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

Application 6.8.

1.Change the refractive index on the outside to 1.1, 1.2, 1.3.

2.Change the radius to 2 times the wavelength and 4 times the wavelength.

APPENDIX 6.1

A6.1.1 Boundary Condition Method Applied to TE Modes of Plane Plate Waveguide

The wave travels in the X direction, but it is sufficient for mode formation to consider only the Y direction. For the fields E and B one assumes wavelike solutions in (1) and exponential decreasing solutions in (2) and (3). Application of the boundary conditions matches the fields. The objective is to calculate the

We have for the solutions of the wave equation in the media:

We have four linear homogeneous equations for the coefficients A, B, C, D and therefore the determinant of the system of the four equations must be zero. The solutions of the resulting equation determine the values of θ for which modes

The exponential factors cancel out, and the determinant may be developed with respect to the first row into a sum of two 3 × 3 determinants

which is the condition for the TE modes (see Eq. (6.49)). Similarly one has for the TM modes

The boundary value method gives us the same result with less effort. One only has to assume that the k values in the media (2) and (3) are imaginary and apply the boundary conditions to the solutions in the three sections.

Calculation of the mode conditions for the TE case, which is Eq. (6.49) for the planar waveguide.

NA1PLTE is only on the CD.

Application A6.9. Derive in a similar way the condition for the TM case, which is the p-polarization.

C H A P T E R

Blackbody Radiation,

Atomic Emission,

and Lasers

7.1 INTRODUCTION

At the end of the nineteenth century electromagnetic theory and thermodynamics were well developed and there was the question of reunification of these theories. Blackbody radiation is the emission of electromagnetic radiation from a closed cavity at temperature T through a small hole (Figure 7.1). It was found that the frequency distribution of the emitted radiation was dependent upon the temperature T . M. Planck formulated the famous blackbody radiation law in 1900. He used electromagnetic theory and thermodynamics, but needed an assumption that marked the beginning of quantum theory. Planck’s law involved a fundamental physical constant, now called Planck’s constant. The emission of light from the blackbody was interpreted as quantum emission of light. In the first half of the last century, the quantum emission of atoms and molecules was studiedand in

FIGURE 7.1 Schematic of a blackbody radiator. The body is surrounded by the heat bath of temperature T . Electromagnetic radiation is emitted and the frequency distribution depends on temperature T .

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the second half of the century the laser was developed. (“Laser" stands for light amplification by stimulated emission of radiation).

7.2 BLACKBODY RADIATON

7.2.1 The Rayleigh–Jeans Law

An example of blackbody radiation is a wire heated by electricity. First it gets red and when the temperature increases it becomes white. Increasing the temperature, one finds that the maximum of the frequency distribution of the emitted electromagnetic radiation shifts to a shorter wavelength. In Figure 7.1 we show a schematic of a blackbody. Radiation is emitted at thermal equilibrium and the energy is drawn from the heat bath at temperature T . The first, but insufficient, analysis of blackbody radiation was done by Rayleigh and Jeans, analyzing the modes of a rectangular box in a heat bath (Figure 7.2). Oscillators were assumed to be at the walls of the box, emitting and absorbing light. The modes of the box are standing waves and in Figure 7.3 we show one standing wave in one direction as we have in a Fabry–Perot. The number of standing waves in three dimensions was analyzed and the number of modes determined with respect to the same energy which is the same frequency or wavelength. It was assumed that in thermal equilibrium, each mode carried the energy kT , where k is Boltzmann’s constant. The energy density per frequency interval du/dν was equal to the energy dE per volume V and frequency interval dν. The energy density per frequency interval du/dν was then calculated to be

FIGURE 7.2 A box as cavity in the heat bath of temperature T ; see Figure 7.1.

FIGURE 7.3 Standing wave pattern in one dimension. The length is lx , the magnitude of the wave is A, and the wavelength λ.

This was the Rayleigh–Jeans law and turned out to be valid for the long wavelength region only. In the short wavelength region, the energy density increased by ν2 and led to very high and unrealistic large energy densities, the so-called “violet catastrophe."

The graph in FileFig 7.1 shows the Rayleigh–Jeans radiation law in the visible spectral region, where one can observe the increase of the radiation density to shorter wavelengths.

FileFig 7.1 (L1RAJEANS)

Graph of the Raleigh–Jean law using units of energy density per frequency interval.

L1RAJEANS is only on the CD.

7.2.2 Planck’s Law

The radiation law of blackbody radiation was discovered by Max Planck and is valid in both the short and the long wavelength regions. Planck’s radiation law agreed with the Rayleigh-Jeans law for the long wavelength region. Many years later Einstein derived Planck’s radiation law using the concept of transition probabilities. One assumes fictitious oscillators to be on the walls of the blackbody cavity. These oscillators are in contact with the heat bath at temperature T . When radiating, they get the energy from the heat bath and transform it into radiation energy. Einstein defined the processes of induced absorption, induced emission, and spontaneous emission for the emission and absorption of radiation by the oscillators (Figure 7.4). The probability of a transition of induced absorption between the levels numbered 1 and 2 is called W12 and is proportional to the

FIGURE 7.4 Schematic of induced absorption and induced and spontaneous emission.