The output frequency of a diode laser depends on the injection current and the temperature.  Diode lasers can emit light over a range of frequencies, and frequency tuning is possible.  To obtain a stable frequency output, it is important to stabilize the diode's temperature and the injection current.  In a diode laser with grating feedback the output beam reflects off the grating, while the first-order diffracted beam is directed back into the laser diode.  The optical feedback from the grating is spectrally narrowed and peaked at a frequency that can differ from the central frequency of the free-running diode laser.  The feedback narrows the laser linewidth to from ~ 50 MHz to ~1 MHz. The central frequency will be very close to that of the feedback signal.  To tune the central frequency of the laser, the grating is adjusted by applying a voltage (0 – ~200 V) to a piezoelectric transducer (PZT).

Many experiments require a laser with a well-defined frequency.  But over time, the central frequency of a diode laser with grating feedback will drift.  This drift is caused by fluctuations in temperature and injection current and mechanical fluctuations.  Stabilizing the laser by locking it to an external reference reduces this drift.  Laser frequency stabilization is based on the generation of a frequency error signal, which passes through zero at the lock frequency.

A narrow peak in a saturated absorption spectrum is often used as an external reference for frequency locking a laser.  A fraction of the laser output is send through an atomic vapor cell.  A saturated absorption spectrum is measured and the laser frequency is tuned to either the side or the peak of a narrow, saturated absorption line.  Side-locking is one of the simplest stabilization methods.  On the side of a narrow absorption line the output voltage V(w) of the differential photo-detector has a steep slope as a function of the laser frequency w.  To lock the laser to a frequency w₀, for which dV(w)/dw|_w0 ¹ 0, a reference voltage V(w₀) is subtracted from the output signal to produce an error signal err(w) = V(w) - V(w₀).  This error signal serves as input to a feedback loop which adjusts the laser's frequency to produce err(w) = 0.  This is accomplished by adjusting the PZT voltage.  Side-locking is widely used in laser-cooling of neutral atoms where a small detuning of the laser frequency is necessary.  A disadvantage of side-locking is that fluctuations in beam alignment and intensity will alter the lock point and cause a drift in the laser frequency.

Peak-locking is less sensitive to these fluctuations.  But on a peak dV(w)/dw|_w0 = 0.  If the laser frequency has been adjusted to equal the peak frequency w₀, and the voltage of the differential photo-detector is being monitored, it is easy to detect a drift in the laser frequency.  A drift towards higher or lower frequency causes a decrease in the output voltage and an error signal of the same sign.  A non-zero error signal alone is therefore insufficient to determine whether the laser frequency should be increased or decreased.

To lock onto a peak, the laser frequency is dithered slowly at a frequency W.  An AC signal with frequency W in the kHz range is fed into the reference channel of a lock-in amplifier and into the controller of the PZT.  The PZT expands and contracts, which changes the length of the laser cavity.  The frequency of the laser light as a function of time varies as

w(t) = w₀’ + Dwcos(Wt).

Here w₀’ is the frequency of the laser when it is not dithered and Dw >> W; w₀’ may differ from w₀ if the laser frequency has drifted.

The output voltage of the differential photo-detector thus varies as

V(t) = V(w₀’ + Dwcos(Wt)).

The output voltage of the differential photo-detector becomes the input signal of the lock-in amplifier.  The lock-in amplifier multiplies the reference signal with the detector signal and outputs a DC signal, which is proportional to the time average of the product.

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Let us assume that V(w) µ exp(-(w-w₀)²/a²), i.e. that we have a Gaussian peak, as shown in the figure below.

The graphs below plot the output voltage of the detector (blue) and cos(Wt) (red) as a function of time for Dw = 5 MHz and w₀’ - w₀ = 0, 2 MHz and -2 MHz respectively.

w₀’- w₀ = 0

w₀’- w₀ = 2 MHz                                                            w₀’- w₀ = -2 MHz

For w₀’ - w₀ = 0  the output has its maximum value when cos(Wt) = 0.  For w₀’ - w₀ > 0 the output has its maximum value when cos(Wt) < 0, and for w₀’ - w₀ < 0 the output has its maximum value when cos(Wt) > 0.

The figure below shows the output signal of the lock-in amplifier as a function of w₀’ - w₀.  The signal is the time average of the product of cos(Wt) and the output signal of the differential detector.

The lock-in amplifier produces an error signal, err(w) which can be used in a feedback loop to lock the laser frequency at w₀.
On resonance err(w) = 0 and for small dither amplitudes derr(w)/dw ¹0.

A disadvantage of the dither-locking method is that the output of the laser is modulated directly, or that expensive electro-optic components must be used to modulate only the light entering the saturated absorption cell. Furthermore, both peak and side-locking techniques have a limited recovery range, since the resonance peaks are very narrow.

The Dichroic-Atomic-Vapor Laser Lock (DAVLL) technique aims to extend the recovery range by employing a weak magnetic field to split the Zeeman components of an atomic Doppler-broadened absorption signal and then generating an error signal that depends on the difference in absorption rates of the two components.

The Zeeman effect removes the degeneracy of atomic hyperfine states.  A magnetic sublevel characterized my the quantum number m_f is shifted in energy by DE = g_Fm_BB₀m_f , where B₀ is the magnitude of the external magnetic field.  The shift in the transition energy for two sublevels is then given by

DE_trans = DE’ - DE = m_BB₀(g_F’m_f’ - g_Fm_f),

where the primed symbols are related to the upper state.

To use the DAVLL technique, a small fraction of the laser light is passed through an atomic vapor cell.  The cell is placed inside a large solenoid.  The magnetic field B generated by the solenoid is parallel or anti-parallel to the wave vector k of the laser light.  The laser light must be linearly polarized.

Let B₀ point into the z-direction.  The Zeeman effect splits each formerly-degenerate hyperfine energy level characterized by the quantum number F into 2F + 1 components characterized by m_f, with m_f ranging from F to -F in integer steps.  For optical transitions the selection rules are Dm_f = 0, ±1.   For Dm_f = 0 the electric field vector must be parallel to the magnetic field B₀.  But the electric field vector E of the laser light oscillates in a plane perpendicular to B₀, so no Dm_f = 0 transitions are induced.  Right-circular polarized light traveling anti-parallel to B₀ induces Dm_f = 1 transitions, and left-circular polarized light traveling anti-parallel to B₀ induces Dm_f = -1 transitions.

The Zeeman effect shifts the transition energies for Dm_f = 1 transitions relative to the transition energies for Dm_f = -1 transitions.  If the Doppler broadened absorption curve is shifted towards higher frequencies for right-circular polarized light, then it is shifted towards lower frequencies for left-circular polarized light.

After the light has passed through the vapor cell, it is split into two beams with orthogonal circular polarizations.  This can be done using a quarter-wave plate and a linear polarizing beam splitter.  The quarter wave plate changes the two orthogonal circular polarization components in the beam leaving the vapor cell into two orthogonal linear polarization components.  The linear polarizing beam splitter directs these component into different photodiode detectors.  The output voltages of the two detectors are proportional the the intensities of the right and left hand circular polarized beams exiting the cell.  By subtracting the two output voltages an anti-symmetric error signal is generated which passes through zero and is suitable for locking.

The advantage of the DAVLL technique over the side-locking or peak-locking techniques is its large tuning range. The tuning range of the DAVLL technique is limited by the width of the Doppler broadened absorption peaks, while the tuning range of the other techniques is limited by the width of the Doppler-free saturated absorption peaks.  Frequency modulation is not required.

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