Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007

2. High frequency limit

3. Low frequency limit

ω

ω

Application 8.5.

1.Check your computer program manual and get familiar with physical unit systems, MKS, and cgs.

2.Modify the calculations for gold, 4.5 · 107 [1/Ohm m].

3.Modify the calculations for silver, 6.3 · 107 [1/Ohm m].

O6SKINS

Skin Depth

1. Skin depth (in meters) for intensity depending on frequency

2.Skin depth (in meters) for intensity depending on wavelength

(For checking: for 1 mm wavelength angular frequency is 2π · 3 · 10 .)

Application 8.6.

1.Derive the penetration depth for the intensity, Eq. (8.61).

2.Check your Mathcad manually and get familiar with physical unit systems, MKS, and cgs.

3.Modify the calculations for gold, 4.5 · 107 [1/Ohm m].

4.Modify the calculations for silver, 6.3 · 107 [1/Ohm m].

5.Modify the calculations for nickel, 1.5 · 107 [1/Ohm m].

6.Modify the calculations for lead, 0.5 · 107 [1/Ohm m].

8.4.5Reflectance at Normal Incidence and Reflection Coefficients with Absorption

We have seen in Section 8.3 that the r component is not zero at the principal angle, which means the angle corresponding to the Brewster’s angle for the case of the lossless dielectric. The reflectance R is equal to the square of the reflection coefficients of Fresnel’s formulas, (Eq. (8.32) and (8.33)) and is valid for both dielectric and metals if the corresponding values of n and K are used. For normal incidence, when the parallel and perpendicular components are the same we have

In FileFig 8.7 we show for normal incidence the dependence of the reflectance R on K. When K 0 we go back to Fresnels formulas, depending only on the real part of the refractive index, as discussed in Chapter 5 for lossless dielectrics. When K is large, we have high reflectance R as observed for metals.

FileFig 8.7 (O7REFNKS)

Graph of the reflectance at normal incidence depending on K.

O7REFNKS is only on the CD.

8.4.6 Elliptically Polarized Light

We mentioned in Chapter 5 that elliptically polarized light may be produced when light is totally internally reflected in a dielectric medium. Reflection on metal surfaces shows a similar phenomenon. The reflection coefficients rp and rs may have arguments depending on the angle of incidence. Since each component picks up a different change in the argument, the difference of the arguments of the reflected components is not the same and corresponds to the angle φ of elliptically polarized light, as discussed in Chapter 5.

In FileFig 8.8 we have plotted the difference of the phase angles after reflection, depending on the angle of incidence.

FileFig 8.8 (O8ARDELS)

Graph of the difference of the arguments of zrp and rzs depending on specific values of n and K.

O8ARDELS is only on the CD.

Application 8.8.

1.Change the optical constants and plot a graph of depending on a range of values n for fixed K and three values of θ, for example, 35◦, 45◦, 55◦.

2.Change the optical constants and plot a graph of depending on a range of K for fixed value of n and three values of θ, for example, 35◦, 45◦, 55◦.

In order to get to n and K we have to determine the real and imaginary parts of the right side of Eq. (A8.9). As done in Born and Wolf (1964, p. 619), one can make an approximation by neglecting in the square root of Eq. (A8.8) the term (sin θ)2 with respect to n 2, and obtain explicit expressions for n and K :

We show in FileFig 8.9, graphs of P , , and ψ depending on the angle of incidence θ. These graphs are for specific values of n and K. A comparison of the exact and approximate calculation, again for specific values of n and K, is shown in FileFig 8.10. In praxis one often uses iteration for the determination of n and K and uses more than two input data for a best fit calculation.

For zp rp exp(iδp) and zs rs exp(iδs ), graphs are shown for P tan ψ with P rs /rp, (difference of the arguments of rs and rp), and of atan(zs/zp).

OA1DELTAFfS is only on the CD.

Application A8.9.

1.Change the optical constants and plot a graph of P depending on a range of values n for fixed K and three values of θ, for example, 35◦, 45◦, 55◦.

2.Change the optical constants and plot a graph of atan(zs/zp) depending

values of on a range of K for fixed value of n and three values of θ, for example, 35◦, 45◦, 55◦.

FileFig A8.10 (OA2METPDS)

Graphs are shown for z n + iK depending on ψ because one has P tan ψ, P rs /rp and is the difference of the arguments of rs and rp. Curves of the exact expressions are compared with the approximations.

OA2METPDS is only on the CD.

Application A8.10.

1.Study the approximation for different values of P and fixed value of .

2.Study the approximation for different values of and fixed value of P .

See also on the CD

PO1. Principal Angle. (see p. 315)

PO2. Oscillator. (see p. 317)

PO3. Sellmeir Expression. (see p. 319)

PO4. Drude Model. (see p. 322)

PO5. Skin Depth. (see p. 325)

PO6. Reflected Intensities. (see p. 328)

PO7. Phase Angle. (see p. 328)

and the Fourier integral transforms S(y) into the Fourier transform G(ν). Using slightly different coordinates from those in Chapter 3, we have that the

Fourier transform of S(y) is G(ν).

−∞

From Fourier transform theory we find that the inverse relationship is also true and we have the Fourier transform of (note the plus sign in the exponent)

−∞

Both variables ν and y have continuous values from −∞ to +∞. The variable y is the variable in the space domain and the variable ν is the variable in the frequency domain and has the dimension of 1/space coordinate. (In infrared spectroscopy one uses as the unit cm−1). Both G(ν) and S(y) are not normalized. When G(ν) is symmetric with respect to zero, the integral may be written as

We now discuss some analytical Fourier transformations.

9.1.3Examples of Fourier Transformations Using Analytical Functions

We present two examples to calculate the Fourier transformation, and the Fourier transformation of the Fourier transformation. We show that the latter is indeed the original function.

9.1.3.1 Gauss Function