CHAPTER 3 3.1 Anomalous chromatic dispersion of a chirped mirror Fig. 3.1 The operation principle of a chirped mirror. (a) An initial pulse propagating in dispersive homogeneous media becomes (b) a positive chirped pulse. When such a chirped pulse enters into the chirped mirror at normal incidence, the different frequency components are reflected by different layers where the corresponding band gap appears. (c) After reflection, the negative dispersion introduced by the negative chirped mirror compensates the positive chirp of pulse so that the pulse is restored to its original shape.
(a)
f
f
E (t , z)e
tf
dt = A(, z )e - io t = A(, z ) ei ( , z ) e -iot - (3.1.2) where =- o and is the phase of spectral components. We also define the group delay Tdelay as it
Tdelay =
= GD GDD· ...
(3.1.3)
The group delay (GD) is related to the position of each frequency component along the pulse. The constant term is called the GD and the coefficient at is called the group delay dispersion (GDD). If the GDD is non-zero, it means that the different frequency components of the pulse dephase, and the pulse is broadening and getting chirped in propagation. Starting from Maxwell’s equations, a wave equation for the propagation of the pulse in a dispersive media can be obtained. If we retain terms up to second order in dispersion, we obtain the parabolic equation for the envelopes of pulses, in the reference frame moving with the group velocity or the pulse:
A G 2 2 A i =0 z 2 t 2
(3.1.4)
where G2 is the group velocity dispersion coefficient. This equation is equivalent to the one obtained for beam
Page 12 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015
t
f
f
f
t
(3.1.1)
t
t (c)
where A is the complex envelope, is the phase along the pulse and o is the carrier frequency. The Fourier Transform gives the frequency content of the pulse: E (, z ) =
tt
f
Before starting the analysis of beam reflection from our flat focusing mirror, it is instructive to take a close look at the problem of managing chromatic dispersion for ultrashort pulses. 2 Dispersion occurs in all materials due to the dependence of the refractive index on frequency and affects strongly the evolution of optical pulses. The pulse broaden as they propagate in dispersive medium in the same way as diffraction broadening occurs for an optical beam. Let us consider a pulse propagating along a given direction in space, z. The pulse is a temporal envelope of a plane wave so we do not consider diffraction effects. Such a pulse can be represented as
E (t , z ) = A(t , z )e - io t = A(t , z ) ei ( z ,t ) e - io t
(b)
t
diffraction in paraxial approximation if we substitute time t by transverse coordinate x . In a completely analogous way, as we did previously, we can obtain the solution of this equation in the frequency domain as:
t
A(, z ) = A(, 0) exp
i
G2 2 z 2
(3.1.5)
amplitude This solution shows that the spectral (equivalent to the far-field in diffraction) does not change when a pulse propagates in a dispersive medium but rather the spectral components acquire the phases, which depends quadratically on the frequency. As a consequence of this phase modulation of spectral components, the temporal pulse profile changes its temporal shape, broadening during propagation. In the region of normal GDD (G2 > 0), low frequency components travel faster than the high frequency components so the pulse will be “red” at the leading edge and the “blue” at the trailing edge (Fig. 3.1a). If the material dispersion is anomalous (G2 < 0), we obtain the opposite effect with blue components at the leading edge. Any ultrashort pulse (with broad spectrum) propagating in a dispersive medium becomes chirped and broadened, which is detrimental in many applications of these pulses. Pulse compression and chirp elimination is possible by using a media which provides a negative GDD. One of the most important techniques to compensate the dispersion effects of ultrashort pulses is using the chirped Bragg mirror. A 1D Bragg mirror consists of a multiple layer structure of alternating materials with high and low refractive indices. The optical thickness of each layer is close to a quarter-wavelength, nHlH = nLlL = /4, for operating wavelength λ. The chirping 3,4 implies that the periods vary spatially along the longitudinal direction. The different frequencies of a pulse can penetrate and reflect at different depths in this multi-layered structure. Consequently, each frequency component of the pulse travels different distances and experiences different optical path lengths inside the mirror. If the chirp is correctly designed, anomalous phase modulation can be achieved, where blue frequencies travel shorter distances than red ones. In this way, a flat chirped mirror has the potential to compress the pulse to its initial value which is completely equivalent to suppress diffractions of an optical beam (detailed discussion in Sec. 3.2).