CHAPTER 3 3.3 Lateral shift and Focal length The chapter 2 proposed that effective propagation length L that can compensate a diffracted beam turning its front into a flat profile is determined by the curvature of the spatial dispersion line. The more curved is the anomalous dispersion, the more far away is the focal length because it needs more distance in free space to compensate the negative phase profile. It can be proven that the concave curvature provides a negative diffraction, with the diffraction coefficient adiffr (corresponding to the curvature)
adiffr = -
1 2 2 k x 2
Eq. (3.3.1) controls how the angular components dephase during the propagation in the unit length. In the case of reflections, the diffraction coefficient can also be considered and the focal length can be calculated by the formula in term of lateral shift s = - kx .
F = k0
2 s = - k0 2 k x k x
It brings to two consequences: Firstly, the second order derivative, equivalent to the curvature of dispersion, should be positive as we mentioned in sec. 3.2. Secondly, the slope of lateral shift should be negative. A simple geometrical approach is proposed for a insight of lateral shift and focal length. To simplify the treatment, the bandgap of a chirped mirror is supposed to sweep linearly with the longitudinal distance z as shown by black dashed
line in Fig. 3.3c. With such approximation, a numerical lateral shift and diffraction length can be calculated: 16 s=
-d 2 1 k2 k = ( k - k0 - x ) x dk x a C 2 k k
L = -k
-d 2 2 3 k x2 = ( k k ) 0 aC 2 k dk x2
The analysis of equation (3.3.4) shows that the diffractive propagation length L is always positive at small angles (as the optical axis kx = 0) for a chirped structure. However, it can become negative at some large incidence angles only for a positive chirped mirror. This means that the beam reflecting at some sufficiently large angle should experience zero or negative diffraction. Therefore, the sign of the chirp needed for the flat focusing mirror is proposed to be positive, opposite to that of the chromatic dispersion compensating chirped mirror used for pulses, spreading in normal dispersion media. The oscillation effect in the chirped mirror does not appear in estimations (Eq. 3.3.4 and 3.3.5) in the simple geometric approach. It comes out that due to this fringing, the negative diffraction length can be obtained for both types of chirped mirrors at the angular values where the resonant peaks appear, which is very different from the prediction of the analytical model. In fact, the fringes even increase the effect of the negative diffraction at particular angular ranges where Focal length becomes strongly positive, also well beyond the estimations of our rough analytical model.
Page 15 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015