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

2.Integration over the wavelength range from 3·10−6 to 3·10−5 meters to obtain the radiance

3·10−5

R : f (x)dx.

3·10−6

Radiance

R1.316 · 104.

3.Multiplication with area times solid angle to obtain the radiant energy Area A, Solid angle SA: A : 1, SA : 4; radiant energy RR: RR : A · SA· R; RR 5.263· 103 watts. RR has the same value as the corresponding value when integrating over the frequency.

Application 7.2.

1.Calculate the radiance for different wavelength intervals.

2.Calculate the radiant energy and choose an area of 5 mm2 and a solid angle of π4.

3.Consider the wavelength range from 1 mm to 100 mm for comparison with the Rayleigh–Jeans law. Calculate the corresponding frequencies and derive from Planck’s law the corresponding energy density for this frequency interval. Do the same calculation for the Rayleigh–Jeans law and give the difference in the numerical values.

Graph of blackbody radiation depending on the frequency. The calculation of the radiance of a particular frequency range and calculation of the corresponding radiant energy by multiplication with area times solid angle.

L3BBFS is only on the CD.

Application 7.3.

1.Calculate the radiance for different frequency intervals.

2.Calculate the radiant energy, and choose the area times solid angle such that the radiant energy is the same as you calculated in Application FF2.

3.Numerical calculation of the Stefan–Boltzmann law. Calculate, using the same units, the integrated radiation from Planck’s law for a chosen temperature T and compare with the Stefan–Boltzmann law (FileFig 7.2).

The Stefan–Boltzmann law is plotted using linear and logarithmic scales.

L4STEFS is only on the CD.

FileFig 7.5 (L5WIENS)

Wien’s law is plotted for two ranges of the temperature.

L5WIENS is only on the CD.

7.3 ATOMIC EMISSION

7.3.1 Introduction

The operation of a laser needs an “active medium” between the two mirrors of the laser cavity. Energy is “pumped” from the outside of the cavity into this medium and produces atoms or molecules in excited states. Here we discuss only excited energy states of atoms in the gas phase and consider the hydrogen atom and atoms with hydrogenlike spectra. The energy states are labeled by letters with subscripts and superscripts and some of the notations have their origin in the “old days” of spectroscopy. At that time, for example, one used for the characterization of some spectral lines: s for “sharp,” and d for “diffuse.” The

letters s and d are still in use for the characterization of the angular momentum. There is some truth to these notations, since we know that the s-state has less degeneracy than the d-state.

7.3.2 Bohr’s Model and the One Electron Atom

In Bohr’s model, an electron with a negative charge circulates around the positive charge in the center of its orbits, determined by the principal quantum number n. The energy of such an orbit is given by

where K is the constant of Coulombs’ law, e the electron charge, m the electron mass, and h Planck’s constant. The principal quantum number n has integer numbers 1, 2.3 . . .. Radiation is emitted when the electron changes its orbits and the energy of the emitted photon is given as

where ni is the quantum number of the initial orbit and nf is the quantum number of the final orbit. In Figure 7.6 we show the series of lines originating from the state nf 1, called the Lyman series, from the state nf 2, called the Balmer series, and from the state nf 3, called the Paschen series. The significant achievement of Bohr’s derivation was that he used fundamental physical constants and could reproduce exactly the empirical constant of the expression of the Balmer series.

7.3.3 Many Electron Atoms

7.3.3.1 Principal Quantum and Angular Momentum Quantum Numbers

The Schroedinger equation and the Pauli principle are needed to understand the atomic energy schematics and transitions. As an example we look at an atom with Z electrons and a nucleus with a positive charge of Ze. A list of some lower energy states is shown in Figure 7.7. For such an atom, we have energy levels labeled by the principal quantum number n and the angular momentum quantum number l. The Schroedinger equation tells us that for each n there are n − 1 different possible states of the angular momentum.

7.3.3.2 Magnetic Quantum Number and Degeneracy

To each state labeled by the angular quantum number l, there are 2l+1 substrates. They only have different energy values if the atom is in a magnetic field. The corresponding quantum number is called the magnetic quantum number m. If the magnetic field is zero, all states have the same energy, and therefore the state is m fold degenerate.

FIGURE 7.7 Energy levels and quantum numbers for a few states of the hydrogenlike atom. The states are shown for each n and labeled by s, p, d, f . Spin states are also indicated.

In Figure 7.8 the number of electrons occupying different states is listed for the elements from Z 1 to Z 25. There are irregularities, explained by quantum mechanics. An example is Z 19, where the 4s state has lower energy than the 3d state.

7.3.3.6 Transitions Between Energy States

Photons may be emitted when the atom changes from a higher energy state to a lower one. In an atom like the sodium atom, there is just one “outer” electron. All other electrons are in “closed” shells. This outer most electron is an “s” electron (Figure 7.9), and therefore the spectrum has a similarity to the spectrum of the hydrogen atom. The columns in Figure 7.9 are labeled with capital letters, referring to compound states of all electrons. Since the K and L shells are closed, they do not contribute and the angular momentum of all electrons is the same as the single electron in the M shell.

There are selection rules restricting the energy levels between which transitions are possible. The general rule is that the angular momentum has to change by plus or minus 1. Transitions are only allowed between the levels of the columns labeled by s, p, and d (Figure 7.9).

FIGURE 7.8 Electronic states in atoms. Electrons having the same n value in “shells,” named K, L, M. For each n we have n − 1 substates of angular momentum with quantum number l, named s, p, d, f . There are 2l + 1 possible values of m, and two spin states. Therefore the maximum occupations of K and L are K : n 1, 2[(2·0+1)] 2; L : n 2, 2[(2·0+1)]+2(2·1+1)] 8 (the M shell is more complicated).

7.4 BANDWIDTH

7.4.1 Introduction

The atom emits light when an electron makes a transition from a higher energy state to a lower one. The emitted light is not monochromatic, since the emission process last only for a short time. Therefore we have a wavetrain of limited length. Such wavetrains are described by the superposition of a large number of monochromatic waves having a certain frequency distribution (see Chapter 4). The frequency spectrum shows a maximum at the transition frequency and the bandwidth of the frequency distribution is related to the time of the emission process.

´

We follow the book Lasers by F. K. Kneub’uhl and M. W. Sigrist, B. G. Teubner, Stuttgart, 1988, and use some of their numerical values in the examples.

FIGURE 7.9 Energy schematic for sodium. The energy level of 3s, that is, the ground state of the outermost electron, has been chosen as 0. The two well-known “Sodium D” lines are indicated.

7.4.2Classical Model, Lorentzian Line Shape, and Homogeneous Broadening

Light is emitted from an atom when an electron leaves an excited state E2 and occupies a state with lower energy E1. One has for the energy of the process

Before the electron can make the transition, it has to be placed into the upper state. This process is called population inversion (see Section 5.2). The electron remains in the upper state for a limited time. On average the life-time of an excited state is 10−8 sec. However, longer lifetimes corresponding to metastable states play an important role in laser action. The emitted light is described by a wavetrain with decreasing amplitude (Figure 7.10), and may be described in first approximation as

The lifetime τ is defined as the time in which the initial amplitude A0 drops to a value of A0/e A0/2.718. The frequency distribution of the wavetrain now