Amplification of Coherent Radiation

Abstract

In the preceding chapter, we discussed the interaction of broadband radiation with an ensemble of two-level atomic systems. Here, we treat the interaction of monochromatic radiation with an ensemble of two-level atomic systems. We will show that the photon density in a disk of light traveling in an active medium increases exponentially with the traveling path length.

Problems

7.1

Amplification of radiation in titanium–sapphire. Given is an active titanium–sapphire medium with a population difference $N_2 - N_1 =10^{24}$ m$^{-3}$.

• (a) Determine the gain coefficient at the frequency of maximum gain.

• (b) Determine the single-path gain factor at the frequency of maximum gain when the crystal has a length of 10 cm.

• (c) Determine the gain coefficient and the single path gain factor of radiation at a wavelength in vacuum of 1 $\upmu$m.

7.2

Gain cross section of Ti $^{3+}$ in titanium–sapphire. Compare the gain cross section of an excited Ti$^{3+}$ ion with the gain cross section of a two-level system that has a naturally broadened line at the frequency of maximum gain coefficient of titanium–sapphire.

7.3

Two-dimensional gain medium. The two-level systems of a two-dimensional gain medium have a gain cross section $\sigma _{21} =1.5 \times 10^{-19}$ m$^2$. The population difference is equal to $N_2^\mathrm{2D} - N_1^\mathrm{2D} =10^{16}$ m$^{-2}$.

• (a) Estimate the modal gain coefficient in the case that radiation propagates along the active medium and that the mode has a height of 800 nm.

• (b) Estimate the gain for radiation traversing the medium.

7.4

Anisotropic media.

• (a) We manipulate the two-level atomic systems of an active medium (for example, by applying a magnetic field, so that the atomic dipoles have an orientation mainly in one direction instead of a random orientation); we assume that $A_{21}$ does not change. Determine $B_{21}$ and $\sigma _{21}$ for radiation of different orientations of the electric field vector of electromagnetic radiation.

• (b) We assume that we orient the two-level systems with their dipoles in a plane. Determine $B_{21}$ and $\sigma _{21}$ for radiation polarized either parallel or perpendicular to the plane.

7.5

Oscillator strength. The classical oscillator model of an atom provides the classical absorption cross section of an atom, $\sigma _\mathrm{{cl}}(\nu )=e^2/(4\varepsilon _0m_0c)g_\mathrm{{L, res}}(\nu )$, according to (9.67)

• (a) Show that the classical absorption strength is

  $$\begin{aligned} S_\mathrm{cl}\equiv \int {\sigma _\mathrm{cl} (\nu )\mathrm {d}\nu }=\frac{e^2}{4\upepsilon _0 m_0c}. \end{aligned}$$

  (7.66)

• (b) Show that the quantum mechanical absorption strength is equal to

  $$\begin{aligned} S\equiv \int {\sigma _{21} (\nu )\mathrm {d}\nu }=\frac{n}{c} h\nu B_{21}= \frac{c^2A_{21}}{8\pi n^2\nu ^2}= \frac{c}{8\pi n^2\nu ^2\tau _\mathrm{s p}}. \end{aligned}$$

  (7.67)

• (c) We introduce the oscillator strength f via the relation

  $$\begin{aligned} S =S_\mathrm{cl}\times f. \end{aligned}$$

  (7.68)

  Estimate the oscillator strengths, which correspond to the absorption and to the gain cross sections of titanium–sapphire.

• (d) Show that in case of a narrow line

  $$\begin{aligned} S = \frac{\lambda _0^2}{8\pi n^3\tau _\mathrm{s p}}, \end{aligned}$$

  (7.69)

  where $\lambda _0 =c/\nu _0$.

7.6

Fluorescence line and absorption cross section.

• (a) Show that we can write, in case of a narrow line caused by transitions in an ensemble of two-level atomic systems,

  $$\begin{aligned} \frac{1}{\tau _\mathrm{s p}}\approx \frac{8\pi c n^2}{\lambda _0^4}\int {\sigma _{12}(\lambda )\mathrm {d}\lambda }, \end{aligned}$$

  (7.70)

  where $\lambda _0$ is the center wavelength, n the refractive index and $\sigma _{12}$ the absorption cross section.

• (b) Show that this leads to the relation

  $$\begin{aligned} \sigma _{12}(\lambda )= \frac{\lambda _0^4}{8\pi n^2\tau _\mathrm{s p}}\,\frac{S(\lambda )\mathrm {d}\lambda }{\int {S(\lambda )\mathrm {d}\lambda }}, \end{aligned}$$

  (7.71)

  where $S(\lambda ) \mathrm {d}\lambda$ is the fluorescence intensity in the wavelength interval $\mathrm {d}\lambda$ at the wavelength $\lambda$ and $\int {S(\lambda )\mathrm {d}\lambda }$ is the total fluorescence intensity. This relation is sometimes called Füchtbauer-Ladenburg relation; in the 1920s, Füchtbauer studied absorption lines [48] and Ladenburg (see Sect. 9.10) fluorescence lines of atomic gases.

7.7

Gain saturation. We consider a four-level laser medium and take into account both pumping and relaxation. Instead of (7.13), we write

$$\begin{aligned} \mathrm {d}N_2/\mathrm {d}t =r - b_{21}Z(N_2 - N_1) - N_2/\tau ^{\star }_\mathrm{rel}, \end{aligned}$$

(7.72)

where r is the pump rate (per unit of volume). We assume that $\tau _\mathrm{rel} \ll \tau ^{\star }_\mathrm{rel}$ and therefore $N_2 \ll N_2$, and find

$$\begin{aligned} N_2 =N_{2, 0} (1+b_{21}\tau _\mathrm{rel}^{\star }Z). \end{aligned}$$

(7.73)

We introduce the intensity $I=cZ h\nu$. It follows that the large-signal gain coefficient is

$$\begin{aligned} \alpha _\mathrm{I} = \alpha /(1+I/I_s), \end{aligned}$$

(7.74)

where

$$\begin{aligned} I_\mathrm{s} =c/\left( B_{21}g(\nu )\tau ^{\star }_\mathrm{rel}\right) \end{aligned}$$

(7.75)

is the saturation intensity.

• (a) Sketch gain curves for $I/I_\mathrm{s} =0$; 1; 10. [Hint: in the case of homogeneous broadening, the whole line saturates.]

• (b) Determine the saturation intensity For Nd:YAG.

• (c) Determine the saturation intensity For titanium–sapphire.

7.8

Saturation of absorption.

• (a) Consider an ensemble of two-level atomic systems and show that the large-signal absorption coefficient is

  $$\begin{aligned} \alpha _\mathrm{abs, I} = \alpha _\mathrm{abs}/(1+I/I_\mathrm{s}), \end{aligned}$$

  (7.76)

  where $\alpha _\mathrm{abs}=-(n/c)h\nu B_{12}g(\nu )(N_2 - N_1)$ is the small-signal absorption coefficient, $I=cZ h\nu$ the intensity of radiation, and

  $$\begin{aligned} I_\mathrm{s} =c/\left( 2B_{12}g(\nu )\tau ^{\star }_\mathrm{rel}\right) \end{aligned}$$

  (7.77)

  the saturation intensity. [Hint: begin with (7.26); take into account that the total population density $N_\mathrm{tot} =N_2 + N_1$ is constant; introduce the population difference $\varDelta N$, with $\varDelta N =N_2 - N_1$; then derive the differential equation for $\mathrm {d}(\varDelta N) / \mathrm {d}t$ and determine the steady state solution; because the lower level remains populated, the saturation intensity is smaller (by a factor two) than in case of a four-level system with a short lifetime of the lower laser level (Problem 7.7).]

• (b) Determine $\varDelta N, N_2$, and $N_1$ for $I=I_\mathrm{s}$.

• (c) Sketch absorption curves of a transition for $I/I_\mathrm{s} =0$; 1; 10.

• (d) Why is the saturation intensity in case of saturation of absorption and in case of gain saturation independent of the populations of the two-level systems?

7.9

Show that the transition probability for stimulated emission induced by monochromatic radiation in a frequency band ${\text {d}}\nu$ is given by

$w_{{21{\text {stim}}}} {\text { }} = {\text { }}B_{{21}} \rho {\text {(}}\nu {\text {)}}g{\text {(}}\nu {\text {)d}}\nu .$

7.10

The photon flux in a beam of monochromatic radiation is $\varPhi = cZ$, where Z is the photon density.

• (a) Show that the transition probability for stimulated emission for an atom in the beam is equal to $w_{{21}} (\nu ) = \sigma (\nu )\varPhi$.

• (b) Determine the photon flux that is necessary to reach $w_{{21}} = 10^{{ - 9}}\,{\text {s}}$ for the laser materials mentioned in Table 7.1.

7.11

Relate the transition probability for stimulated emission to the growth coefficient and to the gain coefficient.

7.12

Test of equations. Compare the dimensions of left and right side of the following equations:

(7.18), (7.23), (7.38), and (7.52).

[Hint: for the dimension of $B_{21}$, see (6.11) or Table 6.1.]

7.13

Determine $B_{12}$ for the transition that is responsible for the absorption band of titanium–sapphire for E || c (Fig. 7.7).

7.14

Optical thickness and self-absorption. The optical thickness of a material is defined as the product $\alpha L$, where $\alpha$ is the absorption coefficient and L the length of the material; a material is optically thick if $\alpha L \gg 1$ and optically thin if $\alpha L \ll 1$.

• (a) Determine the length of a titanium-sapphire crystal for which $\alpha L = 1$ in the center of the pump band of a TiS laser. [Hint: make use of Fig. 7.8]

• (b) Determine the thickness of a TiS crystal at which self-absorption of fluorescence radiation strongly influences the fluorescence spectrum.