Tunable optical cavities for wavelength Indications in Gas Laser

Explain an all-optical tunable detention in gas laser based on wavelength conversion wavelength reconversion. The use of stable, calibrated optical reference cavities to equipping interferometer users with more exactly. Various resonance wavelengths of cavity were resolved by frequency measurements in a vacuum by use of a gas laser comb followed by corrections to account for atmospheric pressure and cavity temperature. A purpose of a new stable cavity is discussed that allows for the longitudinal mode index to be determined by difference-frequency measurements between the S and P polarized modes. For a wide range of signal pulse interval (ps to 10 ns), an output signal wavelength and bandwidth that are the same as that of the input.


Introduction
Optical interferometers measure length in units of a known optical wavelength. Common interferometric systems use frequency-stable lasers, and the wavelength is a calculated parameter that depends on the local air's refractive index, which varies with air density and composition. Consequently, length measurements performed in air are substantially less accurate than similar measurements utilizing an interferometer operating in vacuum [1]. A different approach that has been suggested is to frequency-lock a tunable laser to a mode of a mechanically stable reference cavity that is open to the air [2, 3,4]. Essentially, the frequency of a tunable laser is controlled to fix the wavelength in the medium. The wavelength may be determined by a calibration of some type, and known correction factors (such as the reference-cavity temperature) applied as necessary. Here, calibrated several modes of a prototype reference cavity by way of frequency measurements with a femtosecond-laser frequency comb [5]. Furthermore, by smoothly tuning between two known resonance wavelengths, swept-wavelength interferometry is possible with no associated refractive index measurements. A wavelength calibration of the cavity at a certain temperature can be accomplished indirectly by an optical frequency measurement. Measured a tunable laser's frequency while it is locked to a resonance of the cavity in a vacuum chamber. The wavelength of the resonance in the vacuum may then be calculated with high precision since the limiting factor is not the refractive index but is instead likely to be the residual error of the temperature monitoring or offsets in the frequency-locking process. As discussed below, when air is re-introduced to the chamber the wavelength of each mode remains nominally the same, apart from several small corrections not related to the air's refractive index [6].
Subsequent frequency-locking of a laser to such a resonance will allow the user to know the laser wavelength in the reference cavity air path which, for instance, may be placed adjacent to a length-measuring interferometer. Monitoring the cavity temperature appears to be preferable to controlling it, since temperature control would unduly heat or cool the air in the cavity relative to the interferometer measurement path. The correction for atmospheric pressure requires knowledge of the barometric pressure, but again not as precisely as the present method of using the Edlen equations [7]. The material aging issue refers to a gradual shrinking at the rate of l\l ≤ -5x10 -9 yr -1 in the low-expansion material ULE glass [8]. This characterization is the result of long-term measurements of Fabry-Perot resonance frequencies. Another candidate low-expansion glass material, Zerodur-M, exhibits an aging rate larger by an order of magnitude [9]. Direct measurements of optical frequency as described here are to be distinguished from resonance frequency estimates based on measurements of the cavity free spectral range (FSR). Frequency measurements of a cavity's FSR to the level of 10 -7 have been demonstrated at a single wavelength [l0], but the FSR is wavelength dependent since the dielectric-mirror phase shift upon reflection is wavelength dependent. An accurate ( / ≈10 -8 ) measurement of a resonance frequency by way of FSR measurements does not appear to be practical at this point. However the tunable laser must be brought into coincidence and locked to the same longitudinal mode index (m) that was previously calibrated. Several methods of finding the correct mode are discussed.
The choice of which laser to employ in this research has been based on several factors.
Traditionally, most metrology systems have been built around the red 633 nm wavelength since frequency-stabilized He-Ne lasers have long been available. For this application we need a tunable laser with a line-width narrow enough to be captured and locked by electronic means. While it is true that the longer wavelength is a negative factor with regard to an interferometric measurement, the DFB characteristics including ruggedness, power, cost, eyesafety, fiber compatibility and component availability are positive factors. For instance, for paths on the order of one meter, a very precise l\l≈-3 x10 -8 measurement represents a fractional fringe interpolation of only 1/50 of a wave using the telecom-wavelength lasers.
The actual stability of the laser wavelength in air is a difficult parameter to measure.
Monitoring the laser frequency via a heterodyne beat-note will only record changes of the air's refractive index. By building a rigid test interferometer and monitoring the apparent path length using our "stable wavelength," we are likely to measure a changing phase shift at some level [11].
The aim of this study is discussed that allows for the longitudinal mode index to be determined by difference-frequency measurements between the S and P polarized modes. For a wide range of signal pulse interval (ps to 10 ns), an output signal wavelength and bandwidth that are the same as that of the input.

Stability
The resonance condition of an optical cavity is often approximated as an integer number of wavelengths in the cavity round-trip path length, or m = L. This simplification neglects any phase-shift upon reflection at the mirrors. It also assumes plane-wave propagation, neglecting the phase accumulation that occurs as the optical wave diffracts. We write the resonance Where L is the physical round-trip path length of the resonator, the integer m is the longitudinal mode index, (m, is the total phase-shift upon reflection from the dielectric mirrors per round-trip, and  R is the total Gouy phase shift around the ring cavity. Note that (1). In other words, the resonant wavelength is to first order independent of the air density in the cavity. One may expect a higher-order dependence if the air affects  m (), R , or L in any indirect manner.
Examples would include a change of the cavity length L due to air pressure or mirror contamination. The largest sources of uncertainty appear to come from residuals after correcting for changes of the mechanical length L due to temperature, aging, and atmospheric pressure. Equation (1) defines two distinct sets of modes, s and p polarized, that will in general have different resonance wavelengths since  m , is polarization dependent due to the non-zero angle of incidence on the cavity mirrors. The total Gouy shift is the phase difference between a plane wave and the actual cavity mode wave, over a single round trip in the cavity.
It is a function of the cavity geometry, i.e., the mirror curvatures and spacing. The shift may be calculated numerically for any cavity geometry, and will be a factor on the order of  R ≈ 2π\5 for all geometries considered here. For a two-mirror Fabry-Perot cavity, the Gouy term can be written as [12]  R =2(1+p+q)cos -1 (√ ) Here, p and q are the transverse mode indexes. As we will be interested only in the TEM 00 modes, p = q = 0 for the designs considered in this report. The mechanical length L clearly enters Eq. (1) through the  R term in the denominator, in addition to appearing explicitly in the numerator. However due to the large factor m (typically >l0 5 ) in the denominator of Eq.
(1), this indirect contribution of cavity length drift to wavelength shift is quite small, far below the level of 10 -8 . The  R term can therefore be considered a constant for each cavity design considered here. The phase shift upon reflection at each mirror is a function of mirror stack design. The coatings considered here are fabricated at elevated temperatures by ion-beam assisted RF deposition, which results in nonporous layers that have virtually the same density as the bulk materials Ta2O5 and Si0 2 [13].
A molecular monolayer on each mirror of a ring cavity represents about 1 part in 10 8 of the path length. However, this does not equate to the wavelength instability caused by accumulating a monolayer of contamination. At this point it is worth noting that the initial calibration does not need to be performed in a high-vacuum environment. Indeed, measuring the mode frequency with a background pressure of 0.1 Pa (≈1 mTorr) will cause an error (if uncorrected) of only ≈23*10 -10 [14]. To summarize the contributions of the dielectric mirror phase shift  m (A) and Gouy term  R to the overall wavelength instability, these terms are significantly less than 10 -8 .
Wavelength uncertainty related to temperature, atmospheric pressure, and material aging are discussed in the following section concerning accuracy.

Accuracy
Considering Where  c and T c are the calibration wavelength and temperature, respectively, is the error of the polynomial approximation with respect to the true coefficient of expansion, and T(t) is the temperature at time t. The error in the estimation of (t) will have two independent components which will be combined in a root-sum-square (RSS) fashion to determine the total error caused by temperature.
As the wavelength calibration will be performed in vacuum, any change in the resonant wavelength with air pressure must be properly accounted for. There are two known effects According to theory the change in bulk volume causes a proportional change in length [15] that follows The volume change is related to the pressure change by the bulk modulus K: In Eq  In practice one could employ helium back-filling or simply measure the cavity mode frequency in a known pressure of helium. The refractive index of helium as a function of frequency and temperature is presently known theoretically with an uncertainty of approximately ±3 x10 -9 [16]. At near infrared wavelengths the value is approximately an order of magnitude smaller than that of air (I-n = 2.5 x10 -5 at 10 5 Pa). The temperature

Mode Differentiation
A small cavity is appealing since the free spectral range (FSR) would be large, which would facilitate the unambiguous identification of each mode. For instance, the s polarized TEMoo modes of a ring cavity with an 8 mm perimeter would be separated by 37.5 GHz. At red wavelengths, a laser frequency repeatability of about 50 ppm would be more than sufficient to return unambiguously to a given mode. However, such a small cavity is more vulnerable to environmental effects such as contamination of the mirror surface.
Due to these uncertainties I have considered longer cavities, on the order of 25cm, in order to reduce the potential wavelength instability caused by possible contamination of the mirrors.
However, longer cavities present a challenge in the unambiguous identification of a particular (previously calibrated) mode. There are a number of distinct approaches to accomplish the mode discrimination in a cavity of this size (L = 25 cm round-trip, 1.2 GHz free spectral range). The first is to rely on laser repeatability for a tunable laser to return to the same (previously calibrated) mode. This may be possible with certain types of lasers, for instance solid-state lasers. Secondly, another air-spaced optical reference cavity may be used, with a shorter length and lower resolution, and the coincidences between the two cavities will indicate the correct modes. This vernier approach is introduced in Fig.(1) , below. The shorter cavity must be air spaced since the frequency of the finely-spaced reference cavity modes may shift by more than a FSR with the ambient barometric pressure. This represents the precision that the frequency difference must be measured in relation to the limiting line width.

Cavity Resolves
It is useful then to explore different designs for the cavity and coating with the figure of merit r in mind. Other aspects to consider include spatial-mode shape, fabrication difficulty, and the number of (expensive) optical coating runs required.

Three -mirror cavity
We start with a three-mirror cavity that has the advantage of being a stigmatically Compensated see Fig. (2). We analyze this using the same coating on all three mirrors, to avoid more than one expensive rnirror-deposition run. There are many different coating designs, and we use here H(LH) n because the s and p polarizations have the same phase shift approximately mid-band (H and L refer respectively to the layers of high and low refractive index). This allows the difference frequency v SP between the two polarization modes to be on the order of a megahertz near the coating band center, which facilitates the measurement of kilohertz differences between cavity FSRs. (For instance, it may allow an inexpensive frequency-to-voltage converter to be used, with no intermediate IF stage). It also allows the cavity to be used as a reference for two relatively close (less than 100 MHz) optical frequencies, allowing the interferometer system designer very long synthetic wavelengths (c/ v > 3 m). We calculate the loss per pass by multiplying the reflectivity at each mirror.   The calculated data is for the same H(LH) n dielectric coating on all three mirrors of a three mirror cavity see Fig. (2) . The different slopes of the two phase-shift curves allow the discrimination of the longitudinal modes of the cavity by measuring the s-to-p frequency difference.

Conclusions
The approach of using stable calibrated resonators to deliver accurate wavelengths in air is technically feasible. We estimate that a total wavelength .uncertainty of about ≤ 3.9 x 10 -8 could be delivered to interferometer users by such a reference cavity over a +5 o C temperature range, presently limited by the method of accounting for the mechanical contraction from vacuum to atmospheric pressure. It is possible that this particular uncertainty could be greatly reduced by performing measurements (in air) of a length held fixed by a vacuum interferometer. That concept is not explored in this technical report, but is an area of further research. Other areas for further research include exploring the accuracy versus calibration interval trade-off, and the locking of visible (red) DFB lasers should they become commercially available.