This article will go through the necessary steps about using the using the Pound-Drever-Hall stabilization technique to stabilize a TLB-7000, TLB-6900, or TLB-6300 series tunable diode laser to a reference cavity. If it is your first time attempting active laser stabilization, it is recommended that you should start with a reference cavity linewidth much broader than the free-running linewidth of the laser.
This article should be considered only as an initial reference for laser stabilization, as it requires more advanced servo designs and careful engineering of the reference cavity construction and mounting before truly optimum performance can be achieved.
In atomic, molecular, and optical physics, narrow-linewidth continuous-wave (cw) lasers are now a standard tool. They have been used to attain precision measurements in many fields of science and continue to do so to this day. One downside to their use is that the “free-running” linewidth, or short-term stability of the laser, means that they are unsuitable for many applications without active stabilization of the laser frequency.
This article will go into detail about a powerful and elegant technique, used to control and stabilize the frequency of a cw laser to a high-finesse optical cavity, for some of the most challenging precision measurements in modern optics. This is an important factor, as both Precision spectroscopy and the manipulation of atomic and molecular systems have directly benefited from the resulting improvement in cw laser stability over the past several decades.
For example, atomic clocks built around optical transitions need incredibly stable laser sources to ensure accurate probing of the sub-Hertz linewidths available in laser-cooled samples1. Another area that is critically dependent on the availability of narrow-linewidth laser systems with extremely low frequency and amplitude noise, are interferometric measurements such as those taken in the search for gravitational waves (e.g. LIGO2).
The following section will talk about the primary features of a feedback control system used for laser stabilization and go deeper into its main roles. The systems required components and a simple implementation will then go described in more detail. The final section will form an analysis of the closed-loop performance of the system.
Feedback Loop Basics
Figure 1 shows a schematic of a simple feedback loop3. The start of the loop is the laser that is to be stabilized. The feedback loop uses a split off part of the initial output. The inherent noise of the laser can be characterized in terms of the linear spectral density of frequency fluctuations, defined as the RMS frequency fluctuations (δvRMS) per square-root unit bandwidth (B): SF(f) = dnRMS/B1/2 [Hz/Hz1/2].
Types of laser can differ considerably in free-running linewidth and noise spectrum, depending on the stability and finesse of the resonator design, gain-medium characteristics, and other laser parameters.
For example, in many solid-state lasers with relatively high-finesse resonators, the spectral density of frequency noise is often dominated by pump and mechanical fluctuations, which will typically drop as ~1/f and can be largely suppressed by the feedback loop. In models such as New Focus™ diode lasers on the other hand, the principle frequency/phase noise is often of a quantum nature due to larger spontaneous emission rates7. This results in significant noise processes extending out to higher Fourier frequencies.
Detecting fluctuations in the laser frequency, requires a highly stable reference for comparison. One of the typical methods of achieving this is to use a high-finesse Fabry-Perot cavity, built in such a way as to provides the required stability over the time scale of interest. Using the Fabry-Perot cavity (vm) and the cavity length L as: vm= m(c/2L), we can determine the mth resonant frequency.
This can be extremely sharp when low-loss, high reflecting mirrors are used8. While mechanical cavities may start to drift over longer time scales, when used for a period of seconds they can provide very high stability. This stability can be exploited due to a combination of a linear response to the incident optical field and a sharp cavity resonance.
As opposed to the nonlinear response of atomic transitions that can saturate, it is possible to increase the reference cavity signal until the signal-to-noise ratio (SNR) of the detected cavity resonance is high enough to provide the needed stability for the laser.
To tightly fix the frequency of the laser to a resonance of the Fabry-Perot cavity, rapid detection of a resonance with a SNR must be occur. As it is this that ultimately determines the performance of the system, it is possibly the most important part of the feedback loop. The disparity between the laser frequency and cavity resonance is converted into a voltage, with a discriminator coefficient D given in units of V/Hz.
There a number of methods to obtain the discriminator voltage, or “error signal”. The clearest and most straightforward way is to lock to the side of the cavity transmission fringe. By using the slope on either side of the transmission peak, the side-fringe locking technique can convert frequency fluctuations of the laser into amplitude fluctuations, which are then picked-up by a photodiode.
A simple technique to implement, it nevertheless suffers from several drawbacks. First, amplitude modulation (AM) from the laser directly couples into the error signal; the feedback loop cannot distinguish between frequency modulation (FM) and AM. Adjustments in the laser amplitude will therefore be “written” onto the laser frequency. Secondly, because of the short photon-lifetime of the Fabry-Perot cavity8, fast fluctuations in the frequency of the laser will not be detected in transmission through the cavity. A final restriction is the narrow locking range.
If there is even a small deviation from the locking point, the laser can unlock if the frequency briefly shifts across the cavity transmission peak. These last two limitations present a particularly concerning tradeoff; high-finesse cavities are desirable so as to provide a narrow linewidth for laser stabilization, yet at the same time will restrict the bandwidth of the feedback loop and soundness of the lock9.
A more easily implemented method is “Pound-Drever-Hall” (PDH) stabilization, which also manages to avoid all the above-mentioned issues9. PDH stabilization is not dissimilar to the powerful technique of modulation-spectroscopy used for the sensitive detection of atomic and molecular transitions10. Pound11 first identified this technique as a method for the stabilization of microwave oscillators by introducing phase modulation at a frequency several times greater than the resonance linewidth.
To work around the limitations of AM on the laser beam, PDH stabilization instead quickly probes both sides of the cavity resonance using the rapid modulation of a laser's frequency. The amplitude fluctuations can then be reduced to their shot-noise limited level, if the resonance information is detected at a sufficiently high modulation frequency.
As well as this, PDH stabilization utilizes the light reflected from the Fabry-Perot cavity. This a benefit as the reflected light will be at a minimum on resonance decoupling AM noise from the error signal. Another critical factor of the PDH technique is that the response isn’t limited by the cavity lifetime, allowing for a greater bandwidth in the feedback loop. The next section will talk about the details of this technique and its simple implementation.
After the error signal (e) is generated, it is transmitted through the servo “loop filter” to ensure the feedback is applied to the laser with the appropriate phase. As there is a finite time delay in the feedback loop, none of the Fourier frequencies of the error signal can be sent back to the laser with the proper phase. The frequency-dependent voltage gain (G, with units of V/V) must therefore roll off toward zero at some frequency to prevent positive feedback.
Once the signal is conditioned by the loop filter, the correction voltage is finally applied to the actuator, characterized by a coefficient A in units of Hz/V. The maximum bandwidth of the servo loop is usually determined by the frequency range over which the actuator exhibits a flat frequency response to the applied correction signal.
For instance, a way of correcting the laser frequency is using a piezo-mounted cavity mirror as an actuator, as they often have a resonant frequency on the order of a few kHz. Thus to prevent the excitement of the piezo resonance the servo bandwidth needs to remain much less than this. New Focus TLB-7000, TLB-6000 and TLB-6300 series lasers are made with handy inputs for linear frequency tuning up to several kHz. With appropriate servo designs, these can be used alongside each other to provide rapid laser-frequency corrections5.
Figure 1. Simplified schematic of laser feedback loop. SF: spectral density of frequency noise; E: error signal point; D: discriminator coefficient; G: loop filter gain; A: actuator coefficient.
Pound-Drever-Hall Laser Frequency Stabilization
Experimental Setup and Conceptual Model
The New Focus™ StableWave™, Velocity™ and Vortex™ series lasers offer narrow instantaneous linewidths. This makes them the ideal choice for precision spectroscopy, and by using the PDH locking technique they can be easily stabilized to a high-finesse cavity. The New Focus range also comes with many of the critical components needed for PDH laser stabilization. Figure 2 shows an experimental layout to carry out PDH laser frequency stabilization.
An optical isolator should be used subsequent to the laser to remove back-reflections from coupling into the laser cavity. For quick phase modulation of the laser, a New Focus electro-optic phase modulator (Model 4001 or 4003) can be coupled with the driver Model 3363. These modulators enable an extremely pure phase frequency modulation of the laser beam with especially low residual amplitude modulation (RAM)12. It is important to be aware that if there is unwanted amplitude modulation on the optical beam present at the same frequency at which the laser phase/frequency is being swept will corrupt the detected error signal and result in what could be unstable offset when locking to the cavity resonance13.
Once it has passed through the modulator, the laser beam is spatially mode-matched to a high-finesse Fabry-Perot reference cavity. A low-noise photodiode (e.g., New Focus Model 16X1-AC) detects the light reflected from the cavity. A polarizing beamsplitter and then a quarter-wave plate separates the light reflected by the cavity from the incident light. The power reflection coefficient of the cavity as a function of laser frequency, is shown and 3(a) and expressed in units of the cavity free spectral range for a cavity with a finesse of ~160.
Although there are some components remaining in the setup in Figure 2, we will first discuss how the photodiode signal provides information on the laser frequency fluctuations, forming a conceptual model for the PDH frequency stabilization technique9. Firstly, the phase modulator sweeps the phase of the laser carrier at a frequency greater than the linewidth of the cavity resonance. This phase modulation produces frequency of modulation(Ω) spaced FM sidebands on the laser carrier.
As these sidebands are well outside the cavity resonance, they are directly reflected from the reference cavity input mirror without considerable phase shift. The light reflected from the cavity at the carrier frequency will be in phase with the incident light, when the laser carrier frequency is matched to a cavity resonance. If the reflected carrier and sidebands are detected by the photodiode, two heterodyne beats are yielded at the frequency of phase modulation, Ω, resulting from the beat between the carrier and each sideband.
On resonance, these two beats will be 180º out of phase with respect to each other due to the nature of FM. Hence the two beats will directly eliminate each other and there will be no photodiode signal at the frequency of modulation.
When the laser frequency deviates from the cavity resonance, the reflected light at the carrier frequency undergoes a phase shift, and depending on the direction of the frequency deviation it will be positive or negative [Figure 3(a)]. Also, now that they aren’t exactly 180º out of phase, the two heterodyne beats produced by the photodiode do not cancel each other out. Therefore, by detecting the amplitude of the beat signal at Ω, and error signal is produced that can be used to correct the laser frequency.
Figure 2 demonstrates how the amplitude of the beat is obtained by frequency-mixing the photodiode output with the signal at frequency Ω that drives the phase modulator. To ensure the mixer inputs have the proper phase relationship, a phase shifter is used between the modulation signal source and the mixer. A low-pass filter is used to isolate the DC term from the mixer output, which then gives the error signal that is applied to the actuator for laser-frequency correction, after appropriate conditioning.
Figure 3(b) shows how the phase shifter should be adjusted to match the error signal, as the laser frequency is swept through the cavity resonance. This figure is gained through a modulation frequency Ω equivalent to 6% of the cavity free spectral range. A more detailed analysis of the PDH technique can be found in the appendix, which goes into detail about how the amplitude of the photodiode signal at frequency Ω provides information on laser frequency fluctuations. A key feature of the PDH error signal is its broadband response.
The dashed line in Figure 4(c) represents the log magnitude of the cavity response versus Fourier frequency (Bode Plot). At low Fourier frequencies, the error signal response is constant over frequency. When the frequency increases above the half-width-half-maximum (HWHM) of the cavity resonance, there is a smooth –6 dB/octave roll-off of the cavity response function vs. Fourier frequency.
This change in the cavity response function is due to the storage time of the reference cavity. The phase of the reflected light at the carrier frequency represents a time-average of the incident laser phase and frequency. As directly reflected sidebands do not undergo a shift in phase from the cavity, the heterodyne beat between the reflected sidebands and the carrier is sensitive to rapid fluctuations of the laser phase, letting the laser to effectively be phase-locked to a time-average of itself. It is this ability of to provide corrections for frequency fluctuations over a broad bandwidth that makes the PDH locking technique such a powerful tool for laser frequency stabilization.
Figure 2. Experimental layout for PDH laser frequency stabilization. EOM: electro-optic modulator; PBS: polarizing beamsplitter; λ/4: quarter-wave plate.
Figure 3. (a) Power reflection coefficient and phase shift on reflection vs. laser frequency for Fabry-Perot cavity with finesse of ~160. Frequency scale is in units of the free spectral range. (b) PDH error signal vs. laser frequency for modulation frequency Ω = 6% of free spectral range.
Figure 4. (a) and (b) Useful circuits for the servo loop filter and their corresponding frequency responses. (c) An example of the total servo gain for a stable feedback control loop (solid line), shown on a log-log scale, and the frequency response of the reference cavity (dashed line). fr corresponds to the HWHM of the reference cavity resonance.
Feedback Loop: Loop Filter and Actuator
All that remains after a good error signal is obtained is to send this signal through the servo loop filter and back to the laser's actuator to “close the loop”. As talked about in an earlier section, loop filters job is to alter the error signal such that it is applied to the laser with suitable amplitude, phase, and frequency response. If the phase of the error signal shifts too quickly it can result in positive feedback; to prevent this, it is important that the gain for the overall feedback loop has a slope of less than 9 dB/octave at the unity gain frequency (fo)14.
The value of the unity gain frequency is reliant on the overall loop gain, given by: Gtotal= DGA. An example of a Bode plot for the overall gain of a stable feedback control loop is shown in Figure 4(c) (solid line). Figures 4(a) and 4(b) show two examples of useful circuits often found in the servo loop filter along with their individual frequency responses. If the bandwidth of the feedback loop is much less than the cavity roll-off frequency fr (i.e. fo << fr), the simple integrator circuit in Figure 4(a) is all that is needed to provide a stable overall response function for the loop.
For fo > fr, the circuit of Figure 4(b) can be used to turn off the integral gain at a frequency when the response of the reference cavity begins to roll off (i.e. set f1≤fr). The solid line in Figure 4(c) is the total servo gain for both cases. There are more sophisticated response functions that can be applied to improve performance, especially to provide increased gain at lower frequencies15.
By analyzing the closed loop error signal: δeRMS = δvRMSD (see Figure 1), the ability of the servo electronics to track frequency fluctuations and keep the laser tightly locked to the cavity resonance can be estimated. By running a Fourier analysis on this signal it reveals how well the feedback loop is suppressing detected fluctuations at various Fourier frequencies.
Depending on the laser system the free-running noise spectrum will vary, but one source of noise common to most systems, “1/f noise,” typically drops off as the inverse of the Fourier frequency to some power p. The feedback loop must therefore have a high enough bandwidth so that the servo gain will equal or exceed the rise of this noise at lower frequencies.
It isn’t possible to only use the in-loop analysis to determine the absolute stability or linewidth of the laser. This is because analysis at this point cannot determine the difference between real fluctuations of the laser frequency and noise present in the discriminator itself. For example, if the resonance of the reference cavity is shifting due to gradual thermal expansion or tiny vibrations of the cavity length, the feedback loop will write these fluctuations onto the real output of the laser.
Monitoring the error signal voltage is a reasonable indicator of how well the feedback-loop electronics are performing, but an independently stabilized laser system is necessary to quantify the absolute stability or linewidth of the laser.
As an example, we can suppose that the error signal voltage e is monitored while the laser is locked to the cavity resonance. The RMS voltage measurements at this point (δeRMS, over a bandwidth determined by the range of the voltage meter) can be converted into an RMS frequency fluctuation of the laser using the measured actuator coefficient: δvRMS= δeRMS/D. If a 10 mV signal is measured at e for a given discriminator coefficient of D=1 V/MHz, it follows that the servo electronics are able to keep the laser locked to within 10 kHz (RMS) of the cavity resonance.
- B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible Lasers with Subhertz Linewidths,” Phys. Rev. Lett. 82, 3799 (1999).
- For a more detailed discussion of laser stabilization as a problem of control theory, see Refs. 4, 5, and 6.
- J. L. Hall, “Stabilizing lasers for applications in quantum optics,” in Quantum Optics IV, J. D. Harvey and D. F. Walls, eds. Springer Verlag: Proceedings of the Fourth International Conference, Hamilton NZ, 1986.
- J. L. Hall, M. S. Taubman, and J. Ye, “Laser Stabilization,” in Handbook of Optics: Fiber Optics and Nonlinear Optics, Vol. 4, M. Bass, J. M. Enoch, E. W. V. Stryland, and W. L. Wolfe, eds. (McGraw-Hill, New York, 2001), p. 27.1.
- T. Day, E. K. Gustafson, and R. L. Byer, “Sub-Hertz Relative Frequency Stabilization of Two-Diode Laser-Pumped Nd:YAG Lasers Locked to a Fabry-Perot Interferometer,” IEEE J. Quan. Elec. 28, 1106 (1992).
- C. E. Wieman and L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1 (1991).
- E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1998).
- R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B 31, 97 (1983).
- See for example, New Focus Application Note #7 and references therein.
- 1R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum. 17, 490 (1946).
- See New Focus Application Note #2.
- Ch. Salomon, D. Hils, and J.L. Hall, “Laser stabilization at the millihertz level,” J. Opt. Soc. Am. B 5, 1576 (1988).
- G. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback control of dynamic systems, (Menlo Park: Addison-Wesley Publishing, 1988).
- J. Helmcke, S. A. Lee, and J. L. Hall, “Dye laser spectrometer for ultrahigh spectral resolution: design and performance,” Appl. Opt. 21, 1686 (1982).
- More in-depth descriptions can be found in other review articles, e.g. E. D. Black, “An Introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys. 69, 79 (2001).
This information has been sourced, reviewed and adapted from materials provided by Oriel Instruments.
For more information on this source, please visit Oriel Instruments.