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

FIGURE 7.12 The upper energy level has the width E2, the lower E1. Transitions of higher and lower frequencies are indicated, with respect to the peak frequency of the Lorentzian line shape.

gas also move while emitting light. The frequency shift, related to their particular velocity v, is given as

The distribution of the velocities depends on the temperature and follows Maxwell’s velocity distribution law. One gets

where I0 is a constant, m the mass of the oscillator, c the speed of light, k the Boltzmann constant, and T the absolute temperature. As we did in Section 7.4.2 for the Lorentzian line shape, we define the Doppler line shape as I (ν) I0gd(ν) and get

The line shape of Eq. (7.38) has a Gaussian profile which is different from the Lorentzian, schematically shown in Figure 7.13a.

The halfwidth at half height is calculated to be

The Gaussian line shape may be looked at as the envelope of a large number of Lorentzian line shapes. Each oscillator moves with a different velocity in a different direction and the peak frequencies of the emitted light have a Gauss distribution, (shown in schematically Figure 7.13b). When discussing an exam-

FIGURE 7.13 (a) Comparison of Lorentzian and Gaussian line shapes of approximately same line width at half height; (b) Gaussian line shape is the envelope of Lorentzian line shapes emitted statistically at different velocities in different directions.

ple of the Ne–He laser below, we give numerical values for the halfwidth of Eq. (7.39).

7.5LASERS

7.5.1Introduction

We discussed blackbody radiation in Section 7.2, atomic emission in Section 7.3, the Fabry–Perot in Chapter 2, and modes in Chapter 6. We now discuss the following simplified model for a two-level laser. We consider a Fabry–Perot filled with an active medium of atomic oscillators. First we need population inversion. The oscillators will be excited into the upper state 2 by using the light of frequency νP from the outside. We need the excited states to remain excited for sometime, for example, 10−3sec. Then we need stimulated emission to transfer the energy of the excited oscillators to the modes of the Fabry–Perot. To have laser light emitted from the Fabry–Perot, one must couple out part of the light of the modes. This is done by making one of the mirrors of the Fabry–Perot not totally reflective, such as when there is a small hole in the mirror.

FIGURE 7.14 Schematic of a three-level laser. S23 is the probability of a radiationless transition, Wik and Wki are probabilities of induced absorption and emission, and Aik is the probability of spontaneous emission.

7.5.2 Population Inversion

7.5.2.1 Two-Level System with Stimulated and Spontaneous Transitions

When discussing the blackbody radiation law, we used for the occupation of the different energy levels in thermal equilibrium the Boltzmann distribution Ne−E/kT , where k is Boltzmann’s constant. This distribution law tells us that, at temperature T , the lower states are more occupied than the higher states. For laser action we need just the reverse, which is more electrons in a higher energy state. This is called population inversion and is in contrast to the population of the electrons in thermal equilibrium.

We assume that we have to deal with oscillators having just two levels, as shown in Figure 7.14. We call the upper level E2 and the lower level E1. The change in time of the number of photons N1 of E1 and N2 of E2, the rate equations, is

where u(ν) is the radiation density. The coefficient B12 is the Einstein coefficient of stimulated absorption. The coefficient B21 is the Einstein coefficient of stimulated emission and A21 is the coefficient of spontaneous emission. We have used these coefficients when deriving Planck’s radiation law, (see Section 7.2), and used the relation

as well as B12 B21. This relation has to be slightly changed for atomic oscillators. We have to take into account that the atomic energy levels may be degenerate, which means there are several energy levels having the same energy. For example, if there is a threefold degeneracy, we have to use the weight g 3

7.5.2.2 Changes in the Upper Level Considering Stimulated Transitions

A necessary condition for laser action is population inversion which as stated before, means there must be more photons in the upper state than in the lower state. Using only the stimulated emission and absorption processes, we have for the number of photons in the upperstate.

and one has the condition for laser action, N2/g2 > N1/g1.

7.5.3Stimulated Emission, Spontaneous Emission, and the Amplification Factor

To have the condition N2/g2 > N1/g1 fulfilled, one has to make the stimulated emission larger than the spontaneous emission. The stimulated emission transfers the photons from the upper energy state E2 into the modes of the Fabry–Perot. The bandwidth for the emission process was discussed above and we assume that we have a Lorentzian line shape. The modes of the Fabry–Perot are much narrower and close to delta functions (see Figure 7.15). The probability of stimulated emissions per unit time is calculated by considering Eq. (7.46), written for a number of n photons as

The radiation density is u(ν) hνn. Taking into account the line shape of the emission process, (see Eq. (7.30)), we have to multiply the coefficient of stimulated emission B21 by u(ν)gl(ν), that is, B21u(ν)gl(ν), and get

ν−1 ν0 ν1

FIGURE 7.15 The width of the modes of the cavity at frequencies ν−1, ν0, and ν1 are almost delta functions compared to the bandwidth of the Lorentzian-shaped emission line.

We can express the change of u(ν) with respect to the time dt by considering the length dz in which the light travels in the time dt This gives us dt dz/c . Here we use the speed of light c , modified in the medium of the laser cavity. The change of the radiation density over the interval dz is then

The expression in the curly braces is called the amplification factor ε(ν):

The gain of the beam ε(ν) is “per length” and an example for the Ruby laser is calculated in FileFig 7.7.

´

Gain calculation for the example of the Ruby laser. (See also Kneub’uhl and Sigmst, 1988.)

L7RUBYS is only on the CD.

7.5.4The Fabry–Perot Cavity, Losses, and Threshold Condition

The Fabry–Perot cavity has two plane parallel mirrors with reflectivities R1 and R2, and the modes are standing waves. These standing waves may be considered as two traveling waves, moving in opposite directions. If R1 and R2 have high reflectivity, the traveling waves will pass forward and backward through the volume filled with the oscillators in the excited state E2. By stimulated emission,

296 7. BLACKBODY RADIATION, ATOMIC EMISSION, AND LASERS

replaced by the Doppler line width. The line shape gd(ν) is approximately 1 divided by (2ν0 (2kT /mc2) ln 2) (see Eq. (7.39).)

L8HENES is only on the CD.

7.5.5 Simplified Example of a Three-Level Laser

We consider an energy schematic of the oscillators shown in Figure 7.14 with an upper state 3, a lower state 2, and the ground state 1. We have for transitions between 3 and 1 induced absorption, induced emission, and spontaneous emission, as discussed in Section 7.5.2. The transition probabilities are called W31, W13, and A31, respectively. A similar description holds for the transitions from 2 to 1. The transitions from 3 to 2 are now special. They are radiationless transitions and their probability is called S32. The occupation number of the oscillators in the states 1, 2, and 3 are called N1, N2, N3 and the total number N0 is assumed to be a constant.

For the steady state, the time derivatives are zero. We assume that A31 W31, which gives us for the ratio N2/N1 of the oscillators

From the discussion of blackbody radiation, W13 W31 and W12 W21, and we assume that the radiationless transition probability S32 is much larger than the transition probability W13 (the probability to empty, state 3). We then have from Eq. (7.63)

Using Eqs. (7.60) as N0/N1 and (7.61) to (7.63) we may get after some calculations

The threshold condition for operation is obtained when, in the steady state, there are as many oscillators in state 1 as in state 2, N1 N2. From Eq. (7.65) one has W13 A21. The minimum power for operation is calculated from the condition that the power corresponding to induced absorption of level 3 is equal to the power corresponding to the spontaneous emission of state 2. In FileFig 7.9 we calculate P NA21hν for a process involving a metastable state and spontaneous emission.

FIGURE 7.17 (a) Coordinates for discussion of the radius of curvature of the wavefront of a mode in a confocal cavity; (b) radius of curvature of the wavefront is indicated for different values of the z coordinate.

where k 2π/λ and try to find a solution of the wave equation of the form

Inserting Eq. (7.67) into Eq. (7.66) and assuming for the calculation that the second derivative δ2ψ/δz2 may be neglected, we further consider the paraxial wave equation

Solutions of Eq. (7.68) describe a Gaussian beam profile in the transverse direction, depending on the distance r from the axis, where r2 x2 + y2 (Figure 7.17). A solution of Eq. (7.68) is

with

and

where the two functions q(z) and P (z) are not independent of each other. We are mainly interested in the solution for q(z) and can solve for P (z) if we need it, which is a phase factor.

We write the solution of Eq. (7.70) as