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

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B¨ornstein-Landolt

VIII/1A1 Series New

Table 3.1.7. Fresnel’s formulae for the amplitude (ﬁeld) reﬂection and transmission coe cients.

Case	The four values	Using the angles	sin Θ		is eliminated
	Θ, Θ , nˆ, and nˆ	Θ and Θ only	nˆ
	are considered		nˆ
	are considered		n¯ =
			n¯ =	nˆ

Reﬂection

rs = Es = Es

Reﬂection

rp = Ep = Ep

Transmission

ts = Es = Es

Transmission

tp = Ep = Ep

Application of cases

nˆ cos Θ − nˆ cos Θ

nˆ cos Θ + nˆ cos Θ

nˆ cos Θ − nˆ cos Θ

nˆ cos Θ + nˆ cos Θ

2ˆn cos Θ

nˆ cos Θ + nˆ cos Θ

2ˆn cos Θ

nˆ cos Θ + nˆ cos Θ

Mostly used for

pure dielectric media.

sin(Θ − Θ ) − sin(Θ + Θ )

tan(Θ − Θ )

tan(Θ + Θ )

2 sin Θ cos Θ

sin(Θ + Θ )

2 sin Θ cos Θ

sin(Θ + Θ ) cos(Θ − Θ )

In a stack of ﬁlms, the angles to the axis were calculated previously.

cos Θ − n¯2 − sin2 Θ

cos Θ + n¯2 − sin2 Θ

(3.1.68)

n¯2 cos Θ − n¯2 − sin2 Θ

n¯2 cos Θ + n¯2 − sin2 Θ

(3.1.69)

2 cos Θ

cos Θ + n¯2 − sin2 Θ

(3.1.70)

2¯n cos Θ

cos Θ + n¯2 n¯2 − sin2 Θ

(3.1.71)

See remark

in Sect. 3.1.5.5.

131] .p .Ref

optics Linear 1.3

100				3.1.5 Optical materials			[Ref. p. 131
		x				x
	k”	H”			k”	E’
		H”	k’		H”	E’	k’
		E’	k’				k’

	E”
	E”				E”	H’
		’			E”	H’
		’	H’			’
			z				z
	k			E	k
E				E
E					H
	H				H
	H
a	Index n	Index n ’		b	Index n	Index n ’

Fig. 3.1.18. Refraction at an interface, represented in the plane of incidence: (a) Es-case, (b) Ep-case. The commonly used convention is shown for the orientation of the relevant vectors (k: the wave number

vector, E: the electrical ﬁeld, and H: the magnetic ﬁeld) ensuring that k, E, and H are a right-handed system in every case. The E-ﬁeld is important for the action on a nonmagnetic material.

Polarization:

–E perpendicular to the plane of incidence: s-polarization (TE-case or σ-case [88Kle]), the corresponding E-component is called E [99Bor] or Es (s: “senkrecht” (German) which means “perpendicular”) [88Yeh] or index E [97Hua] or index x [90Roe, 77Azz, 91Sal].

–E parallel to the plane of incidence: p-polarization (TM-case or π-case [88Kle]), the corresponding E-component is called E [99Bor] or Ep [88Yeh] or index M [97Hua] or index y [90Roe, 77Azz, 91Sal].

Snell’s law :

nˆ sin Θ = nˆ sin Θ

(3.1.72)

with

nˆ, nˆ : refractive indices of both media, respectively, Θ, Θ : see Fig. 3.1.18.

Other convention than Fig. 3.1.18b [58Mac, 89Gha, 91Ish] (electrical engineering) on the orientation of the E-vectors: E and E point into the same direction for Θ → 0, H changes sign; application: E- interferences.

Remark :

–

nˆ is real and nˆ is complex (absorption [76Fed, 77Azz] or gain [88Boi]).

nˆ

< 1 and (¯n2 − sin2 Θ) < 0 (total reﬂection). Then n¯2 − sin2 Θ =

–

nˆ and nˆ are real and n¯ =

nˆ

sin2 Θ − n¯2

yields for (3.1.68) and (3.1.69) rs = exp (i δs) and rp = exp (i δp) (modulus = 1,

all energy reﬂected) and tan

δs

−

sin2 Θ − n¯2

and tan

δp

−

sin2 Θ − n¯2

cos Θ

n¯2 cos Θ

The intensities in the media are calculated with help of the z-component of Poynting’s vector [88Kle, 90Roe, 76Fed].

Reﬂectance (reﬂected part of intensity):
Rs,p = \|rs,p\|2 .	(3.1.73)

Landolt-B¨ornstein

New Series VIII/1A1

ΘB + ΘB = 90◦ ,

Ref. p. 131]			3.1 Linear optics
Transmittance (transmitted part of intensity):
Ts,p =	Re (ˆn cos Θ )	\|ts,p\|	2
	Re (ˆn cos Θ)

with

Re : real part.

Energy conservation:

Ts,p + Rs,p = 1 .

101

(3.1.74)

3.1.5.6 Special cases of refraction

3.1.5.6.1 Two dielectric isotropic homogeneous media (nˆ and nˆ are real)

n − n

−

(3.1.75)

n + n

(The negative sign of r

results from the convention of Fig. 3.1.18 that E

is di racted into

−

E ).

n − n

= R

and T

= T

= 1

n + n

−

Example 3.1.8. n = 1, n

= 1.5 (glass): Rs = 0.04 .

3.1.5.6.2 Variation of the angle of incidence

3.1.5.6.2.1 External reﬂection (n < n )

Brewster’s angle (angle of polarization) ΘB :

Rp = 0 , tan ΘB = n .

Example 3.1.9. n = 1, n = 1.5, ΘB = 56.3◦ . See Fig. 3.1.19.

3.1.5.6.2.2 Internal reﬂection (n > n )

Critical angle of total reﬂection:

n sin ΘC = n .

Total reﬂection: Θ > ΘC with |rs| = |rp| = 1 and the phases of the reﬂected waves: rs and rp = exp (i Φp) .

Brewster’s angle:

n tan ΘB = n .

(3.1.76)

(3.1.77)

(3.1.78)

= exp (i Φs)

(3.1.79)

Landolt-B¨ornstein

New Series VIII/1A1

102	3.1.5 Optical materials	[Ref. p. 131

	1.0
	1.0
	0.8
	0.6
s,p	0.4
r	0.4
coefficient	0
	0.2
Reflection	0.4
	0.2
	0.6
	0.8
	1.0
	0

		s,p
	rp	R
	rp	Reflectance
	rs	Reflectance
	rs

10	20	30	40	50	60	70	80	90
		Angle of incidence [°]						b

1.0

0.8

0.6

0.4

R s

0.2

R p

0	10	20	30	40	50	60	70	80	90
			Angle of incidence [°]

Fig. 3.1.19. (a) Reﬂection coe cients rp and rs and (b) reﬂectances Rp and Rs for n = 1 and n = 1.5.

	1.0
	1.0
	0.8
	0.6
s,p	0.4
r	0.4
coefficient	0
	0.2
Reflection	0.4
	0.2
	0.6
	0.8
	1.0
	0

p s

10	20	30	40	50	60	70	80
		Angle of incidence [°]

s,pPhase

s,p R

Reflectance

1.0
1.0
0.8


0.6


0.4


0.2

			R s		R p
			R s		R p
0
0	10	20	30	40		50	60	70	80	90
			Angle of incidence [°]

Fig. 3.1.20. Internal reﬂection (n = 1.5, n = 1). (a) Reﬂection coe cients rp and rs for Θ < ΘC and phases Φp and Φs for Θ > ΘC. (b) Reﬂectances Rp and Rs (= 1 for Θ > ΘC).

Example 3.1.10. n = 1.5, n = 1, ΘC = 41.8◦, ΘB = 33.7◦. See Fig. 3.1.20.

Penetration depth in Fig. 3.1.21 [88Kle, p. 67]:

dpen =	λ0		.	(3.1.80)

	n2 sin2
2 π		Θ − n 2

Goos–H¨anchen shift [88Yeh, p. 74], see Fig. 3.1.22:

dG.−H.,s,p =	d Φs,p	(3.1.81)
	d Θ

with Φp and Φs from Fig. 3.1.20. For a more precise treatment of the Goos–H¨anchen shift for Gaussian beams see [05Gro1, p. 100].

Landolt-B¨ornstein

New Series VIII/1A1

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

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