o

11 n(w) w

d n(w)

) ng (w) C •

In most applications optical pulses propagate in a region of normal dispersion, i.e. at dn I dw > 0. Then according to Eq. (3.28) it holds that v g < Vc7j = ^{c n(w).}

Red frequency contributions propagate faster in a medium than blue ones, but the pulse keeps its shape as long as the group velocity is constant ('dispersion-free'); this is a favourable condition for optical telecommunications, where a transmitter injects digital signals (IDA currents') in the form of pulses into optical waveguides, which have to be decoded by the receiver at the other end. In optical fibres this situation is similar to that in BK7 glass at A = 1.31.im, which can be seen in Fig. 3.13 for zero passage of the material parameter M(A) and will be discussed in the next section.

Example: Phase and group velocities in glasses

We can use the specifications from Tab. 3.1 to determine the index of refraction and the group refractive index as a measure of the phase velocity and group velocity in important optical glasses. The wavelength 850nm is of substantial importance for working with short laser pulses, because, on the one hand, GaAs diode lasers with high modulation bandwidth exist in this range (up to pulse durations of 10 Ps and less) and, on the other, the wavelength lies in the spectral centre of the Ti-sapphire laser, which is nowadays the most important primary oscillator for ultrashort laser pulses of 10-100 fs and below. There, with the Sellmeier formula (1.6) and the coefficients (3.28)

Tab. 3.1: Indices of refraction of selected glasses.

Abbreviation BK7 SF11 LaSF N9 BaK 1 F 2

Index of refraction at 850nm

1.5119 1.7621 1.8301 1.5642 1.6068 Group index of refraction

ng 1.5270 1.8034 1.8680 1.5810 1.6322 Material dispersion

cM(A) (.1m -1 ) —0.032 —0.135 —0.118 —0.042 —0.075

from Tab. 1.1, we calculate the values for Tab. 3.1. The values for the group refractive index are always larger than the values of the (phase) refractive index by a few per cent.

For shorter and shorter pulses, the bandwidth increases according to eq. (3.25), and the frequency dependence of the group velocity influences the pulse propagation as well. This is specified as a function of frequency or wavelength by one of two parameters: the group velocity dispersion (GVD) D v (v) and the material dispersion parameter MN:

Du( v ) _ 1 d2 A (27 ) 2

M

( ) - ddA vlg

d ( 1 )

day

^Vg

2

2wnc Dv (v)' Like before, we gain the pulse shape from

E(z,t) = e -i(wot- 00z)

X

f 00 ^6.0,0

^eipv(w - w0)2 z/ 2 C i(w - u-'° )(t-tg ) 27E • -

This time the pulse is not only delayed, but also distorted in shape. We cannot specify this modification generally any more, but we have to look at instructive examples.

Example: Pulse distortion of a Gaussian pulse

At z = 0 the optical pulse E(0,

t) = E0 e21

2(t/t ) 2 e—ot with intensity half-width tp has the spectrum

s _(O , w)

⁼ eo e —[(w—wo)t p ] 2 /81n 2 .

At the end of the propagation distance at z = f, the spectrum is deformed according to eq. (3.29). For the sake of simplicity we introduce fp = t p2 /41n 2D, and find

e0 e—[(w—wo)tp]2/81n 2 e i(UeD)[(w—wo)t p 1 2 /81n 2 .

Inverse Fourier transformation yields the time-dependent form E(e,t) = ^{70E0 e}-i(2nvot-00,e)

2 exp f 21n 2(t - t g ) 2 x exp t2p[1 (wD)

fiD t2p [1 + (L/eD) 2 } ) •

(3.29)

3.4

Light pulses in dispersive materials

87

Hence not only is the pulse delayed by tg , but it is also stretched,

e p ( ^{z=f) = tp} V 1 + (fteD) 2 , (3.30)

and furthermore the spectrum exhibits the so-called 'frequency chirp', where the fre- quency changes during a pulse:

1

d

1 t — zlv g

v(t) = —4)(t) = +

2n

dt

it

eD ti23 [1 + (/ eD) 2 ] .

Fig. 3.14: Pulse distortion manifests itself as pulse broadening and frequency chirp. The red frequency components run ahead (left-hand part of the pulse), whereas the blue ones lag behind

(right-hand part). The neither distorted nor delayed pulse is also indicated for comparison.

Now we can determine how far a pulse propagates within a material without sig- nificant change of shape. For example, according to Eq.

(3.30)

it holds that the pulse duration has increased at

= ^(3.31)

by a factor of

.\/.

This propagation length is also called the 'dispersion length' and plays a similar role in the transmission of pulses as the Rayleigh zone does for the propagation of Gaussian beams (see p.

40).

For

BK7

glass from Tab.

3.1

it holds that

D(A=850 nm) = 0.04 ps 2 /m.

Then one finds for a GaAs diode laser and a conventional Ti—sapphire laser

GaAs diode laser:

tp = 10

Ps

fp=--- 200 m,

Ti—sapphire laser:

tp = 50 fs fp = 5 mm.

It turns out that a short pulse is heavily distorted even by a

5mm BK7

glass window!