Stabilizing an Interferometric Delay with PI Control

Transcription

1 Stabilizing an Interferometric Delay with PI Control Madeleine Bulkow August 31, 2013 Abstract A Mach-Zhender style interferometric delay can be used to separate a pulses by a precise amount of time, act as a frequency comb, and measure lengths to extreme precision. The accuracy and precision for each application is dependent on the length stability in the interferometer itself. While passive length stabilization is enough for many applications, especially on a short time scale, relying solely on passive stability is not always sufficient or practical. Instead, this paper explores the effectiveness of active length stabilization, using a feedback loop from the interferometer s output intensity at one port to a piezo-mounted mirror. 1 Introduction 1.1 Interferometers and Their Applications An optical interferometer operates by sending light on two or more different paths, then recombining the light and observing the constructive or destructive interference that results. In an interferometric delay, there is a significant difference between two path lengths, so that light traveling the longer path not only experiences some phase shift relative to the shorter one, but also arrives at the recombination site a non-negligible amount of time later. The path length difference in such an interferometer will have a related free spectral range, meaning that the setup can be used as a frequency comb, as in Ref [1] where it is used to separate a carrier signal from its sidebands. In order to effectively filter light, or make precision measurements, the path length difference in the interferometer must be known and constant. To achieve this, it is necessary to stabilize the path length difference against environmental perturbations on the order of nanometers. Ref. [2] shows an example of such stabilization in a balanced Mach-Zehnder interferometer, where the path lengths differ by at most a few micrometers, while Ref [3] shows its application in a more complicated setup. 1

2 Figure 1: The Mach-Zehnder style interferometric setup, with feedback from one photodiode to a piezo for length correction. 1.2 PI Control The active stabilization of our interferometer used PI or Proportion-Integral control to turn an error signal, related to path length deviation, into a correction signal, which was used to change the path length as needed. Proportional- Integral control is so called because it creates a correction signal which is a weighted sum of the instantaneous error (the proportional part) and the integral of the error over time. The correction signal looks like u (t) = K P e (t) + K I t 0 e (τ) dτ (1) where coefficient K P is called the proportional gain and K I is the integral gain. The proportional part corrects for error on a short time scale, but may maintain a constant non-zero error if its gain is too low, or start oscillating from overshooting if its gain is too high. The integral part will correct any error that exists on a long-term scale, but has little effect on error on shorter time scales. Used together, they can correct for error occurring at both low and high frequencies. 2

3 2 Methods 2.1 Interferometer Setup The Mach-Zhender delay, Figure 1, operates by separating light at Beamsplitter A, and sending it on two different paths - one of them longer by a path length difference l - then recombining them at Beamsplitter B to produce interference patterns. If the light is initially in phase φ, then after traveling the top path and arriving at Photodiode 1, it has phase: φ 1t = π π l + π π d = 2π + 2π d + 2π l + 1 2π l + π 2 2 while the light traveling the bottom path has phase shift (2) (3) φ 1b = 2π d resulting in an effective phase difference of (4) φ 1 = 2π + 2π l between the light from each path. Assuming that the laser light has initial intensity I 0, the fraction of the initial light sent along the top path and reflected toward Photodiode 1 is α 1t, and the fraction transmitted straight through the bottom path to Photodiode 1 is α 1b, then the intensity observed at Photodiode 1 from the constructive interference of these beams is ( ) 2π l I 1 = max (α 1t I 0, α 1b I 0 ) + min (α 1t I 0, α 1b I 0 ) cos 2π + (6) ( ) 2π l = max (α 1t, α 1b ) I 0 + min (α 1t, α 1b ) I 0 cos (7) The voltage produced across Photodiode 1 from this incident light is then ( ) 2π l V 1 = V 1,offset + V 1,pp cos (8) where V 1,offset and V 1,pp are dependent mostly on the initial intensity I 0. For the purposes of this stabilization setup, we will assume that both the intensity and wavelength of the initial laser light are constant, so that variation in V 1 is attributed entirely to variation in l. The voltage measured here is then used as input for the feed back loop, to correct perceived changes in length. (5) 3

4 2.2 Error Signal and Correction Since we are attributing the voltage at Photodiode 1 solely to length changes, fixing the length to a specific l means fixing V 1 to a specific set voltage, V stab. To observe length changes with the greatest precision, we picked a set voltage at a point where V 1 has the greatest sensitivity to changes in l: at approximately V 1,offset, when the incident light halfway between a bright fringe and a dark one. V 1,offset was estimated by observing the maximum and minimum voltages produced by full oscillations in the incident light, and taking their mean. This estimate was entered as V stab into a commercial PI control box, and subtracted from the input voltage V 1 to produce an error signal, e (t). With this error as input, the PI control box produced a correction signal V out = K P e (t) + K I t 0 e (τ) dτ (9) where the integral and proportional gain were varied systematically until the error signal got locked. 3 Results When the interferometer was successfully locked, the noise in the input voltage, and therefore the length, was visibly transferred into the correction signal to hold the input voltage approximately stable, as shown in Figure 2. The error voltage can be translated into a error length, by solving ( ) π 2π δl ( π ) V 1 V stab = V 1,pp cos + V 1,pp cos 2 2 ( ) 2π δl = V 1,pp sin (10) for the length displacement δl at a given instant. This conversion assumes that V stab is exactly V 1,offset, but since sin is approximately linear in this region, the assumption is valid to first order. Taking samples of stabilized and unstabilized voltage variation as in Figure 3 and converting these voltages into length shows the typical range of length displacement, as in Figure 4. The root-mean-squared length displacement for the samples depicted is typical: x rms,unstab = 27.0nm and x rms,stab = 3.0nm. In an interferometer where l is 10.5cm, this amount to lengths stability about on the order of one in 10 7 We can also examine the effectiveness of stabilization as a function of frequency, by taking a Fourier transform of the output voltage and converting voltage displacement density into length displacement density. Figure 5 shows this frequency response, where the unstabilized interferometer exhibits a large amount of noise in the Hz range, including a number of spikes presumably corresponding to some vibration or mechanical resonance in the interferometer setup. The stabilized frequency response shows that this noise has been killed 4

5 Figure 2: The voltages V 1 and e (t) with noise before stabilization, and the (scaled) voltage V stab correcting for this noise during stabilization. almost entirely at low frequencies (< 100Hz), and damped for higher frequencies (at least up to 400Hz). 4 Discussion To consider how the stabilization might be improved, we must first check our assumption that the observed error is actually due to variation in length, and not to variation in the frequency or intensity of the incident light. Noise on the order of a part in 10 7 is still significantly larger than any noise in the frequency of the laser, so frequency variation can be safely ignored. Measuring just the intensity of the laser and converting it to perceived length changes gives us the histogram in Figure 6. Comparing root-mean-squared (perceived) displacement x rms,intensity = 1.5nm to x rms,stab = 3.0nm, and standard deviation σ intensity = 0.4nm to σ stab = 0.8nm, it is clear that intensity noise rather than length variation accounts for a significant portion of the error signal during stabilization. To eliminate this problem, ideally, we would use an error signal with less or no dependence on intensity. One option is to use the signals both from Photodiode 1 and Photodiode 2, which have mean voltages V 1,offset and V 2,offset, the ratio which should be constant and intensity independent. By stabilizing V 1 V 2 at V 1,offset V 2,offset = R fixed, although the magnitude of nonzero error would still be intensity dependent, the path length difference l corresponding to zero error would now be intensity independent. Other improvements could involve changing the gain in the PI controller specifically to kill off the apparent resonant frequencies in the Fourier transform 5

6 Figure 3: Samples of the error signal e (t), with and without stabilization. Figure 4: The distribution of length displacement, in nanometers, during the samples shown in 3. 6

7 Figure 5: Length displacement density against frequency. Most of the lowerfrequency noise in the unstabilized interferometer is successfully eliminated by stabilization. Figure 5, and systematically determining what gains would be optimal for stabilizing the system without prompting overshoot and over-oscillation. Another extension of this work might introduce the ability to stabilize l to an arbitrary value, instead of exclusively at l = ( n + 2) 1 for integer n. In Ref. [1] shows that this is possible in a balanced Mach-Zehnder interferometer, varying the l continuously over the range of a few micrometers. 5 Conclusion The goal of this project was to create an interferometer whose path length difference could be maintained at a constant value over long time periods of time. The interferometer was built with materials to maintain a reasonable amount of passive length stability, then a feedback loop was introduced which achieved stability on the order of a few nanometers over long time scales. To make the interferometer significantly more stable (i.e. an order of magnitude or more), it would be necessary to change the design of the feedback loop, to eliminate or decrease its dependence on the intensity of the stabilization laser. 6 Acknowledgments Many thanks to Wes Campbell, Andrew Jayich, Sylvi Haendel, Michael Ip, Danilo Dadic, Sam Freitas, and Anna Wang for helping me around the lab. Thanks to the NSF for funding this REU, and to Francoise Queval for coordinating it. 7

8 Figure 6: With one arm of the beam interferometer blacked so that no interference is occurring, the voltage on Photodiode A still varies from its expected value, the same as if l were changed by a couple nanometers. References [1] Bateman, J. E., et al. HnschCouillaud locking of MachZehnder interferometer for carrier removal from a phase-modulated optical spectrum. JOSA B 27, (2010). [2] Freschi, A. A., and J. Frejlich. Adjustable phase control in stabilized interferometry. Optics letters 20, (1995). [3] Krishnamachari, Vishnu Vardhan, et al. An active interferometerstabilization scheme with linear phase control. Opt. Express 14, (2006). 8