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

polarized and has an effect on the oscillator under consideration. The local field at the oscillator must be corrected. This is called the Lorentz correction and the effective field at the site of the charges is

Similar to the low density case, using Eq. (8.9), the square of the refractive index for the denser medium is

and is called the Clausius–Mossotti equation, or with the parameters of our model,

We have in a solid that the interaction between the oscillators becomes very strong. The oscillation frequencies are modified and the damping constants become large. In addition, in crystals the periodicity must be taken into account.

8.3 DETERMINATION OF OPTICAL CONSTANTS

8.3.1 Fresnel’s Formulas and Reflection Coefficients

The determination of the two parts n and K of the complex refractive index may be accomplished by using reflection measurements. In Chapter 5 we found that Fresnel’s formulas relate the reflection coefficients of the p- and s-polarization cases to the real index of refraction. In a similar way to that of Chapter 5, one can show that the complex reflection coefficients are related to complex refractive indices through Fresnel’s formulas. Replacing n2 by n2 n2 − iK2 we have for the reflection coefficients

In order to represent r and r depending on the angle of incidence. We need the law of refraction in complex terms (see FileFig 8.4 (M4SNELL) of Chapter 5). The law of refraction is

The angle θ must be a complex quantity since the left side of Eq. (8.31) is real. Introduction of Eq. (8.31) into Eqs. (8.29) and (8.30) gives us

In FileFig 8.1 we have graphs of the absolute value and the argument for reflected amplitudes of the parallel (zrp) and perpendicular (zrs) cases depending on n and K. For K 0 we see from the first graph that for the parallel case the minimum corresponding to the Brewster angle is not zero. The angle at the minimum is called the principal angle. The second graph shows the phase jump at the Brewster angle, which is now a smooth transition. In the application of FileFig 8.1, Section 3, we plot the parallel (zrp) and perpendicular (zrs) cases for several different values of n and K on the same graph.

FileFig 8.1 (O1FRNKPSS)

Graphs are shown for reflected amplitudes of the parallel (zrp) and perpendicular (zrs) cases depending on n and K for n1 1. For both cases the absolute value of the reflected amplitudes and the phase angle are plotted for n1 1, n2 1.5, and K 2.

O1FRNKPSS is only on the CD.

Application 8.1.

1.Make a graph and find the values of the principal angle for one value of n and three values of K.

2.Make a graph and find the values of the principal angle for one value of K and three values of n.

3.Observe that the curve of arg(zrp(θ)) is not continuously approaching the curve of arg(zrs(θ)) when K is going from 0.01 to zero. What is the remaining phase difference?

8.3.2 Ratios of the Amplitude Reflection Coefficients

Equations (8.32) and (8.33) give the dependence of the reflection coefficients on n and K. We have the general problem that we cannot represent n and K as functions of rs and rp. This is only possible for approximations, (see Appendix 8.1), and other methods have to be considered.

One method is to apply reflection measurements at several angles of incidence to determine the absolute values of the ratio rs /rp. The ratio rs /rp is calculated similarly to that done by Born and Wolf (1964, p.617).

From Chapter 5 the ratio r /r from Fresnel’s formulas is

Using the law of refraction and the trigonometric formula for the sum of angles

In FileFig 8.2 we show graphs of the real parts of rp and rs for various values of n and K. The third graph shows the ratio rp/rs and the fourth rs /rp. From these two graphs it appears that the ratio rp/rs is much more useful for the determination of optical constants than the ratios rs /rp, because they are smooth and do not show a resonance, related to the appearance of the Brewster angle. The optical constants may be obtained by measuring values of |rp/rs | for two different angles of θ, and solving the two equations for the unknowns n and K.

FileFig 8.2 (O2FRSOPS)

Graphs are shown for the real part of the ratios rs /rp and rp/rs , calculated with the expressions used in FileFig 8.1, for n1 1, n2 1.5, nn2 1.5, K 0.1, and K 0.01, KK 0.5, KKK 2.

O2FRSOPS is only on the CD.

8.3.3 Oscillator Expressions

8.3.3.1 One Oscillator

To fit experimental data of a narrow frequency range, in which we have a resonance feature, one may use for n + iK a similar expression derived in Eq. (8.23) but extended to four parameters,

where A is a general constant, S the oscillator strength, γ the damping constant, and ν0 ω0/2π the resonance frequency. An example is shown in Figure 8.2 and a calculation given in FileFig 8.3.

FIGURE 8.3 The dependence of the polarizability on frequency for the microwave, infrared, and ultraviolet regions.

representation of n and K (schematically shown in Figure 8.3). Having measured n and K over a large range of frequencies, the experimental data are fit to formulas such as

where fj , γj , and ωoj are empirical constants. The constants in these expressions are determined by a “best fit" calculation over a large range of frequencies with respect to the measured values of n and K

8.3.4 Sellmeier Formula

Similarly to what has been discussed for the oscillator expression, one may fit experimental data to represent the dependence of n and K on the wavelength by using a polynomial approach of the type

This is called a Sellmeier-type equation and has been used, for example, to fit the data for potassium bromide in the spectral region from .2 to 42 microns using 11 empirical constants.1 An example is given in FileFig 8.4.

When fitting experimental data one has to keep in mind that n and K are not independent. They are related by the Kramer–Kroning model.2 In some spectral regions, for example in the x-ray region, one obtains the K value from absorption measurements and calculates the corresponding n value with the Kramer–Kroning model.

FileFig 8.4 (O4SELMRS)

A graph of a Sellmeier expression n(λ) for fused quartz is shown for the range of λ from 4000 to 8000 Angstrom using parameters ci with i 1 to 3.

O4SELMRS

Graph for Demonstration of the Sellmeier Presentation of the Refractive Index

l

l

Application 8.4. Determination (backward) of the constants c1, c2, and c3. Read from the graph three values of λi and the corresponding value of n(λi ). Consider c1 to c3 as unknown and formulate a system of linear equations. One

1E. D. Palik, Handbook of Optical Constants of Solids II, Academic Press, New York, 1991.

2Charles Kittel, Introduction to Solid State Physics, John Wiley & Sons, New York, 1967.

would be n(λi ) c1 + c2/λ2i + c3/λ4i . Solve the system of these inhomogeneous linear equations with one of the available computer programs. Make an estimate of the error.

8.4 OPTICAL CONSTANTS OF METALS

8.4.1 Drude Model

In Section 8.2 we discussed the optical constants of dielectrics and in Section 8.3 their determination. We now discuss metals and how their optical constants are also described by a complex refractive index. The determination of the n and K values for metals is similar to what has been discussed before, but the model representing the material is different.

Metals show high reflectivity in the visible and infrared spectral regions and their attenuation increases with lower frequencies. The values of real and imaginary parts of the refractive index of metals are in the lower frequency region which is much higher than one usually finds for dielectrics.

The interaction of the electromagnetic wave with the metal is described by the Drude model. The electrons are assumed to move almost freely in the metal and there is no restoring force to make the electrons vibrate, as discussed for the dielectric. For an isotropic medium with free conducting charges we write Maxwell’s equations as

where vj is called the drift velocity. The electrical field of the light and the current density in the material are related by the wave equation. We assume E and j are vibrating in the y direction and propagating in the x direction and from Eq. (8.41) we get for the wave equation

As in Section 8.2, we now find an expression for jy in terms of the parameters of the damped oscillator model and the vibrating electrical field E0. The differential equation of the model is now without the force term,

The general solution of Eq. (8.44) is the sum of the solutions of the homogeneous and inhomogeneous equations. For the homogeneous equation,

Using the trial solution u u0e−t/τ , we get for γ τ −1. Typically one has a value of 10−13 for τ and we neglect this solution.

Equation (8.48) relates the current density j0 of our model to the electrical field E0 of the light, vibrating with angular frequency ω.

We now turn to the wave equation, (see Eq. (8.43)), using the trial solutions

and introducing j0 from our model, Eq. (8.48), we obtain for the complex wave vector k

Equation (8.52) relates the refractive index to the static conductivity of the metal σ , the frequency of light ω and the relaxation time τ , which is a parameter of our model and is related to the metal. In FileFig 8.5 we show graphs over a large frequency region of the real and imaginary parts of Eq. (8.52).

8.4.2 Low Frequency Region

For low frequencies, that is, when ωτ 1, we may neglect ωτ with respect to i (see Eq. (8.52)), and get

1. General Expression