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

In FileFig 7.12a we show graphs of the first four modes from (0,0) to (1,1). In FileFig 7.12b we show graphs of the next five modes from (1,2) to (2,2). Contour plots were chosen for better reference to Figure 7.21, and identification of the zero-intensity lines and mode numbers.

FileFig 7.12 (L12MOCY1to4S)(L12MOCY5to9S)

Modes for confocal cavity and circular mirrors. Contour plots of generalized Laguerre polynomial for indices l and p equal 0, 1, and 2. Scaling factor is X.

L12MOCY1to4S

Cylindrical Coordinates for Circular Mirrors in Confocal Resonator

Field distribution as contour plot for graph 00, 10, 01, and 11. The L(l, p) functions are written out for 00 to 22. The constant in the exponential is X:

M01i,j : (cos(1 · β(xi , yj )) · q(xi , yj ) · L10(xi , yj ))2

M11i,j : (cos(1 · β(xi , yj )) · q(xi , yj ) · L11(xi , yj ))2.

L12MOCY5to9S

Cylindrical Coordinates for Circular Mirrors in Confocal Resonator

Field distribution as contour plot for graph 02 to 20. The L(l, p) functions are written out for 00 to 22. The constant in the exponential is X:

i : 0 . . . N j : 0 . . . N N 40

Constant X : X ≡ 2. There h stands for l and p runs from 0 to 2.

Lh2(x, y) : [1/2(h + 1)(h + 2) − (h + 2)u(x, y)] + (1/2)u(x, y)2

L21(x, y) : 3 − u(x, y)

L20(x, y) : 1.

M02i,j : (cos(2 · β(xi , yj )) · q(xi , yj ) · L20(xi , yj ))2

M20i,j : (cos(0 · β(xi , yj )) · q(xi , yj ) · L02(xi , yj ))2.

M12i,j : (cos(2 · β(xi , yj )) · q(xi , yj ) · L21(xi , yj ))2

M21i,j : (cos(1 · β(xi , yj )) · q(xi , yj ) · L12(xi , yj ))2.

M22i,j : (cos(2 · β(xi , yj )) · q(xi , yj ) · L22(xi , yj ))2.

Application 7.12.

1.Compare with Figure 7.21 and the number of zero intensity lines with the mode numbers.

2.Convert surface to contour plots and try to identify the modes.

PL1. Rayleigh-Jeans Law and Planck’s Law (see p. 271).

PL2. Graph of Black Body Radiation depending on Wavelength and Frequency.(see p. 275–277).

PL3. Lorentzian Line Shape with angular Resonance frequency ωo and Lifetime τ (see p. 284).

PL4. Calculation of (N2 − N1)/No (see p. 292). PL5. Paraxial Wave Equation (see p. 294).

PL6. Calculation of w2(z) and R(z) for Confocal Cavity (see p. 296). PL7. Modes for Confocal Cavity and rectangular Mirrors (see p. 300). PL8. Modes for Confocal Cavity and circular Mirrors (see p. 305).

8.2 OPTICAL CONSTANTS OF DIELECTRICS

8.2.1The Wave Equation, Electrical Polarizability, and Refractive Index

We write Maxwell’s equations for an isotropic and nonmagnetic material without free charges, which means we assume · P 0 and ρ 0 and have

where E is the electrical field vector, B the magnetic field vector, j the current density vector ρ the charge density, and ε0 8.854×10−12 F/m, the permittivity of vacuum. We now study the effect of an outside electrical field on the bound charges in an isotropic material. The outside electrical field is assumed to be a harmonic wave and will act on the bound charges and make them vibrate. For the current density N of such vibrating charges we have

where N is the number of charges per unit volume, e the charge of the electron, and v the velocity vector. Since we assume an isotropic medium, we only have to take into account one direction of vibration, which we call y, and one direction of motion, which we call x. With v dx/dt we have

The induced dipoles are ex, and P N(ex) is called the polarization. The current density is then

The wave equation for vibration in the y direction and propagation in the x direction is

where the right side of Eq. (8.5) is called the source term.

In the first approximation, we set Py proportional to the incident electrical field and write

where α is the atomic polarizability, a constant characteristic for the material. We introduce into the wave equation a trial solution Ey A cos(kx − ωt), and get

We associate the velocity v with the the phase velocity ω/k in the medium and obtain for n

We have obtained a relation between the optical constant n and the material constant α, the atomic polarizability.

8.2.2 Oscillator Model and the Wave Equation

8.2.2.1 Less Dense Medium

To study the dependence of the refractive index on frequencies and losses, we relate the refractive index to the parameters of an oscillator model.

The polarization vector P of the medium is defined as the number of electrical dipoles per unit volume. The induced electrical dipole moment is eEy , which we now call (eu), where u is the displacement of an electron in an atom. The number of dipoles per unit volume is N and we consider only one component of

We describe the displacement u of the charges by the vibrations of a damped oscillator,

where u is the displacement of the charge from its equilibrium position, m is the mass, f the force, and γ the damping constant. The frequency without the damping term is ω012 f/m, and the resonance frequency for the damped oscillator is ω0 ω0. The electromagnetic wave of the light drives these oscillators. The forced damped oscillator equation is

where E(t) E0e−iωt . The driving electromagnetic wave produces polarization

P (t) Neu(t) [Ne2εoE(t)]/[mεo(ω02 − ω2 − iγ ω)] χ ε0E(t), (8.14)

where ε0 is the permittivity of free space. As a result of the imaginary damping term in u(t) one has a complex susceptibility χ, indicated by a star. In Eq. (8.14) we have related the polarization P (t) and the electrical susceptibility χ to the parameters of our model and the electrical field E(t) of the light.

From the wave equation, we have another expression of P (t) (see Eq. (8.6)) and introducing P (t) ε0NαE(t) into Eq. (8.14) we get

where α is the atomic polarizability and ωp2 Ne2/mε0 the plasma frequency. One should not confuse this with ω02, the square of the angular frequency of the oscillator model, representing the dielectric. We have obtained a relation between the material constant of the atomic polarizability α and the parameters of our oscillator model. For the square of the refractive index, (see eq. (8.9)) we get

Since (n )2 is a complex number, we marked it with a star. It is customary to call the real part of the complex refractive index n, and the imaginary part K. The imaginary part K may be called the extinction index. We have n n − iK and

where ε is the real part of the dielectric constant and ε is the imaginary part. For optically thin media, such as gases, one has n close to 1 and K is small and has the approximation

The damping term of the damped oscillator equation appears in the imaginary part of n , indicating the losses present in the description of our model. When the damping term is zero we get

where ω0 is the resonance frequency of the case without losses. The refractive index depends on the frequency and the model parameters. In Figure 8.1a the dependence of n on the frequency is shown schematically. When γ 0 we

FIGURE 8.1 (a) Dependence of n on frequency. For no damping we have a singularity; (b) normal and anomalous dispersion; (c) dependence of K on frequency. The maximum is not at infinity if

γ 0.

have singularities at the resonance frequency, and for γ 0 these singularities are avoided. On the left side of Figure 8.1a the refractive index increases with higher frequencies. This region is called normal dispersion, shown on a prism on the left in Figure 8.1b. The reverse case is on the right side of Figure 8.1a, called anomalous dispersion and shown on a prism on the right in Figure 8.1b. In Figure 8.1c we show an absorption curve, which is the dependence of K on the frequency.

8.2.2.2 Dense Medium

So far, in the preceding discussion, it was assumed that the local field E0 at the site of the oscillators is the same as the applied field. This is only true if the density of the oscillators in the medium is low, as it would be for a gas. For a dense distribution of the oscillators in a solid, the surrounding area is also electrically