Ref. p. 131]	3.1 Linear optics	107

	Elementary
	waves
	se	Extra-
	Rays	ordinary
	so
	Wavefronts
		Ordinary
	Optical
	axis
Index n	Indices no and ne	Index n

	x
Optical axis	k oII	so
			k e
			se
	o
			e
			ew
			z
k II s
Index n	Indices no and ne

Fig. 3.1.28. Huygens’ tangent construction of bire-

Fig. 3.1.29. Refraction for normal and ray direc-

fringence in a crystal slab for transversal-limited

tions. η : angle between z-axis and optical axis.

beams.

Θe = arcsin (n sin Θ/nΘ e) ,

(3.1.88)

Θew = arctan

tan η − C

(3.1.89)

1 + C tan η

with

n2

nΘ2 e

−

n2 sin2 Θ tan η + n sin Θ

C =

o

×

.

n2

2

− n2 sin

2

e

nΘ e

Θ − n sin Θ tan η

Application: Θe, Θo, no, and nΘ e phase di erences (interferences) and reflection coe cient, Θo and Θew ray separation in a crystal.

Example 3.1.13. Calcite: no = 1.658, ne = 1.486, η = 45◦: #1: Θ = 0◦: C = 1.244822, nΘ e = 1.565,

Θo = Θe = 0◦, Θew = −6.224◦; #2: Θ = 45◦: nΘ e = 1.636, C = 0.438329, Θo = 25.23◦, Θe = 25.6◦, Θew = 21.33◦.

General formulation of (3.1.85)–(3.1.89): see [76Fed, Table 9.1] for more detailed discussions.

3.1.5.8 Photonic crystals

Starting with the forbidden (stop) bands in case of multi-layer Bragg reflection [88Yeh, p. 123] a material class is under development which stops light propagation along as many directions and for as many wavelengths as possible. This suppresses the spontaneous emission for laser applications and opens new possibilities in the microand nano-optics [95Joa, 01Sak, 04Bus]. Photonic crystal fibers [04Bus] can be designed for special light propagation properties and high-power fiber lasers [03Wad].

Landolt-B¨ornstein

New Series VIII/1A1