An exited atom can emit a photon via stimulated or spontaneous emission.  The photon energy will be equal to the energy difference between the final and the initial state.  However, the uncertainty principle prevents us from knowing this energy difference precisely.

DEDt > ~h

Emission lines therefore have finite widths.  Excited atomic states have a typical lifetime of 10^-8 s.  This corresponds to a natural line width for emission lines of ~6 10^-8 eV or ~15 MHz.  Absorption lines also exhibit this broadening.

Lifetime broadening is not the only reason emission lines have finite widths.  The resolution limit for atomic spectra in the visible region is often determined by Doppler broadening.  The frequency of the light emitted by an atom will be Doppler shifted because of the thermal motion of the atoms.  A stationary detector measuring the frequency of the light emitted by atoms traveling towards or away from the detector with speed v will measure a Doppler shift towards a higher or lower frequency, respectively, when compared to the frequency of light emitted by the same species of atoms at rest.  The light from a stationary source absorbed by atoms traveling towards or away from the source with speed v will have a lower or higher frequency, respectively, than that absorbed by atoms at rest.  In an atomic vapor the atoms have a velocity distribution due to their thermal motion.  Since the thermal velocities are non-relativistic, the Doppler shift in the angular frequency of the light absorbed by atoms traveling towards or away from a stationary source is given by the simple form

w = w₀(1 ± v/c),

where w₀ is the angular frequency of the light absorbed by atoms at rest.

At high pressures, perturbations of the energy levels by collisions (pressure broadening) can become the limiting factor for the resolution.  Power broadening occurs because of the shortening of the lifetime of the excited states due to stimulated emission.

Natural, pressure, and power broadening produce Lorentzian line shapes, and Doppler broadening produces a Gaussian line shape.  A convolution of a Lorentzian and Gaussian line shape can be approximated by a Voigt profile.

The high-resolution details of atomic spectra, such as the hyperfine splitting, are often obscured by the sources of line broadening.  In a low-pressure gas, the main source of broadening is Doppler broadening from the thermal motion of the atoms or molecules of the gas.

Assume that we want to measure the absorption spectrum of an atomic vapor.  Assume that for a particular transition the energy difference between the initial and final state is E_f – E_i = hw₀.  When a laser beam propagates through an atomic vapor cell, the number of atoms moving with velocity v in a small velocity interval between v and v + dv in the direction of propagation of the light is

n(v)dv = N (m/(2pkT)^1/2 exp(-(mv²/(2kT)))dv,

where N is the total number of atoms and m is the atomic mass of the atoms.  An atom moving with velocity v in the direction of propagation of the light will absorb photons with frequency w = w₀(1 + v/c). We have

(w - w₀)/w₀ = v/c.

The intensity of the absorbed radiation as a function of frequency is therefore given by

I(w) = I₀ exp[-mc²(w - w₀)²/(2kTw₀²)] .

I₀ is the absorbed intensity when w = w₀.

This profile has the shape of a Gaussian with FWHM

Dw_Doppler = (2w₀/c)(2 ln2 kT/m)^1/2,

or

Dl/l = 2(2 ln2 kT/(mc²))^1/2,

since Dw/w = Dl/l.

If the frequency of a laser beam propagating through the atomic vapor is scanned across the absorption profile for a particular transition, the intensity of the absorbed radiation as a function of frequency will reproduce the full Doppler-broadened line shape.  

5S_1/2 to 5P_3/2 transitions in ⁸⁵Rb and ⁸⁷Rb
Doppler broadened line shape
The hyperfine structure is obscured.

At any given laser frequency, however, only a subset of the available atoms, whose velocity along the laser beam is such that the Doppler condition is fulfilled, absorbs the laser light.

"Doppler-free saturation spectroscopy" using tunable diode lasers can be use to minimize the effects of the Doppler broadening and to measure narrower line shapes.  The light from the laser is split into two beams, a saturating beam and a probe beam, arranged so that they cross in a region of a gas cell containing the atomic vapor.  The intensity of the saturation beam must be high enough to appreciably lower the population difference between atoms in the lower and in the upper state.  On resonance, the saturation parameter s is defined as the ratio of the number of photons involved in induced transitions from the upper to the lower state to the number of photons emitted in spontaneous transitions.
The population rate equations for the upper and lower state can be written as

dN₁/dt = GN₂ + aI(N₂ - N₁),  dN₂/dt = -GN₂ - aI(N₂ - N₁),

where 1/G is the excited state lifetime.  The stimulated transition probability is proportional to the intensity I of the laser beam, and a is the proportional constant.

Each photon stimulating a transition from the upper to the lower state produces a second, identical photons.  Therefore

s = 2aIN₂/(GN₂) = 2aI/G.

In the steady state dN₁/dt = dN₂/dt = 0 and N₂/N = (aI/G)/(1 +2aI/G) = (s/2)/(1 + s).
[GN₂ + aI(N₂ - N₁) = 0, (G + aI)N₂ = aIN₁, (G + 2aI)N₂ = aI(N₂ + N₁), N₂/(N₂ + N₁) = aI/(G + 2aI)]
If s = 1, then 1/4 of the atoms are in the upper state.  The saturation intensity I_s is the laser intensity for which s = 1.  I_s is given by 2p²hcG/(3l³).  The intensity of the saturating beam should be greater than I_s.  For the 5S_1/2 to 5P_3/2 transitions in Rb the saturation intensity is approximately 1.6 mW/cm².

If we monitor the transmission of the probe beam as the frequency of the laser is scanned, then the transmission is not affected by the presence of the saturating beam at most frequencies.  The two laser beams interact with completely different, independent groups of atoms.  But when the frequency of the laser beam is very close to the resonance frequency, then both beams begin to interact with the same atoms, those whose velocity component along the direction of propagation of the light is approximately zero.  Both beams will induce transitions from the initial to the final state in the same atoms.  But the strong, saturating beam will quickly deplete the number of atoms in the initial state, leaving very few for the probe beam to interact with.  For the probe beam there is no excess of initial-state atoms left to interact with because the saturating beam has equalized the populations of initial and excited-state atoms.  The weak beam travels through the vapor cell without being depleted even though it is at resonance.  This effect is known as "saturation."

Saturation spectroscopy eliminates Doppler broadening by directing the saturation beam and the probe beam through the gas in opposite directions.  The only atoms, which are in resonance with both beams, are atoms, which have no component of velocity in the direction of propagation of the beams, and therefore absorb at the frequency associated with atoms at rest.  In the rest frame of atoms with velocity components along the beams, the frequencies of the probe beam and the saturation beam are shifted in different directions, and the atoms can only be in resonance with one of the beams, not both.

5S_1/2 to 5P_3/2 transitions in ⁸⁵Rb and ⁸⁷ Rb
"Doppler-free saturation spectroscopy"
The hyperfine structure is resolved.

Example:

The absorption spectrum due to 5²S_½  - 5²P_3/2  transition in rubidium (Rb)
The 5²P_3/2 excited state lies an energy equivalent of 780 nm above the 5²S_½ ground state.

The ground electron configuration of rubidium (Rb) is:1s²; 2s², 2p⁶; 3s², 3p⁶, 3d¹⁰; 4s², 4p⁶; 5s¹.  Rubidium has a 5s electron outside of closed shells.  Its energy-level structure resembles that of hydrogen.  The first excited state is produced when the single 5s electron becomes a 5p electron, and no core electrons are excited.  Natural rubidium has two isotopes with different nuclear spin quantum numbers I.

The allowed 5s to 5p transition in the two isotopes are show in the figure below.

A transition between some hyperfine sublevel in the 5²S_½ state and a hyperfine sublevel in the 5²P_3/2 manifold must satisfy the selection rules Df = 0, ±1, Dj = 0, ±1,  Dl = ±1, Dm_f = 0, ±1.

A typical experimental setup is shown in the figure below.

A beam from a diode laser is split into three beams: a saturation beam, a probe, and a reference beam.  The probe and reference beams are of the same intensity and are passed through a low-density rubidium vapor cell.  The transmitted intensities of the probe and reference beams are recorded by a differential photo-detector.  This detector can output the amplified difference between the two signals.  The much more intense saturating beam enters the rubidium vapor cell from the opposite side and overlaps the probe beam.

If only the probe beam is propagating through the cell, we expect to measure the full Doppler profile of the 5S_1/2 to 5P_3/2 transitions in ⁸⁵Rb and ⁸⁷Rb, as shown in the figure below.

If the saturating beam is on when the laser is resonant with a particular transition, then the probe and saturating beams will both excite atoms with approximately zero velocity along the axes of the incident beams.  Since the saturating beam has a higher intensity than the probe beam, it will excite most of the atoms,  allowing the probe beam to pass through virtually unabsorbed.

The reference beam, however, does not overlap the saturating beam and it is almost completely absorbed at resonance.  If the transmitted intensity of the reference beam is subtracted from the transmitted intensity of the probe beam, then the difference in the intensities when plotted as a function of laser frequency shows the hyperfine structure transitions as a series of spikes.

Let us examine the ⁸⁷Ru F = 1 to F' transitions.  The level spacings and a measured difference spectrum are shown in the figures below.

The saturated absorption spectrum exhibits the three expected peaks corresponding to the three hyperfine transitions F' = 0, 1, and 2.  Between each pair of hyperfine peaks we observe a crossover peak.  Crossover peaks occurs when there are several hyperfine transitions under the Doppler profile. 

Assume that the laser frequency w is tuned exactly half way between two hyperfine peaks with frequencies w₁ and w₂ (w₁ < w₂)  that share a common lower state.  Assume that the saturation beam is Doppler shifted into resonance at w₁ for atoms which move with speed (w - w₁)/w₁ = v/c in the direction of propagation of the saturation beam.  Then the probe beam will be Doppler shifted into resonance at w₂ for atoms which move with speed (w₂ - w)/w₂ » v/c towards the probe beam, i.e. in the direction of propagation of the saturation beam.  The atoms interacting with each of the beams are in the same velocity class.  Therefore there will be fewer atoms of this class in the ground state available to to absorb the weak probe beam, and it will travel through the vapor cell without being depleted.  An additional absorption peak is produced.  The crossover transitions are often more intense than the normal transitions. 

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The output wavelength and the linewidth of diode lasers depend on many factors, such as the cavity length of the laser chip, the bandgap, and the temperature and injection current.  To obtain a stable frequency output, it is important to stabilize the diode's temperature and the injection current, because the refractive index, the cavity length and the gain curve all change slightly with temperature and current.  

Since diode lasers emit light over a narrow wavelength range, wavelength tuning is possible.  External cavity tunable laser diodes come in various forms.  One type is a Littman-Metcalf cavity in which a diffraction grating in conjunction with a PZT tuning mirror is used to achieve a single mode tunable output beam. 

The zeroth order diffraction peak, i.e. the reflected beam, becomes the output beam, while the first-order diffraction peak is send back into the diode laser chip.

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Laser Diodes