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FUNDAMENTALS OF LIGHT
SOURCES AND LASERS

Mark Csele

A JOHN WILEY & SONS, INC., PUBLICATION

Page 174

Assuming that the distance between the cavity mirrors is L, the condition for a

standing wave in the cavity is

m
l

2
¼ L (6:3:1)

where m is an integer, so the spacing (FSR) of resonant modes in the cavity is

Dn ¼ m
c

2L
(6:3:2)

where Dn is the spacing in hertz. For any interferometer the sharpness of the reso-

nant peaks for an interferometer (measured as a FWHM defined in Section 2.2) is

described by

d ¼
nf

F
(6:3:3)

where d is the spectral width (FWHM) in hertz, nf is the frequency of the first mode

(m ¼ 1), and F is the finesse of the interferometer; it is a ratio of the mode separation

(FSR) to the spectral width. Finesse is a function of the reflectivity of cavity mirrors:

F ¼
p

ffiffiffi
R

p

1� R
(6:3:4)

where R is the total reflectivity of both mirrors. As the reflectivity of cavity mirrors

increases, the spectral width of the peaks becomes very narrow, and in most lasers

the reflectivity of the cavity mirrors is very large.

Example 6.3.1 Argon Laser Spectral Width Computation An argon laser with a
1-m cavity spacing has an HR with a reflectivity of 99.99% and an OC with a reflec-

tivity of 98.0%. Compute the expected spectral width of a single mode of the cavity.

FSR

Figure 6.3.1 Response of an interferometer.

RESONATOR AS AN INTERFEROMETER 163

Page 175

SOLUTION Begin by computing the FSR as

nFSR ¼ m
c

2L
¼

3� 108

2
¼ 150MHz

This is the spacing between resonant peaks for the interferometer. Now the finesse of

the cavity may be computed first by calculating the total reflectivity of the cavity as

Rtotal ¼ RHRROC ¼ (0:9999)(0:98)

¼ 0:9799

This reflectivity is then substituting into equation (6.3.4), allowing the calculation of

finesse, in this case, 154.7. Knowing that finesse is a ratio of FSR to spectral width,

spectral width can be solved by substituting into equation (6.3.3) to yield an answer

of 969 kHz. This corresponds (using the 488-nm blue line of the argon laser) to a

spectral width of 7.7 � 10
27

nm. This is an extremely narrow spectral width; how-

ever, special techniques (outlined in this chapter) are required to isolate a single

mode from adjacent modes (which in this case are 150 MHz apart). As we shall

see in this chapter, most lasers have a much larger spectral width, originating

from the fact that many modes can oscillate simultaneously in most lasers.

6.4 LONGITUDINAL MODES

In Chapter 4 we introduced the notion of linewidth and stated that real laser gain

media do not have sharp, defined wavelengths but rather, amplify over a relatively

wide range. In a gas laser, Doppler broadening leads to the existence of a gain curve

in which the gain of the laser (and hence the output as well) peaks at a center wave-

length as shown in Figure 6.4.1. At all points on the gain curve where the gain is

sufficient to overcome losses in the laser (and the laser cavity is resonant), the

laser may oscillate and have output. We now know that the cavity itself is an

interferometer and is resonant only at wavelengths spaced apart by the FSR of the

O
p
tic

a
l G

a
in

lpeak

Figure 6.4.1 Gain curve for a practical laser.

164 CAVITY OPTICS

Page 348

infrared, 77

of an unknown gas, 23

Spectroscopic notation, 57

Spectrum, see Emission spectrum

Spherical-plane resonator, 180, 182

Spin, electron, 64–66

Spin-spin effect, 66

Spontaneous emission, 34

as noise, 96–98

rate, 92, 94–95

Spontaneous lifetime, 34, 124, 132

Spot size, 181

Stability, HeNe output, 245–246

Stability, resonator, 178

Stability, semiconductor laser, 322–324

Stefan–Boltzmann law, 2–3

Stern–Gerlach experiment, 63–65

Stimulated emission, 90

rate, 92, 95

Stinger electrodes, 253

Strength, transition, 132, 144

Stripe contact, 316, 322

Superradiance, 104

in a HeNe laser, 238

in dye lasers, 331

in nitrogen lasers, 264

Symmetry, in a nonlinear material, 228

Table of elements, 67

TEA lasers, 93, 267, 290

TEM modes, see Modes, transverse

Thermal equilibrium, 14, 86–87, 89

Thermal lensing, 297

Thermal light, 1, 12

Thermal population:

in carbon dioxide, 289

of levels, 127

Thompson, G.P., 52

Three-level laser, 119, 136–139

Three-phase power, 256

Threshold, gain, 106

and cross section, 145

and inversion, 146

Threshold, inversion, 146

Threshold, pumping, 105, 149

calculated, 150

diode laser, 106

four-level laser, 143

three-level laser, 139

Thyratron, 268, 274–275

Transitional probabilities, 131–132

Transitions:

allowed, 63, 68

electronic, 73

forbidden, 63, 68

radiative, 12

selection rules, 63, 68

Transition strength, 132, 143–144

Transmission, EO modulator, 205

Transmission line, 265–266

Transverse electrodes, 93, 265. See also TEA

lasers

Transverse modes, see Modes, transverse

Triplet absorption, 328, 333

Triplet quenching, 328

Triplet states, 71

in dye, 328, 333

Two-level system, 133

Unstable resonators, 183–184

in an excimer laser, 274

UV catastrophe, 7

UV lasers, 261–281

Valence band, 38

Vanadate, 302

VCSEL, 317–318

VECSEL, 318

Vernier scale, 24–25

VFD (vacuum fluorescent display), 36

Vibrational energies, 73–76

in carbon dioxide, 74–75, 284

in hydrogen, 73

in nitrogen, 262

Vibronic levels, 74, 76

Vibronic transitions, 262

Voltage, half-wave, 205

Voltage multiplier, 242

Walk-off loss, 172

Wave, electron as a, 52

Wavefunction, 53–54

Waveguide lasers, 290

Wavelength, 10

Wavelength, DeBroglie, 51

Wavelength selection, 166

using a prism, 168–169

Wavenumber, 78

Wave-particle duality, 52

Wein’s law, 4–5

White-light (krypton) laser, 249, 259

Windows, Brewster, 161

Xenon chloride, 272

Xenon flashlamp, 118

as ruby pump, 296

as YAG pump, 303

output spectrum, 118

INDEX 343

Page 349

Xenon fluoride, 272

X-ray laser, 105

YAG (yttrium aluminum garnet) laser, 301

absorption spectra, 118

diode-pumped, 225–226, 304–306

energy levels, 121

host materials, 302

Q-switched, 303

thermal effects, 127–128

Young, double slit experiment, 52

Zeeman effect, 63

in an ion laser plasma, 252

ZnSe (Zinc selenide), 285

344 INDEX

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