Real-time precision refractometry: new approaches

Transcription

1 Real-time precision refractometry: new approaches Mark L. Eickhoff and J. L. Hall We introduce two new approaches for near-real-time, high-precision tracking of the refractive index of the ambient atmosphere. The methods can be realized at low cost and are expected to have important practical application in those accurate dimensional metrology applications employing interferometry in air. A valuable potential application is the control of step-and-repeat mask positioning for integrated circuit production in which temporal stability time scales over days can be crucial. Extension of the methods to absolute index measurement is discussed Optical Society of America 1. Motivation and Overview Interferometry can provide an accurate and convenient method for measuring lengths or displacements in terms of the wavelength of light. A Michelson interferometer controlling the displacement of some physics apparatus or crossed-axis materialfabricating machine is a common example of this type of measurement. With current laser technology and servo control systems, the controlled stage can be positioned reproducibly within an air environment to within a few micrometers precision: However, variability in the index of refraction limits the achievable accuracy and reproducibility. For applications requiring the greatest precision, a vacuum would be the preferred environment for making these interferometric measurements because the variability of the index of refraction of the surrounding environment would then no longer be a source of noise and drifts. However, it is clearly inconvenient to make these measurements in a vacuum, and the unbalanced forces associated with atmospheric pressure changes can work easily through mechanical compliance to degrade the obtainable precision seriously. A practical and customary recourse is to measure the index of refraction precisely and utilize that information to When this research was undertaken both authors were with the JILA, University of Colorado and the National Institute for Standards and Technology, Boulder, Colorado. Mark L. Eickhoff is now with Ophir Corp., Littleton, Colorado J. L. Hall is also with the Quantum Physics Division, National Institute of Standards and Technology, Boulder, Colorado Received 11 September 1995; revised manuscript received 25 March $ Optical Society of America correct the errant displacement-controlling interferometer. Previous absolute measurements of refractivity have been made at high accuracy with various traditional or innovative interferometric systems, but the cost and convenience levels are not well suited for practical applications. For example, a recent publication reported an international intercomparison of index of refraction determinations by several major standards laboratories with interferometers geared specifically for index of refraction measurements. 1 Agreement was in general quite good , but the methods appear inconvenient for general application, such as in a semiconductor production clean-room facility. Indeed, the excellent agreement was achieved only when the gas flow system to each refractometer was identical. Furthermore, each refractometer took some significant time to come to equilibrium with the gaseous sample, thus delaying any prospective corrective actions. A second approach to providing index of refraction information is to employ a miniature weather bureau to determine the local pressure, temperature, humidity, and CO 2 concentration. Edlén s refractive-index formula, 2 which reproduces the extensive data of Barrell and Sears, 3 is based on these simpler environmental measurements and gives the index of refraction of standard dry air for a wide range of temperature and pressure. Refractive contributions can be usefully included for the water vapor and CO 2 content of the air. Edlén s formula has been updated numerous times since it was first developed 1,4 8 and is generally regarded as a realistic approach to obtain the needed refractivity corrections. 1 This Edlén formula approach certainly works, but 20 February 1997 Vol. 36, No. 6 APPLIED OPTICS 1223

2 it requires care and potentially expensive equipment to measure the environmental parameters accurately enough to produce the sub distance measurement accuracy level that is usually sought. For example, considering that the atmospheric refractivity at STP is , a refractive-index measurement objective of requires pressure to be known to 37 Pa 0.04%, temperature to be known to 0.12 C, and relative humidity RH to be known to 10% RH. These tolerances can be difficult to achieve with generic transducers and generally require calibration against working standards. Furthermore, for a measurement objective at plus or minus a few 10 8, one would need to employ an appropriate means to determine the concentration of CO 2, for example by means of gas chromatography or infrared absorption spectroscopy, which at the least increases the complexity of the index of refraction measuring system. Thus some direct scheme employing optical interferometry becomes attractive. Both of our new instruments simplify the measurement of refractivity by precisely measuring a single observable quantity in real time. The first method measures optical phase as obtained from the fringes of an open parallel-plate Fabry Perot interferometer. A frequency-stabilized He Ne laser illuminates this mechanically stable interferometer, and a portion of the ring pattern of fringes is imaged onto an array detector and digitized. A personal computer PC then fits the pattern for the interference phase, from which the refractive index can be calculated. The second method uses a rf beat frequency measurement for the readout. A tunable laser is locked onto a mechanically stable cavity that has its intracavity space exposed to the environment. Any changes in the index of refraction within the cavity mode volume are mapped onto the frequency of the laser. These changes are detected by a heterodyne beat frequency measurement relative to a frequency-stabilized laser: The rf photosignal is measured by a counter and recorded with a PC. A useful feature of this second method is that it results in a laser output of constant wavelength in the local air. A feature common to both systems is the high resolution achieved within one optical wavelength, thus enabling the use of a relatively short étalon while still maintaining the desired overall precision. In turn, this gives our methods a cold-start capability in which a tracking measurement can be resumed unambiguously after an interruption. In this paper we explain, discuss, and contrast our two new approaches with each other and with the Edlén s formula approach. To help the reader evaluate the situation, we present rather detailed descriptions of our implementations of each of these approaches in the context of an index-tracking application and compare the results obtained. For example, although all refractivity-measuring approaches should yield the same results under steady-state conditions, they may well demonstrate different responses under dynamic conditions. Several extensive data records are presented that show the accuracy and dynamic response of each of the three methods as we attempt to measure presumably the same physical quantity. The recorded physical environmental input data for Edlén s formula approach temperature, pressure, and the relative humidity give some insight into possible systematic problems unique to each of the three methods. We conclude with a discussion of several unresolved questions and areas for future work, particularly relating to absolute refractivity measurement. 2. Weather Bureau Approach With Edlén s Formula To establish the quantitative validity of our new techniques for index of refraction determination, it is appropriate to make a careful comparison with the existing methods in wide use. In practice, this means our using Edlén s formula. The major contributions to the refractive index of air are pressure, temperature, RH, and CO 2 content. A convenient form of Edlén s formula is given by Ref. 1: n 1 D P T F, (1) where D C 300, and P is the pressure in pascals, T is the temperature in degrees Celsius, F is the pressure of water vapor in pascals, and C is the CO 2 content in parts in 10 6 ppm. For the Edlén method to be sufficiently accurate for its use as a reference standard in our experiments, it is necessary to have sufficiently accurate knowledge of the input data. The pressure may be monitored conveniently and accurately by a resonant cylinder pressure gauge. 9,10 One commercially available instrument has an RS-232 port for communication with apc which controls the entire experiment. The manufacturer s stated accuracy of the instrument is 0.01%, which could introduce an error of in the refractive index as calculated by Edlén s formula. The instrument is unaffected by normal changes in the composition of the ambient air to this level of accuracy. 11 To eliminate any calibration drift questions, this instrument was recalibrated by the manufacturer shortly before our experiments. Air temperature is measured by a precision, calibrated thermistor 12 that has a resistance near 6 k at 20 C. The readout scheme places this thermistor in a bridge circuit that compares the voltage drop across a precision resistor with the voltage drop across the thermistor. The excitation current is less than 10 A manufacturer s calibration current, which keeps self-heating K W in the thermistor below K. A mercury-wetted relay flips the polarity of the bridge excitation voltage, which allows suppression of spurious thermal electromotive force within the circuit. The various bridge voltages are measured by a digital voltmeter 14 with a general-purpose interface bus capability, which allows communication with the PC. The resolution of this scheme is approximately 0.05, which translates into K and which would 1224 APPLIED OPTICS Vol. 36, No February 1997

3 lead to a resolution in the air index from Edlén s formula of However, temperature readout accuracy is also limited by the calibration accuracy of the reference resistor and possible nonlinearity of the digital voltmeter. With the specifications of the present system, the estimated inaccuracy in the temperature measurement is K, which contributes a refractive-index uncertainty in the Edlén s formula result. The air s water vapor content is measured by a chilled mirror hygrometer, 15 which outputs a current linearly related to the dew point. This current flows through a precision resistor, and the resultant voltage is measured in one channel of a fast analog-todigital converter that resides inside the PC. The partial pressure of the water vapor is then calculated from the dew point with the Goff Gratch formulation for the saturation vapor pressure of pure water over a plane surface. 16 The small difference between the saturation pressure of pure water at a given temperature and the saturation pressure of water vapor in moist air is negligible for our purposes. The manufacturer s specification on the dew point measurement is 0.6 K, which corresponds to an uncertainty in the vapor pressure measurement of approximately 110 Pa. With Edlén s formula, this gives a contribution to the overall uncertainty of The quadratic combination of the above-noted uncertainties impacts the index of refraction reference determination at the level of In addition to these sensitivities, a variation of n results from a variation of 100 ppm in the CO 2 concentration of ambient air. 1 Although this level of increase occurs rather easily in an enclosed, occupied space, our tests were conducted in a well-ventilated laboratory. Therefore in the following calculations, we continue with metrological tradition and assume a fixed value for the CO 2 concentration of 300 ppm, a value actually approximately 10% below the nominal atmospheric content. Numerically, the resulting index change is insignificant. All the instruments are under the control of a compiled QUICKBASIC 17 program running on an IBMcompatible computer. Because one of the objectives of this research is to track changes in the refractive index over time, the program samples each instrument and records the data on the hard disk every 5 minutes. 3. Fabry Perot Refractometer with CCD Readout 18 For light propagating along the normal axis of a parallel-plate Fabry Perot interferometer, the general interference condition can be described by 2nd 0 m e, (2) where n is the index of refraction of the medium between the interferometer mirrors, d is the mirror separation, 0 is the vacuum wavelength of incident light, m is the integer order, and e is the fractional order. When e 0 we have the resonance condition, leading to axial transmission of light. In its most general form, Eq. 2 should contain some term involving the phase shift after reflection, especially if different colors are used. For measurements made at only one color, the phase shift after reflection can be absorbed into the mechanical spacing. So the parameter d is not merely the mirror separation, but rather the mirror separation plus lambda times a phase shift. This distinction can be pursued usefully for wavelength intercomparisons, but it becomes unimportant when one uses the interferometer to track changes in the index of refraction over time. To apply Eq. 1 successfully, the constancy of the parameter d over long periods of time must be demonstrated clearly. In other research, we have shown that the stability of at least two materials is sufficient so that one can reasonably employ such a material system as a working length standard. 19 In practice this stable material is used as the spacer of a Fabry Perot interferometer in which the mirrors are attached by optical contact; see below. An approximate expression can be derived from Eq. 2 for the change in the index at two different times: n 2 n 1 0 2d e 2 e 1. (3) For convenience here, we suppose the length d is not excessive 10 cm, and the index does not vary too much in this equation because then the integer orders for the two measurements are equal and they cancel each other. In our software implementation, however, this assumption of equal integer orders is not invoked; see below. Usually we are dealing with the general tuning case e 0, leading to a dark or partially illuminated region in the center of the well-known Fabry Perot rings that are measured to obtain e, as described below. As noted above, the index exceeds unity by approximately 270 ppm at STP, so to enable interferometry to be performed in ambient air with an accuracy of , we need to know each of the quantities appearing in Eq. 3 to an accuracy better than At this precision, the mirror phase shift term wrapped up in d becomes negligible, and we may even obtain d for this expression from the mechanical length as measured by a micrometer. To summarize, by tracking changes in the optical phase as measured by the observable fractional part of the interferometer order, we can measure and track the changes in the refractive index. We discuss below other problems that occur when one attempts to use this interferometer to measure the index of refraction absolutely. We turn now to a detailed discussion of the experimental system. The layout of the refractometer is shown in Fig. 1. The overall arrangement of the focusing mirror and folding mirror makes for a compact design and has the added advantage that the CCD array with its heat-producing electronics is far away from the interferometer body. The entire refractometer is built on 20 February 1997 Vol. 36, No. 6 APPLIED OPTICS 1225

4 Fig. 1. Layout of the Fabry Perot refractometer. Collimated output from the fiber coupler is converged in the vertical plane before entering the plane plane Fabry Perot interferometer. As discussed in text, the direction of fringes and the CCD detector line are perpendicular to the plane of the figure to eliminate astigmatism. Curvature of the R 100-cm focusing mirror is exaggerated for clarity. an aluminum breadboard that rests on rubber feet on top of an optical table. Light from a frequencystabilized He Ne laser is delivered to the refractometer by way of a single-mode optical fiber. A lens collimates the light from the fiber, and a cylinder lens then produces a horizontal focal line in the interferometer. Instead of our realizing a full circular ring pattern, only a limited diametral zone of the ring system is illuminated, thus conserving the available laser light. The resulting fringe pattern is imaged onto a CCD linear array detector of 1024 elements. To avoid astigmatism from the curved folded mirror, the fringe pattern is organized and detected along a vertical axis. A small pump approximately 1 L min slowly draws the outside air into the space between the interferometer mirrors. The interferometer itself is constructed to approximate an ideally rigid mechanical spacer. The body is made from a cylindrical disk of Zerodur 1.00 in cm thick 2.00-in cm diameter with a thermal expansion coefficient of less than K. The endfaces are ground and polished to be parallel to less than 2 arcsec of wedge and flat to 10. The disk center is drilled out along its axis to provide a clear interferometer path gas sample chamber, and a diameter is drilled out to form the input and output air-sampling vents. The mirrors are fabricated from flat Zerodur substrates, coated only over the central 2-cm region with MgF 2 - protected Al. The reflectivity is approximately 0.8, giving a finesse of approximately 14. The mirrors are attached to the interferometer body by optical contact. The interferometer is mounted inside an Al bracket with azimuthally corrugated strips of stainless-steel shims. This provides a buffer to retain the interferometer firmly, but still allows some leeway for thermal expansion or contraction of the Al bracket. Invar or Zerodur would be a more ideal material for this mounting bracket, but see below. The video signal from the CCD array detector is Fig. 2. a Typical scan of a Fabry Perot fringe system imaged onto a CCD array photodetector. Data points are filled dots and the nonlinear least-squares best fit is the solid curve. The signalto-noise ratio is approximately 50. b Residual between data points and best fit. Obvious extraneous noise at one-half of the sampling frequency is due to the offset between separate odd and even sampling circuits within the CCD chip. The current software filter does not compensate fully. Systematic errors are discussed in the text. processed by a sample hold circuit, monitored on an oscilloscope, and digitized with a 12-bit analog-todigital converter, which is on an expansion card in the PC. For the 30 W of light from the fiber, we obtain optimum utilization of the CCD dynamic range for an integration time of 15 ms frame. A typical data set is shown in Fig. 2. Software can readily control the data acquisition and then process the fringe pattern to fit for the physical parameter of interest which is the fringe phase. The fitting algorithm is described in Appendix A, along with our tests using synthetic data to estimate random and systematic effects. The fitting program has an estimated accuracy of approximately of an order as established from tests with synthetic data with various phases and levels of random additive noise. Of course, establishing the integrity of a computer fitting program by testing with synthetic data does have limitations: Synthetic data fitting can only give an indication of how well the computer fits what we think the data should look like. It does not model unknown systematic effects that have been built accidentally into the apparatus. Indeed, Fig. 2 clearly shows some excessive amplitude noise associated with the prominent central peak, perhaps because of stray reflections or nonuniformity of the CCD pixelto-pixel response. A necessary additional family of tests involves the production of real data with a fixed phase or, better yet, several data sets with their phase relationships known a priori. To realize this, we used a dye laser that is locked tightly to a stable external control cavity. This cavity has a longitudinal-mode spacing of approximately 320 MHz so that locking on successive cavity passbands establishes a set of equally spaced optical frequencies with this common step. These essentially divide up one order of the 1-in cm interferometer into 1226 APPLIED OPTICS Vol. 36, No February 1997

5 Fig. 3. a Plot of output phase from the computerized fringefitting algorithm versus the order number from the dye laser control cavity. Dots are experimental data points. Solid line is best-fit line to those points. One free spectral range of the Fabry Perot is orders of the dye laser s control cavity. b Plot of residual from best fit above versus control cavity order. Error bars are from diagonal elements of the covariance matrix after convergence in the fringe-fitting algorithm. Although some systematic trends are clear, these data show that the optical fringe phase is recovered correctly within orders by the current algorithms. approximately 20 evenly spaced steps, so the phase output of the fringe-fitting code as a function of control cavity order should yield a straight line. These results are shown in Fig. 3. The fitted straight line in fringe-phase space has residuals of the order of a few This high-precision fitting to real, well-calibrated known data provides us a realistic basis for high confidence in the phase values recovered by our apparatus, starting from our actual hardware with whatever actual optical defects are present, proceeding through the data-acquisition and data reduction software to trial output data in the form of interference phases versus time. In spite of this excellent performance documented in Fig. 3, it is of course interesting to wonder about the origin of the large discrepancy between the estimated fitting accuracy from synthetic data and from the residuals of the experimental straight-line fit to the real calibration data. Several factors may contribute: Our interferometer is certainly not mechanically ideal. To begin with, the mirrors are not held perfectly parallel to each other. We measured an approximate residual wedge of approximately 1.5 arcsec. This nonparallelism leads to additional peaks in the Airy pattern that are down by a factor of approximately 150 from the main fringes. Furthermore, the main fringes themselves are skewed and distorted slightly. 20 One can minimize this problem by rotating the interferometer body so that the laser light sees a minimum residual wedge in the plane of interest, i.e., the vertical plane that is illuminated by the cylindrical lens. Unfortunately, this choice produces another problem. Because the laser beam in the interferometer has some nonzero width, the left side of the beam sees a slightly thinner interferometer than the right side. The resultant fringe pattern is the convolution of the usual Airy function with a linearly changing interferometer length. We generated synthetic fringe patterns based on this model and then analyzed them with the fitting code. The size of the phase error is approximately for input phases near 0 and less than for input phases between 0.2 and 0.7. We also found that the presence of a prominent, distorted central peak in the fringe pattern excessively biases the fitting algorithm. Although this model does not explain fully the detailed behavior of the residuals shown in Fig. 3, it does display phase errors of the basic size that we observed. For this reason, we believe that imperfect parallelism of the mirrors is near the heart of the small but clear systematic errors shown in the residuals of Fig. 3. On a smaller level, another problem concerns placement of the detector in the exact focal plane. In practice, we use a one-dimensional translation stage to position the array detector in the focus direction. The fringes appear skewed toward the center when the detector is in front of the focal plane. They appear skewed away from the center when the detector is behind the focal plane. The detector is set normally to the center of two positions of equal but opposite skew, so this effect becomes unimportant. Another class of small effects depends on refraction at the entrance and exit faces of the interferometer plate. The idealized Airy function fringes are formed between the two mirrors and are given in terms of angles in the air. When the light enters the substrate, a Snell s law modification is imposed toward smaller angles in the medium, which ideally would be reversed when exiting. However, to avoid parasitic fringes from within the mirror substrates, they are made from optics with a 15-arcmin wedge. This means that the angular fringe pattern is distorted slightly by the bias in the exit angles before the focusing mirror. This last effect leads to problems of a still smaller scale than observed, but in future units we will try to make the mirror wedge direction be horizontal. Several problems may also affect the accuracy of the refractometer system as it tracks changes in the index over extended periods of time. These problems are mostly mechanical in nature. For example, the corrugated strips that hold the interferometer may expand or contract differentially, thus tilting it. Any small tilt of the interferometer would be perceived as an increase in the thickness, but it is sensitive only quadratically around the desired normal incidence condition. In the data reported here we believe this effect to be unimportant. Another minor problem area lies in the Al coatings. It can be difficult to generate semitransparent Al coatings that are fully dense. We believe that porosity of these coat- 20 February 1997 Vol. 36, No. 6 APPLIED OPTICS 1227

6 ings may be involved in some low-level memory correlations that we saw with humidity when the coatings were young. Temperature dependence of the reflection phase is expected to be negligible. These problems should not affect the accuracy of the computer fit to the Airy function because the time scale for these errors is so much longer than the time scale for a single computer fit. Furthermore, the research discussed in Section 4 shows that dielectric mirrors would be fully suitable for this application. 4. Fixed Air Wavelength Laser Another scheme for measuring the index of refraction of air optically relies on frequency measurements. Frequency counters generally offer a simple, convenient way of making precision measurements. Therefore we designed an experiment that maps the index of refraction into a frequency, basically that of a laser servo-controlled in air wavelength, to keep the resonance condition e 0. Of course this optical frequency is too high for direct counting, and so for the comparisons it must be heterodyned into the rf domain with a frequency-stable reference laser. The use of a passive, optical resonator to stabilize a laser apparently was first discussed by White. 21 We use the changing index of refraction of air within the cavity mode volume to scan the laser cavity detuning. After locking the laser to a cavity transmission peak as described in detail below, we have the resonance condition of Eq. 1, with e 0. The index of refraction can then be written as n m FSR, (4) where m is the cavity order, FSR c 2d is the free spectral range of the cavity, and is the laser frequency. Note that when the atmospheric refractivity changes, the frequency changes in such a way so as to keep constant the wavelength in air, measured as so many orders that fit resonantly into the fixed cavity mechanical length. Just as a laser that is tightly locked to an isolated, evacuated cavity realizes a vacuum wavelength frequency standard, the present setup realizes an air wavelength standard. Again, the phase shift upon reflection is incorporated into the cavity FSR. It is a small correction that may be neglected for the accuracy needed here. The laser s frequency can be read out conveniently by heterodyning it with a frequency-stabilized laser frequency 0 on a fast photodiode to yield a rf beat note. Changes in the index of refraction over time can be approximated by n 2 n 1 n (5) 0 With this approach, it may also be feasible to make measurements on the index of refraction absolutely, a possibility discussed below. As indicated, the idea here is to lock a tunable laser to a passive cavity that has a fixed mechanical length, Fig. 4. a Layout of the air wavelength standard prototype: F.I., Faraday isolator; P.D., photodetector; AOM, acousto-optic modulator that is driven by a voltage-controlled oscillator VCO. Analog 25-kHz applied voltage generates 500-kHz FM excursion of the 80-MHz output, enabling phase-sensitive detection of lock signal. Note that the laser output is unmodulated and has a constant wavelength measured in laboratory air. b Sketch of top and side views of the Zerodur bar cavity that was used in the constant-air-wavelength standard prototype. Curvature of the mirrors is exaggerated for clarity. Mating surfaces are tinned individually with indium solder before the mirrors are attached firmly with indium solder flowed onto the cylindrical surface. but whose geometry is open enough to allow thorough sampling of the surrounding ambient atmosphere. Because the servo makes the laser frequency change so that the waves remain in resonance with the cavity length, the wavelength in air becomes related to the stable cavity: Based on the resonance condition, the frequency of the laser changes to accommodate the varying index of refraction within the cavity mode volume. The setup is shown in Fig. 4 a. The laser selected for this experiment uses a short internal mirror He Ne gain tube with a large FSR of 1.36 GHz. 22 One can accomplish laser tuning with a piezoelectric transducer cylinder epoxied to the glass tube of the laser, which suffices to scan the frequency over the entire FSR of the laser. The cavity-locking scheme at present is a first derivative lock. The laser frequency is modulated at 25 khz by an acoustooptic modulator and the transmission of the cavity is fed to a lock-in amplifier that generates the error signal. To obtain the rf domain frequency readout, a portion of the beam is split off before the acousto-optic modulator and combined into one geometric mode with the output of a fixed frequency laser. The rf signal is detected with an avalanche photodiode, measured with a frequency counter, and recorded on the PC. We emphasize that in an actual interferometric application of the stable air wavelength source, these added complexities of frequency reference laser, fast detector, and counter would be absent: Merely a portion of this laser s output before 1228 APPLIED OPTICS Vol. 36, No February 1997

7 the modulator would provide the stabilized source beam to the air-immersed control interferometer. The cavity itself has a somewhat novel construction and is sketched in Fig. 4 b. Two dielectric mirrors high reflectivity at 633 nm, 43.7-cm radius of curvature are soldered to a bar of Zerodur 2.5-cm diameter 15 cm long with indium solder. The ends of the Zerodur bar are specially machined to accept the mirrors. A milled channel along the length at the top allows unimpeded propagation of the beam between the mirrors. The cavity sits on a stainless-steel V block with pieces of O ring serving to cushion it and also serving as a flexible mechanical buffer. As noted above, the rf beat note between this cavity-stabilized laser and a frequency-stabilized reference laser tracks the index of refraction according to Eq. 4. However, as the index changes, the locking electronics could try to push the laser frequency beyond the laser s own gain curve. Because the laser has a limited tuning range, ultimately it may scan itself out of resonance as the output power decreases and lasing action finally stops. To overcome this problem, the laser can be tuned and relocked on another air-cavity order that happens to lie closer to the center of its gain curve. Thus, for the purpose of tracking large changes in the index of refraction, it becomes necessary for one to measure accurately the FSR of the air cavity. We can then use this FSR as an additive offset of the rf beat note before calculating the change in the index of refraction. For an index of refraction tracking accuracy at approximately the level, it is more than sufficient to know the FSR of the cavity approximately 1 GHz with an inaccuracy of 5 MHz. However, to make an absolute measurement of the index of refraction discussed below, it is desirable to know the FSR to even greater precision: To determine unambiguously the integer order for this cavity, the uncertainty in the FSR needs to be at the 1-kHz level. Frequency modulation techniques enable a precise measurement of the FSR of an optical cavity. The technique used here is a straightforward simplification of a more refined technique that is described in detail elsewhere. 23 An electro-optic phase modulator is used to modulate laser light at a frequency in the rf domain. When the rf power feeding the modulator is small enough, the physical picture of the laser power spectrum after the modulator is one of a strong carrier at the unshifted optical frequency, flanked by two weak sidebands at and. This light is coupled into the cavity and the carrier is locked to the mth cavity order. The rf is then scanned so that the sidebands couple resonantly into the cavity at the m 1 th and m 1 th order. By measuring the dc component of the transmitted light as a function of the rf frequency, we can record a picture of the cavity line shape. The FSR is determined as the center frequency of this cavity line shape. In our implementation, a precise rf synthesizer 24 drives a rf power amplifier that supplies approximately 1Wofrfpower at 1 GHz to a LiTaO 3 traveling-wave modulator. This produces sidebands each with approximately 7% of the available optical power coupled out of the carrier modulation index 0.5. With the carrier locked to the cavity, the sidebands are scanned under computer control, and a voltage generated from the dc photocurrent from the transmitted light is measured by a digital voltmeter and recorded in the computer. Because the sidebands are modulated with the same frequency dither used for locking the carrier to the cavity, the resultant line shape is not simply Lorentzian. 25 A computer fit to this line shape taking into account the frequency dither, the sideband frequency, the modulation indices, the cavity linewidth, a dc offset, an amplitude, and a linear baseline drift gives for the center frequency and thus the FSR a value of MHz in vacuo. Another useful experiment was to measure the FSR when the cavity was open to the air. One correction to the vacuum FSR is then a reduction by a factor of the refractive index n. An interesting effect is the increase by 0.5 ppm that is due to the crushing of the rod s axial length by the atmospheric pressure loading, which is discussed below. First we turn to a presentation of our main data. 5. Experimental Comparison of Three Approaches to Air Index Measurement Data for this experiment are recorded in the following manner. The computer goes through its dataacquisition routine every 5 min. First it reads the output from the humidity transducer, the pressure meter, the thermometer, and the counter that measures the rf beat note from the air-cavity setup. Then a fringe pattern from the Fabry Perot refractometer is digitized and written to the hard disk. Third, each of the weather variables is recorded a second time and averaged with the first. The counter is read a second time, and the reading is averaged with the first. An Edlén s formula calculation is carried out with Eq. 1, and all the variables and the result are recorded into data files on hard disk. Finally, the fringe pattern is fit for the phase. In this scheme, the final parameters describing the previous fringe pattern are used as initial values for the present fringe pattern. An advantage of the present fitting program is that neither the initial value for the fringe fraction e nor the final fitted value are constrained to be within the range 0, 1. Put succinctly, the program can interpret a total order of as plus a fractional order of In this way the output phase may move gracefully an integer order by itself. This elementary memory property of the fringe-fitting program thus suppresses nonphysical data jumps that would result otherwise. The data run presented here began 10 December 1992 and lasted for over 4 days. Figure 5 shows the measured pressure, temperature, and humidity over this time. Figure 6 shows the calculated index of refraction based on Edlén s formula, along with the output of the two refractometers. All three methods 20 February 1997 Vol. 36, No. 6 APPLIED OPTICS 1229

8 Fig. 5. Output of the weather bureau showing a ambient pressure, b temperature, c relative humidity as functions of time. are found to agree at approximately the level over the course of the experiment in which the index varies by approximately However, the residuals between the three methods show some structure and offer some insight into each of the optical methods. It is instructive to examine the correlations between the environmental measurements of Fig. 5 and the differences between the index output data shown Fig. 6. Comparison of the refractometers. The change in the index of refraction of the ambient air is shown as a function of time. Output from each refractometer is plotted on the same axis from 0 at t 0. Agreement for index changes is for some hours and for the entire run. Accidental choice of starting time during epoch of rapid change leads to apparent offset during quiescent times. FP, Fabry Pérot. in Fig. 6. Consider the index as calculated from Edlén s formula and the index as monitored by the parallel-plate interferometer: This visual inspection reveals an important correlation with the air temperature. In the two direct measurements that are due to some thermal accommodation with the interferometer structures, the air temperature inside cannot change so rapidly, so the optical measurements smooth out the rapid time structure. See 15 and 25 h in Fig. 5. In other places, the differences follow the temperature only loosely. The intrinsic change that is due to thermal expansion of the interferometer spacer material is too small by a factor of 10 to explain the correlation. We suspect the offsets from hours 30 to 48 and hours 75 to 85 may be linked to the varying adsorption of water into the somewhat porous MgF 2 Al coating. In early data runs, we did observe a strong correlation of the residual with RH, as noted above. We note that the starting point of this long run occurred fortuitously at a time of major weather change in Boulder, Colorado, leading to a high slope of index versus time. Taken with the small time delay associated with the thermal mass of the interferometers, one has a simple explanation of the small differences in the plateau regions for the indicated index changes reported by the three approaches. If one decided instead to bring the three curves into coincidence in these stationary regions, the rms differences of the indicated index changes, , are redistributed toward the regions of rapid temporal change. We note, however, that an expanded display and comparison reveal that a variation of seems to affect all three readouts on a time scale of a few tens of hours. Perhaps these variations are associated with air flow temperature patterns arising when lasers and other heatproducing equipment were used on nearby laboratory tables: The separation of the three sensing systems was 30 cm. For the most part, the performance of the stable air wavelength system may be somewhat better than the Fabry Perot refractometer system, linked probably to the more open construction and the use of pure dielectric mirrors. However, one advantage at present for the Fabry Perot system is the predictability of its long-term aging properties. A Zerodur cylinder with optically contacted mirrors has small and well-established aging properties that we have observed in other experiments over many years. 19 No comparable measurements have yet been made on the novel cavity with indium solder construction used to realize our air wavelength-stabilized laser. But we observed no unexpected length changes over several days in which we tested the stability of this cavity in an evacuable chamber. This testing included several cycles of low vacuum-to-atmospheric pressure. Because of the openness of the design and its cost-effective fabrication, we will monitor drifts of this unit in the future APPLIED OPTICS Vol. 36, No February 1997

9 6. Absolute Index of Refraction Measurements Both of these two new instruments can be used to make absolute measurements of the index of refraction, as well as to track changes in the index. But some sort of absolute calibration procedure is required before this can be realized. It is clearly possible to have absolute knowledge of 0 in Eq. 1 by way of comparison with an I 2 -stabilized He Ne laser. But knowledge of the mechanical length d and the integer order m is also required. The method of exact fractions 26,27 makes it possible to measure m. It may be optimally effective to determine d in vacuo with d 0 m e 2. (6) The method of exact fractions usually is stated for a parallel-plate interferometer, but sideband techniques similar to those mentioned above can be used to determine the FSR, and from that the integer order can be calculated unambiguously for our nonplanar cavity. 28 The vacuum wavelength on resonance times this integer order, along with corrections for the diffractive and reflective phase shifts, then gives twice the mechanical spacing in a vacuum. The length d can also be determined in air. After correcting for the air index, we still must worry about the fact that d in vacuum is not the same as d in the air because of hydrostatic compression produced by atmospheric pressure. It would appear that reasonable knowledge of the Zerodur s bulk properties notably Young s bulk modulus and Poisson s ratio would enable adequate calculations of the hydrostatic compression under atmospheric pressure. However, we believe that the compression induced by atmospheric pressure is in fact the limiting factor in the estimated absolute accuracy for both of these refractometers. The expected length compression that is due to applied external pressure P is given approximately by the expression l l P 1 2 E , (7) where E is Young s modulus and is Poisson s ratio. This expression is evaluated for Zerodur at Pa 625 Torr, the average pressure in Boulder, Colorado. Now the question arises: How accurate are these corrections? Finite element analysis techniques allow more detailed calculations to be performed, but the symmetry of the open cavity is not ideal. Importantly, the boundary condition imposed on the flat Fabry Perot spacer by the optical contact to the thick mirror substrates appears to affect our preliminary modeling results and so leads to ambiguity. Thus until now, we have not addressed these questions fully. Still, in a laboratory the ambient pressure changes are only a few percent, whereas the compressible étalon effect 29 is below 0.5 ppm for a full atmosphere. So it is clear that within the context of an index-of-refraction-tracker application, this compressibility problem is not of an important magnitude. 7. Results, Conclusions, and Recommendations for Future Research The above discussion shows clearly the potential of both of our approaches to track the in situ index of refraction in a measuring and manufacturing environment. We note that our results presented in this paper are derived from measurements on laboratory air with variable purity and composition, not a pure, standard sample with fixed composition. Furthermore, other research activity in the laboratory during this run led to a variable pattern of temperature gradients. Therefore we emphasize that fine agreement is observed between the three methods, even under conditions in which colleagues were working around the apparatus, making noise, stirring up dust particles, and perhaps changing substantially the CO 2 content in the sampled air. In spite of these perturbations to the ambient atmosphere, which indeed may be typical occurrences in a nonideal measuring and manufacturing environment, our comparison shows that the index of refraction tracking is sure to the level and for several hours, closer to There are several ways in which the Fabry Perot device could easily be improved. Perhaps the easiest and most important would be to substitute hardcoated dielectric mirrors for the currently employed dielectric-overcoated aluminum mirrors. This would solve the humidity memory effect noted in some early runs. The chamber of this interferometer body could also be sculpted to provide a smoother and more uniform air flow pattern inside. The mounting of the interferometer body should be made within a ring of low-expansion material, such as Zerodur. The present azimuthally corrugated radial shim method would then probably be fully satisfactory for floating the interferometer body within a firmly mounted outer Zerodur plate. Another welcome improvement would be to reduce the asymmetry in the profile of the interference fringes, which arises from a residual wedge in the interferometer spacer. Two simple approaches are available for a next-generation device. The simplest approach is to obtain the services of a more sophisticated optics shop that can prepare the desired subarcsecond parallelism between the polished faces. In addition, or as an alternative, it would be useful to determine the direction of the residual wedge in the spacer body after the central optical access hole was drilled and after the final polish was prepared, but before the sampling passageway was drilled. One would then organize the passageway to be drilled perpendicular to the direction of the minimum wedge, which would be mounted vertically in the apparatus to match the measurement axis. Improved software would deal with the residual fringe asymmetries. On the subject of improving the constant-air wavelength device, the ever-accelerating progress in diode lasers leads to an elegant near-future prospect: One could use a diode laser, with fast locking onto the open reference cavity to spectrally narrow the laser 20 February 1997 Vol. 36, No. 6 APPLIED OPTICS 1231

10 frequency width. The output beam would be taken as the transmission beam from the resonator to provide a clean output beam in a single spatial mode. The easy and wide tunability of the diode laser would challenge the electro-optical and mechanical designer of such a system relative to providing the necessary long-term stability and robustness for industrial application, but the system would be able to track any pressure changes encountered. The observed, extremely smooth year length drift rate 19,30 of specially prepared samples of ultralow expansion would lead to an attractive capability of calibrating the device at the factory, for example, with optical heterodyne techniques relative to an I 2 - stabilized He Ne reference laser. The drift rate would be completely insignificant in any foreseeable index-tracker application, and one could enjoy use of the device without recalibration for the greater part of a decade for absolute index determination at the 10 7 level. Appendix A: Computer Algorithms for Fabry Perot Fringes The purpose of this code is to find the fractional order at the center of a Fabry Perot fringe pattern assuming that the integer order is known a priori. The fringe pattern in our realization is imaged onto a linear CCD array of 1024 pixels. The signal is digitized and then stored in the computer s memory as an array of 1024 elements. All the calculations are carried out with double-precision variables. The overall fitting strategy is based on a two-stage process. The first stage involves a fast fit to find approximate values, and the second stage subsequently improves the first stage s output. The fast-fit code is built around the fringe centerfinding algorithm given by Snyder. 31 This algorithm essentially finds the first derivative of the fringe pattern with respect to position in an elegant and rapid manner. The zero crossings of the first derivative, interpolated from the discrete domain, indicate the positions of the peaks in the fringe pattern. This procedure is encoded as a fast subroutine that takes the fringe pattern array with typically bright maxima as input and returns an array elements long of fringe peak positions as output. Once the peak positions are located, a nonlinear least-squares loop is used, based on the fitting function x i a 1 f 2 0 d i 1 a 2 (A1) for the position of the ith bright Fabry Perot fringe. 32 Outputs are the symmetry center a 1 and the fractional order a 2. Here, f is the focal length, 0 is the wavelength, and d is the interferometer spacing. In practice, the original fringe positions may be pulled due to fringe asymmetry caused by the imaging process or by the Gaussian envelope of the input laser beam. Thus the value for the fractional order after convergence can be shifted somewhat and is not accurate enough for our purposes. But the fitted parameters enable a second, improved stage of fitting. One of two general procedures can be adopted for the second stage. In the first procedure, one transforms the linear domain of the fringe pattern as realized by the CCD detector to a domain that is proportional to the optical phase as realized by the angular exit space of the interferometer. The advantage here is that the fringes are uniformly spaced in this domain. The other procedure is a nonlinear least-squares loop that is based on an eightparameter model. In the first procedure, we transform the fringe pattern to an axis that inherently is well suited to computerized fitting. In particular, the output of the peak-finding subroutine mentioned above will be more accurate if the fringes are symmetric and uniformly spaced. However, considering the formula for the intensity versus position I x 1 F sin 2 m e cos x x 0 f 1 (A2) for fringes imaged in the far field by a curved mirror or lens, we see that the fringes actually have cosine dependence. This makes the fringes appear inherently asymmetric when plotted versus x. However, a fringe pattern expressed as a function of cos where is the angle between the input ray direction and the interferometer normal is much nicer for computerized fitting. The fringes are equally spaced and symmetric along the cos axis, which is proportional to the optical phase. A useful misnomer that we applied to this idea is linearization of the fringe pattern. A convenient axis transformation is then x min 3 cos min x center 3 1 (A3) x max 3 cos max. New fringe data points need to be interpolated from the original axis x onto this new axis with equally spaced steps on this phase axis. Our approach uses a cubic spline interpolation to ensure the smoothness of the first and second derivatives of the new fringe pattern. So in this new axis system, the fringes are symmetrical and the output of the Snyder fringe position subroutine with the new data is much less prone to fringe pulling. A linear least-squares fit to a line of the fringe positions in this space determines the distance along the cos axis per order slope and the fractional order in units of the cos axis intercept. The fractional order is then determined from these quantities. An added bonus occurs if the two sides set of fringes on one side of the symmetry center are fit independently. The average of the resulting two fractional orders tends to cancel any small error arising in the determination of the center, a 1 above. On our 10- MHz 286 PC, the compiled code runs in less than 7 s. The first version of this code, including the fast first 1232 APPLIED OPTICS Vol. 36, No February 1997