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Ref. p. 131]	3.1 Linear optics	103

	x
	d pen
		d G. H.	z
			z

	z
			RP
			RP
		n	n ’
n	n ’

Fig. 3.1.21. Total reﬂection of plane waves with an inhomogeneous wave in the medium with the refractive index n (dpen : amplitude 1/e).

Fig. 3.1.22. Goos–H¨anchen shift of a total reﬂected beam with ﬁnite (exaggerated small) cross section (RP: e ective reﬂection plane).

3.1.5.6.3Reﬂection at media with complex refractive index (Case nˆ = 1 and nˆ = n + i k )

In Fig. 3.1.23 the refractive index n and the attenuation index k of gold (Au) is shown, in Fig. 3.1.24 the reﬂectance for both polarization cases of gold is given.

	8
	8	Au
		Au
Refractiveindex n	6
	6
	Attenuationindex k
	4			k
				k
	2
	2			n
				n
	0
	300	400	500	600	700	800	900

Wavelength [nm]

Fig. 3.1.23. Refractive index n and attenuation index k of gold (Au).

	1.0
	1.0	Au
		Au
		= 400 nm
	0.8
	0.8			R s
				R s
s,p	0.6
R	0.6
R
Reflectance	0.4
Reflectance	0.4				R p
					R p
	0.2
	0.2
	0
	0	30	60		90
		Angle of incidence [°]

Fig. 3.1.24. Reﬂectance for both polarization cases

of gold (Au). There is a minimum of Rp which is connected with a pseudo Brewster angle.

Landolt-B¨ornstein

New Series VIII/1A1

104		3.1.5 Optical materials	[Ref. p. 131
	x
	Planes of
	constant
	amplitude

	T	z
	T
n	n ’ ik ’
	Planes of	Fig. 3.1.25. Refraction at a medium with absorption: generation
	constant
	phase	of an inhomogeneous wave.

Inhomogeneous wave (Fig. 3.1.25): Snell’s refraction law is modiﬁed:

sin ΘT =	n	sin Θ	(3.1.82)

	nT

with

2 n2T = n 2 − k 2 + n2 sin2 Θ + n 2 − k 2 − n2 sin2 Θ 2 + 4 n 2 k 2 (Ketteler’s formula) .

The e ective refractive index nT determines the direction angle ΘT of planes of constant phase in Fig. 3.1.25 via (3.1.82) [88Kle, p. 78], [41Str, p. 503], [99Bor, p. 740]. The full inhomogeneous wave can be calculated using [99Bor, p. 740].

Example 3.1.11. Θ = 45 ◦, Au: λ = 800 nm, n = 0.19, k = 4.9, nT = 0.73, ΘT = 75.1 ◦ (see [28Koe, p. 209]).

Intensity attenuation in the case Θ = 0 ◦:

I = I0 exp {−2

(ω/c) k z} .

−

(3.1.83)

1/e − depth = 13 nm.

◦

, Au: λ = 800 nm, n

= 0.19, k

= 4.9,

I = I

exp

Example 3.1.12. Θ

= 0

7.7

104 z[mm]

p −

|rs|

Ellipsometry: δ

and moduli

|rp|

of the reﬂected light r

exp (i δ

) and r

exp (i δ

)

can be measured. The complex refractive index of a material results [77Azz, 90Roe]. Application: Measurements for the optical constants of metals, semiconductors, and thin-ﬁlm systems.

3.1.5.7 Crystal optics

3.1.5.7.1 Classiﬁcation

The dielectric tensor εr = εij in (1.1.8) is symmetrical and real in the case of a nonabsorbing medium.

In Fig. 3.1.26 vectors connected with wave propagation in crystal optics are depicted. In Table 3.1.8 optical crystals are listed. In Table 3.1.9 three of the eight surfaces for visualization of wave propagation in crystals are presented.

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 131]				3.1 Linear optics		105

E	D
E

	Beam edge
				Fig. 3.1.26. Vectors connected with wave propagation in
		s		crystal optics [99Bor]: s : ray direction unit vector Poynt-
				ing vector E × H, n : unit vector in the normal direction
B, H	Beam edge	n =	k	k and phase planes, orthogonalities: B, H E, D, n, s ;
B, H	Beam edge	n =	k	k and phase planes, orthogonalities: B, H E, D, n, s ;
				E s ; D n.
Table 3.1.8. Optical crystals.

Classiﬁcation:	Refractive index			Optical type	Example	Values of the
system (syngony)	in the main axis			of crystal		refractive index
of crystal	system					for λ = 589.3 nm

triclinic,	nx = ny = nz = nx			biaxial crystal, no	NaNO3	nx = 1.344 ,
monoclinic,				ordinary waves		ny = 1.411 ,
orthorhombic						nz = 1.651
trigonal,	nx = ny = no			positive uniaxial	SiO2	no = 1.544,
tetragonal,	(ordinary			crystal: no < ne	(quartz)	ne = 1.553
hexagonal	index)
	nx = nz = ne			negative uniaxial	CaCO3	no = 1.658,
	(extraordinary			crystal: no > ne	(calcite)	ne = 1.486
	index)
cubic	nx = ny = nz = n			isotropic crystal	NaCl	n = 1.544

Table 3.1.9. Three of the eight surfaces for visualization of wave propagation in crystals.

Surface	Given	Found by construction are the

Index ellipsoid (indicatrix)	normal direction n	D-vectors for the two polarization cases
(one-shell surface)		and the two refractive indices for phase
		propagation
Index surface, wave vector	normal direction n ray directions s, which are perpendicular
surface (two-shell surface)		to the surface for both polarization cases
Ray surface, wave surface, representing	ray direction s	normal direction n, which is perpendicular
Huygens’ elementary wave for both		to the surface
polarization cases (two-shell surface)

Main feature of crystal optics: s is not parallel with n for wave propagation, mostly.

–s is essential for description of the energy propagation (edges of bundles, rays),

–n is essential for description of the interferences of inﬁnite broad waves.

References: [28Szi, 54Bel, 58Shu, 61Ram, 76Fed, 79Wah, 84Yar, 04Ber, 99Pau, 99Bor]. A detailed comparison between that surfaces is given in [28Szi].

Landolt-B¨ornstein

New Series VIII/1A1

4B2

n2θ e

= BA +

106	3.1.5 Optical materials	[Ref. p. 131

3.1.5.7.2 Birefringence (example: uniaxial crystals)

Uniaxial crystals in the plane of incidence:

–Refraction of the normal direction n of wavefronts: The wavevector surface is shown in Fig. 3.1.27.


sin Θo =	n	sin Θ (ordinary wave (ko))			(3.1.84)

	no
(no does not depend on the angle of incidence),
		n
sin Θe =			sin Θ	(extraordinary wave (ke))	(3.1.85)
	nθ e (Θe(Θ))

(ne depends on the angle of incidence).

–Refraction of rays (Poynting vector): se and so are given by tangent construction in Fig. 3.1.28.

–Algorithm for the calculation of ko ( so), ke, se of Fig. 3.1.28 with n, no, ne, η, θ of Fig. 3.1.29 [86Haf]:

n2(n2o − n2e )2 sin2 Θ sin2(2η) 2 B2

n2)2

sin Θ sin (2η)

× n2 sin2 Θ

(n2

n2)2

sin2(2η)

n(no

−

− 1 +

(3.1.86)

(refractive index for the extraordinary wave)

with

A = (n2e − n2o) n2 sin2 Θ cos (2η) − n2o n2e ,

B = n2o + (n2e − n2o) sin2 η ,

where the decision on the ± sign in (3.1.86) can be made by controlling the satisfaction of

n2θ e n2o + (n2e − n2o) sin2(η + Θe) = n2e n2o .

The resulting angles are:
Θo = arcsin(n sin Θ/no) ,							(3.1.87)
	x
Optical axis
				TE
		k o			k e	TM
			o e		k e		z
			o e


	k
Polarization							Fig. 3.1.27. Construction of wavefront birefringence with
TE and TM							Fig. 3.1.27. Construction of wavefront birefringence with
Index n				Indices no and ne			the wavevector surface: The wavefronts show no transversal
Index n				Indices no and ne			limitation.
							limitation.

Landolt-B¨ornstein

New Series VIII/1A1

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Файл: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo .pdf

3.1.5.7 Crystal optics

3.1.5.7.2 Birefringence (example: uniaxial crystals)

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