94 |
3.1.4 Di raction |
[Ref. p. 131 |
3.1.4.5.2 Fresnel’s di raction through lens systems (paraxial di raction)
Given: a system of lenses and the field distribution E(x, y) to be propagated.
The sequence of steps easily taken is: |
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– |
Given: E(x, y) in the plane z = 0. Required: the field in the plane z = z. Solution: (3.1.39). |
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– |
Given: E(x, y) in the plane z = 0 and near to this plane a lens. Required: the field in the plane |
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z = z. Solution: modification of (3.1.39) by an additional factor L(x , y ) to: |
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E |
Fre |
(x, y, z) = |
i exp {−i kz} |
A |
E(x , y , 0) L(x , y ) |
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λd |
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× exp −i π |
(x |
− |
x |
) |
2 + (y |
− |
y |
)2 |
d x d y , |
(3.1.54) |
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λ z |
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2 + y |
2) |
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L(x , y ) = p(x , y ) exp {−i kn tL} exp |
i k(x |
(3.1.55) |
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2f |
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with
n : refractive index of the lens, tL : thickness of the lens,
f : focal length of the lens,
p(x , y ) : amplitude part, which can describe a marginal aperture or a Gaussian apodization.
A general complex function L(x , y ) can model di ractive optical elements.
Cases of integration:
–No transversal limitations (without stops) and quadratic arguments of the exponential functions due to analytical results. The Collins integral is the closed form of such a calculation (see Sect. 3.1.7.4).
–One stop (finite integration limits): The result includes the error function [70Abr].
–Two and more finite integration limits are not useful. Then, (commercial) numerical field propagation programs through systems should be consulted.
Examples: [68Goo, 91Sal, 71Col, 85Iiz, 92Lug, 68Pap].
The Beam Propagation Method (BPM) in integrated optics (many “infinitely thin lenses”) is the generalization of this method [95Mae, 91Spl, 99Lau, 98Hec].
3.1.4.6 Fourier optics and di ractive optics
Fourier optics results from the transformation of the temporal frequency methods of electrical engineering to spatial frequency methods in optics, see Figs. 3.1.9, 3.1.10 and (3.1.41), (3.1.43), (3.1.44).
References: principles of Fourier optics: [68Goo, 78Loh, 83Ste, 85Iiz, 89Ars, 93Sto, 98Hec, 99Lau], filtering: [92Lug], filtering in connection with holography: [96Har, 71Col], noise suppression: [91Wyr].
Example 3.1.5. Spatial spectral filtering
In Fig. 3.1.16 low-pass filtering of a laser beam with a four-f -setup is shown.
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Ref. p. 131] |
3.1 Linear optics |
95 |
Incident beam |
Low-pass |
Filtered beam |
||
Lens |
filter |
Lens |
||
High |
||||
frequency |
||||
f |
f |
f |
f |
Fig. 3.1.16. Low-pass filtering of a laser beam with a four-f -setup [92Lug]. The mask is a low-pass filter, which transmits a zero mode only and suppresses the higher modes. The incident beam can also be modified by a transmission element which changes amplitude and phase.
Di ractive optical elements influence the propagation of light with help of amplitudeand/or phasechanging microstructures whose dimensions are of the order of the wavelength mostly. They extend the classical means of optical design. References: [67Loh, 84Sch, 97Tur, 00Tur, 00Mey, 01Jah].
Example 3.1.6.
–Gratings generated by mechanical or interference ruling [69Str, 67Rud] on either plane or concave substrates for the combination of dispersive properties with imaging [82Hut, 87Chr].
–Fresnel’s zone plates acting as microoptic lenses of [97Her].
–Mode transformation optics (“modane”) for transformation and filtering of modes of a laser [94Soi].
–Generation of theoretical ideal wavefronts for optical testing with interferometrical methods [95Bas, Vol. II, Chap. 31].
–Mode-discriminating and emission-forming elements in resonators [94Leg, 97Leg, 99Zei].
For pure imaging applications, refracting surfaces are still preferred, even in the micro-range [97Her]. Tasks with special dispersion requirements and special optical field transformations are the main application of the di ractive elements with increasing share.
The technology of dispersion compensation and weight reduction in large optical systems by special di ractive elements is partially solved, now.
Medium with absorption: |
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εˆr = nˆ2 |
(3.1.56) |
|||||
with |
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εˆr : complex relative dielectric constant (or tensor), |
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nˆ : complex refractive index, |
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weakly absorbing isotropic medium: |
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α k0 : nˆ = n − i ke = n − i n κ = n − i |
α |
, |
(3.1.57) |
|||
2 k0 |
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damped plane wave (unity field amplitude): |
n − i 2 k0 |
z |
||||
exp {−i kz} = exp {−i k0(n − i κ) z} exp −i k0 |
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α |
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= exp −i k0z − |
α |
|||||
z , |
(3.1.58) |
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2 |
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96 |
3.1.5 Optical materials |
[Ref. p. 131 |
intensity: |
||
I(z) = I(0) exp {−α z} |
(Lambert-Beer-Bouguer’s law ) , |
(3.1.59) |
amplification in pumped media: |
||
I(z) = I(0) exp {g z} |
(3.1.60) |
|
with
α [m−1] : (linear) absorption constant (standard definition [95Bas, Vol. II, Chap. 35],
[99Bor, 91Sal, 96Yar, 05Hod]) or extinction constant or attenuation coe cient, g [m−1] : gain,
ke [m−1] : [88Yeh, 95Bas] (or κ [m−1] : [99Bor, 04Ber]) extinction coe cient, attenuation index.
Di erent convention after (3.1.6): α, g, ke and κ are defined with other signs, for example nˆ = n (1 + i κ) if the other time separation (1st convention) is used [99Bor, Chap. 13], [95Bas, Vol. I, Chap. 9].
Measurement of α : see [85Koh, 04Ber, 82Bru], [90Roe, p. 34], [95Bas, Vol. II, Chap. 35].
In Fig. 3.1.17 the realand imaginary part of the refractive index in the vicinity of a resonance in the UV are shown.
Single-resonance model for low-density media [99Bor, 96Ped]:
N e2
nˆ = n − i ke = 1 + 2 ε0m (ω02 − ω2 + i γ ω)
= 1 + |
N e2γ |
ω02 − ω2 |
i |
N e2γ ω |
(3.1.61) |
||||||
2 ε0m [(ω02 |
ω2)2 + γ2 |
ω2] |
− |
2 ε0m [(ω02 ω2)2 + γ2 ω2 |
|||||||
] |
|||||||||||
− |
− |
||||||||||
with
e = −1.602 × 10−19 C : elementary charge, m = 9.109 × 10−31 kg : mass of the electron,
ω = 2 π ν [s−1] : circular frequency of the light,
ω0 [s−1] : circular resonant frequency of the electron,
|
n |
|
|
part |
k |
|
real |
part |
|
Refractiveindex, |
imaginary |
UV- |
Visible |
resonances |
range |
n |
|
IR- |
|
k |
resonances |
Resonance |
0
Wavelength
0
Circular frequency
Fig. 3.1.17. Realand imaginary part of the refractive index in the vicinity of a resonance in the UV. The principal shape is explained by the classical oscillator model after J.J. Thomson, P. Drude, and H.A. Lorentz [99Bor, 88Yeh].
Landolt-B¨ornstein
New Series VIII/1A1
Ref. p. 131] 3.1 Linear optics 97
γ [s−1] : damping coe cient, N [m−3] : density of molecules,
ε0 = 8.8542 × 10−12 As/Vm : electric permittivity of vacuum.
Examples see [96Ped, 88Kle], generalization to dense media see [96Ped, 88Kle, 99Bor].
The Kramers-Kronig relation connects n(ω) with k(ω) [88Yeh].
Dispersion formula [95Bac]: |
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n2 |
B1λ2 |
B2 |
λ2 |
B3 |
λ2 |
|||||||
(λ) = 1 + |
+ |
+ |
(Sellmeier’s formula) . |
(3.1.62) |
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λ2 − C1 |
λ2 |
λ2 |
||||||||||
− C2 |
− C3 |
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The dimensions of the constants are given in example 3.1.7. The available wavelength range is given by the transmission limits, usually.
Example 3.1.7. [96Sch]: Glass N-BK7: λ [µm], B1 = 1.03961212, B2 = 2.31792344 × 10−1, B3 = 1.01046945, C1 = 6.00069867 × 10−3 [µm2], C2 = 2.00179144 × 10−2 [µm2], C3 = 1.03560653 × 102
[µm2], n(0.6328 µm) = 1.51509, n(1.06 µm) = 1.50669.
Other interpolation formulae for n(λ) are given in [95Bac], [95Bas, Vol. II, Chap. 32], [05Gro1, p. 121].
Further information is available from glass catalogs (see Sect. 3.1.5.10) and from subroutines in commercial optical design programs:
– relative dispersive power or Abbe’s number νd = |
nd − 1 |
with nd(587.56 nm = yellow He- |
nF − nC |
||
line), nF(486.13 nm = blue H-line), nC(656.27 nm = red H-line) [95Bac, 80Sch]; application: achromatic correction of systems [84Haf],
–spectral range of transmission,
–temperature coe cients of n and νd,
–photoelastical coe cients,
–Faraday’s e ect (Verdet’s constant),
–chemical resistance, thermal conductivity, micro hardness etc.
Sellmeier-like formulae for crystals are available in [95Bas, Vol. II, Chap. 32]. Information in connection with laser irradiation damage is presented in [82Hac]. Specific values of laser glasses are given in tables in [01I ].
The parameters can be calculated from the dispersion interpolation (3.1.62) [91Sal, 96Die]:
β(ν) = n(ν) |
2π ν |
(propagation constant [m−1]) , |
(3.1.63) |
||
c0 |
|||||
cph = |
c0 |
(phase velocity [m s−1]) , |
(3.1.64) |
||
n(ν) |
|||||
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3.1.5 Optical materials |
[Ref. p. 131 |
v = |
2 π |
= |
1 |
(group velocity [m s−1]) , |
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d β |
d β |
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d ν |
d ω |
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1 |
d2 β |
d2 β |
d |
1 |
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Dv = |
= 2 π |
= |
(group velocity dispersion (GVD)) |
|||||||||||||
2 |
π |
d ν2 |
d ω2 |
d ν |
v |
|||||||||||
with
ν : frequency of light,
c0 : velocity of light in vacuum.
(3.1.65)
(3.1.66)
Application: Temporal pulse forming by the GVD of dispersive optical elements [96Die, 01Ben].
The refractive index of metals is characterized by free-electron contributions (ω0 = 0 in (3.1.61)). One obtains from [67Sok, 72Woo], [95Bas, Vol. II, Chap. 35] with a plasma resonance (here collisionfree: γ = 0):
n2(ω) = 1 − |
ωp |
2 |
|
(3.1.67) |
|||
ω |
|||
with |
ωp [s−1]: plasma frequency, depending on free-electron density [88Kle]. From (3.1.67) follows
–n(ω) < 1 for ω > ωp, which means λ < λp (example: λp = 209 nm for Na): transparency,
–pure imaginary n(ω) for ω < ωp, λp < λ .
Other e ects change the ideal case (3.1.67) [88Kle].
The complex refractive index of semiconductors is determined by transitions of electrons between or within the energy bands and by photon interaction with the crystal lattice (reststrahlen wavelength region). It depends strongly on the wavelength and is modified by heterostructures and dopands [71Pan, 95Kli], [95Bas, Vol. II, Chap. 36].
3.1.5.5 Fresnel’s formulae
Fresnel’s formulae describe the transmission and reflection of plane light waves at a plane interface between
–homogeneous isotropic media: [99Bor, 88Kle] and other textbooks on optics,
–homogeneous isotropic medium and anisotropic medium: special cases [99Bor, 86Haf] and other textbooks on optics,
–general case of anisotropic media: [58Fed],
–modification by photonic crystals: [95Joa, 01Sak].
Fresnel’s formulae for the amplitude (field) reflection and transmission coe cients are listed in Table 3.1.7.
Plane of incidence: plane, containing the wave number vector k of the light and the normal vector n on the interface.
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