Материал: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo

1.1.6.1 Normalized shape functions

Normalized line shapes are introduced, which determine the relative strength of interaction.

The line shape depends on the specific interaction process. Two standard line shapes, easy to handle, are the Lorentzian and the Gaussian profiles [92Koe], shown in Fig. 1.1.12. They can be normalized di erently.

I ω

∆ω *DXVVLDQ /RUHQW]LDQ

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1.1.6.2 Mechanisms of line broadening

1.1.6.2.1 Spontaneous emission

The spontaneous emission decay time Tsp of quantum dot lasers can be influenced by the geometry [97Scu], but for all macroscopic laser systems it is equal to the free-atom decay and related to the dipole moment (see Sect. 1.1.5.2). The line width of the power spectrum is ∆ ω = 1/Tsp . The line shape is Lorentzian for undisturbed systems.

1.1.6.2.2 Doppler broadening

In thermal equilibrium the particles in a gas have a Maxwellian velocity distribution of the velocity v:

with

mA : atomic mass,

κT : thermal energy of the particles.

The resonance frequency of a transition is shifted by the Doppler e ect

∆ ω = ωAv/c0 .

Replacing the velocity in (1.1.76) by the frequency, delivers for the resulting spectral distribution a Gaussian line shape (1.1.74) with the width

Some numbers are compiled in Table 1.1.5.

Table 1.1.5. Doppler and collision broadening for a thermal energy of κT = 1 eV. The Doppler broadening refers to ωA = 1015 s−1, the collision broadening holds for a pressure of p = 133 Pa (1 torr) [81Ver, 01Men].

1.1.6.2.3 Collision or pressure broadening

Elastic collisions between radiating atoms imply no energy loss, but a discontinuous jump in the phase of the emitted field. The average temporal length of the wave trains, in the undisturbed case

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given by the spontaneous life time Tsp, is reduced to the collision time τ . The Fourier transform of these shortened waves gives a Lorentzian line shape with the spectral width ∆ ωC = 2/τ or

√32 σC p

∆ ωC = (1.1.97)

π mA κT

with

σC : collision cross section of the atom, p : pressure of the gas.

The collision broadening is proportional to the gas pressure. Some numbers are given in Table 1.1.5.

1.1.6.2.4 Saturation broadening

A strong field of intensity J , comparable with the saturation intensity Js, depletes the upper laser level. The gain is reduced according to (1.1.58a), (1.1.58b) and the gain profile becomes flatter and broader with the spectral width (see Fig. 1.1.13) [81Ver]:

∆ωS = ∆ωA 1 + J/Js .

1.1.6.3 Types of broadening

The interaction of the field depends strongly on the type of broadening. Two idealized cases are the homogeneous and the inhomogeneous broadening [00Dav].

1.1.6.3.1 Homogeneous broadening

All transitions have the same resonance frequency ωA. The gain is saturated for all atoms in the same way as given by (1.1.58a) and shown in Fig. 1.1.13. Examples for this type of broadening are:

–spontaneous emission,

–collision broadening,

–saturation broadening,

–thermal broadening in crystals by interaction with the lattice vibrations.

g

Frequency of the radiation field

Fig. 1.1.13. Saturation of homogeneously and inhomogeneously broadened systems by a radiation field of frequency ω.

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1.1.6.3.2 Inhomogeneous broadening

Groups of atoms with spectral density h(ωR, ωA) and di erent frequencies ωA produce a resulting line profile with center frequency ωR and width ∆ ωR as shown in Fig. 1.1.14. A strong monochromatic field of frequency ω interacts mainly with the group ωA = ω and saturates this particular group. A dip appears in the profile, which is called spectral hole-burning. Examples of inhomogeneous broadening are:

–Doppler broadening,

–Stark broadening in crystals due to statistical local crystalline fields.

h ( , R)

f ( , A)

∆ R

∆ A

A R

Frequency

Fig. 1.1.14. An inhomogeneously broadened profile.

The resulting line profile is a convolution of the individual group profiles and the broadening process, which results in complicated integrals. The saturation process for inhomogeneously broadened lines is quite di erent, as will be shown by a simple example. In this case (1.1.58a) holds only for one group of atoms with the spectral density h(ωA, ωR). Integration over all groups results in the total gain coe cient ginh:

−∞

If the width ∆ ωA is much smaller than the total width ∆ωR, the function h(ωA, ωR) can be taken outside of the integral at ωA = ω. Assuming a Lorentzian profile for the single group, (1.1.98) becomes:

The gain saturates slower than in the case of homogeneous broadening, but the maximum gain is lower by the ratio of the line widths. Inhomogeneous gain profiles can also be caused by spatial hole burning in solid-state laser systems. The standing waves between the mirrors produce an inversion grating and holes in the spectral gain profile [86Sie].

The spectral characteristics of lasers depend strongly on the type of broadening, see Fig. 1.1.15. In steady state the gain compensates losses and the gain profile saturates to fulfill the condition GRV = 1. A homogeneously broadened gain profile saturates till the steady-state condition is fulfilled for the central frequency. The bandwidth ∆ ωL,h is very small and depends on the thermal and

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Fig. 1.1.15. Spectrum of an inhomogeneously and homogeneously broadened laser transition in steady state. Total gain factor GRV vs. frequency of the radiation field.

mechanical fluctuations [00Dav]. In the case of solid-state lasers spatial hole burning will influence the spectral behavior and can produce even for homogeneous transitions multi-mode oscillation [63Tan, 66Men]. In the case of inhomogeneous broadening each spectral group of atoms saturates separately and many modes will oscillate, which produces a large lasing bandwidth ∆ ωL,inh. If single-mode operation is enforced by suitable frequency selecting elements, the left → right and the right → left traveling waves produce two symmetric holes, due to the Doppler e ect. This e ect can be used for frequency stabilization (Lamb dip [64Lam]).

1.1.6.4 Time constants

The line profile of a real laser transition is in most cases a mixture of homogeneous and inhomogeneous profiles, depending on the temperature and the pressure. The following time constants are used in literature:

Some examples of decay times are given in Table 1.1.6.

1.1.7 Coherent interaction

Radiation field and two-level system are two coupled oscillators. Without stochastic perturbations the stored energy is permanently exchanged between these two systems.

If the interaction time of the radiation field with the two-level system is small compared with all relaxation times, including the pump term, the stochastic processes can be neglected and

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