Файл: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo .pdf
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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 |
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Θ, Θ , nˆ, and nˆ |
Θ and Θ only |
nˆ |
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are considered |
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n¯ = |
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nˆ |
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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
99
100 |
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3.1.5 Optical materials |
[Ref. p. 131 |
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x |
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k” |
H” |
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k” |
E’ |
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k’ |
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H” |
k’ |
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E’ |
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E” |
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E” |
H’ |
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H’ |
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z |
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k |
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E |
k |
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E |
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H |
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H |
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a |
Index n |
Index n ’ |
b |
Index n |
Index n ’ |
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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 : |
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nˆ is real and nˆ is complex (absorption [76Fed, 77Azz] or gain [88Boi]). |
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nˆ |
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< 1 and (¯n2 − sin2 Θ) < 0 (total reflection). Then n¯2 − sin2 Θ = |
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nˆ and nˆ are real and n¯ = |
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nˆ |
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i |
sin2 Θ − n¯2 |
yields for (3.1.68) and (3.1.69) rs = exp (i δs) and rp = exp (i δp) (modulus = 1, |
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all energy reflected) and tan |
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= |
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sin2 Θ − n¯2 |
and tan |
δp |
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sin2 Θ − n¯2 |
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cos Θ |
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n¯2 cos Θ |
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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): |
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Rs,p = |rs,p|2 . |
(3.1.73) |
Landolt-B¨ornstein
New Series VIII/1A1
Ref. p. 131] |
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3.1 Linear optics |
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Transmittance (transmitted part of intensity): |
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Ts,p = |
Re (ˆn cos Θ ) |
|ts,p| |
2 |
Re (ˆn cos Θ) |
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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)
r |
s |
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n − n |
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p |
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(The negative sign of r |
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results from the convention of Fig. 3.1.18 that E |
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is di racted into |
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E ). |
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n − n |
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and T |
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n + n |
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Example 3.1.8. n = 1, n |
= 1.5 (glass): Rs = 0.04 . |
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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 :
n
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 |
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s,p |
0.4 |
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r |
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coefficient |
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0.4 |
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Reflectance |
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Angle of incidence [°] |
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1.0
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R s
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Fig. 3.1.19. (a) Reflection coe cients rp and rs and (b) reflectances Rp and Rs for n = 1 and n = 1.5.
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coefficient |
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Angle of incidence [°] |
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π
s,pPhase
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90
s,p R
Reflectance
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0.4 |
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0.2 |
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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 = |
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n2 sin2 |
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2 π |
Θ − n 2 |
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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