A novel technique is demonstrated for stabilizing an intra-cavity etalon used for single-mode selection in a laser cavity. By appropriate polarization analysis of the reflection from an etalon designed as a quarterwave plate an electronic signal can be derived, that enables the implementation of an electronic stabilization scheme. This scheme obviates the need for any modulation of the etalon in order to ensure stable single mode operation of a cw tunable laser.
©2004 Optical Society of America
The single frequency tunable laser has been a key technology in the development of high resolution laser spectroscopy  and in particular the recent spectacular advances in laser cooling of atoms and ions [2–4]. Any such applications, however, rely heavily on the ability to select a single mode of the laser cavity and maintain it for an extended period of time. For one of the most popular lasers on the spectroscopy market today, the extended cavity diode laser , this is achieved by a combination of a short cavity and thereby large cavity mode spacing and a wavelength dependent feedback from a grating. The selected mode can be tracked by adjusting the grating angle as the laser cavity length is scanned in order to change the output frequency. The strongly wavelength dependent feedback from a grating, however, is too weak to provide sufficient feedback for the lower gain tunable lasers such as the dye laser and the Ti:Sapphire laser. These sources are based on an inherently low-loss cavity, where the mode selection is normally carried out with a combination of optical elements inserted into the cavity. These may include birefringent filters , etalons , Michelson mode selectors  or a combination of these. The requirements for the selection elements are particularly stringent in the case of widely tunable lasers. Firstly, the desired mode is one of a great number of possible ones. Secondly, the need to tune the laser frequency implies that the selecting element has to be tuned as well, typically rotated around one of its axes. This non-solid mounting subsequently makes it prone to drifts.
The dye laser and the Ti:Sapphire laser offer tuning ranges from tens to more than a hundred THz. The laser cavity modes of which a single one has to be selected are spaced by typically a few hundred MHz. The inserted optical elements each introduce a loss, which is a periodic function of laser frequency, where the period is referred to as the free spectral range (FSR) of the element. The elements chosen for the single frequency selection have successively smaller free spectral ranges corresponding to successively narrower regions of low insertion loss, until only one laser mode is capable of oscillating at a frequency corresponding to a loss minimum of all inserted elements. The exact requirements for the mode selecting elements depend on the amount of inhomogeneous to homogeneous broadening in the gain medium as well as any spatial hole burning effects.
In a tunable single frequency laser a coarse selection is typically achieved using a birefringent filter. This may consist of one or more plates made of a birefringent material and is rotated to select a laser bandwidth of typically 200 GHz. At this point it is often sufficient just to insert a fused silica etalon with a free spectral range of approximately 200 GHz and a reflectivity of 20–30% in the cavity to ensure single-mode operation [9–10]. However, the stability requirements are extremely stringent. The rotation of the etalon by an angle of order one thousandth of a degree is sufficient for the laser to jump to the next cavity mode.
Two main methods have traditionally been employed to prevent this mode jumping: 1) a passive technique involves the addition of yet another etalon with an even smaller free spectral range and 2) an active technique whereby a feedback is applied to the rotation of the etalon so as to keep it locked to the laser mode. This is achieved by modulating the angle of the etalon around the reflection minimum and deriving a suitable error signal by demodulating the detected reflection from the etalon. While offering the clear advantages of a simpler optical layout this technique carries with it a number of significant penalties. Firstly, the modulated etalon introduces a loss in the cavity at twice the modulation frequency and hence an undesirable intensity modulation. Secondly, the etalon sets up acoustic vibrations in the cavity, which, if not compensated for electronically, will lead to a substantial frequency modulation of the laser.
In this paper we present new technique for deriving a robust electronic error signal for stabilizing a single etalon to a laser cavity mode. A birefringent etalon is inserted with a slight angle between one of its axes and the laser polarization and an error signal is derived from the differential reflection and phase shift of the polarization components along the two axes. The technique has some similarities with a standard technique for stabilizing an external cavity to a laser .
2. The birefringent etalon
The reflection coefficient Ar(δ,R) for an electric field incident on a solid etalon is given by the expression 
where R is the intensity reflection coefficient and δ=4π dn cos(θ)/λ is the phase retardation for a roundtrip of the light of wavelength λ in the etalon with thickness d and refractive index n, which is tilted at an angle θ to the incident beam. This reflection represents a periodic loss with a period of c/(2 n d cos(θ)).
For an etalon made of a birefringent material there are two refractive indices n1 and n2 corresponding to the two principal axes of the material. Hence there are two different values δ1 and δ2 for the phase delay. In general this will result in different reflectivities for the two polarizations. Specifically, if the difference δ1-δ2 is π modulo 2π one polarization will experience a reflection maximum when the other has a minimum. This is equivalent to the etalon being designed as a quarter-wave plate.
The essence of this technique is to insert the etalon in such a way that the direction of the laser polarization forms a slight angle, α, with one of the optic axes as shown in Fig. 1. Specifically, the majority of the light (intensity of this component proportional to α 2) is polarized along this axis while a component proportional to β 2 has orthogonal polarization (α 2+β 2=1). We shall refer to these components as the 1 and 2 component respectively. The incident electric field can be resolved in its two components along these two axes
where E 0 is the amplitude and ω the optical frequency. The reflected electric field is then
At resonance for the 1-component (δ1=0 modulo 2π) the reflected light is linearly polarized along the 2-axis. At either side of this point it is elliptically polarized with opposite helicity.
This polarization can be analyzed by the combination of a quarter-wave plate and a polarizing beamsplitter. A number of implementations of this are possible, but the conceptually simplest one is shown in Fig. 1. If the analyzing quarter-wave plate is aligned with the axes of the birefringent etalon, the light will be linearly polarized after the waveplate. The direction of the polarization will depend on the frequency of the mode relative to the etalon transmission frequency. The varying linear polarization can now be analyzed with a polarizing beamsplitter set at an angle of 45° relative to the axes of the analyzing quarter-wave plate. The two output electric fields from the beamsplitter are:
From the qualitative arguments regarding the polarizations we expect the corresponding intensities I 1(δ 1,δ 2,R) and I 2(δ 1,δ 2,R) detected by a pair of photo detectors to show an asymmetric imbalance around the minimum loss point of the etalon. Thus, the quantity
will cross zero at the etalon resonance and provide an ideal discriminant for an electronic stabilization circuit. Furthermore, it is insensitive to laser intensity fluctuations and relatively easy to compute electronically either by using analogue circuits or digitally.
Using Eq. (1) and the fact that δ 1-δ2 is π modulo 2π for an ideal quarter-wave plate we can now plot the signal S(δ 1, R). Fig 2(a) shows this function plotted over a free spectral range of the etalon for reflectivities from 4% to 20%. It is worth noting that the general shape of the signal does not vary and that the slope through zero only depends weakly on the reflectivity as shown in the inset in Fig. 2(a). This is a particularly useful feature for a practical implementation as the etalon reflectivity can then be optimized with only the optical performance of the laser in mind.
Generally waveplates are only exact for a particular wavelength. For applications in a tunable laser system it is therefore relevant to consider the effect on the signal S of a deviation from an exact quarter-wave retardation of the etalon. The widest bandwidth for an etalon (i.e., the slowest variation of the phase retardation with respect to wavelength) is obtained with a true zero-order plate. That is a waveplate, where the difference in optical thickness experienced by light polarized along the two optic axes is exactly a quarter of a wavelength. For a waveplate made of quartz this corresponds to an extremely thin plate (tens of microns), which will typically be too thin for an etalon. Assuming that a thickness of order 0.5 mm is required for the etalon to perform its purpose in the laser cavity we need to use a higher-order plate, i.e. one where the optical thickness difference is qλ±λ/4, where q is an integer. For a quartz waveplate the retardation will wary by less than ±λ/8 when the laser wavelength is varied by ±20 nm around the design wavelength. Figure 2(b) shows the calculated signals for a variation of ±λ/8 and a reflectivity of 20%. The signal develops a slight asymmetry, but the zero-crossing remains at the correct point and the slope at the zero-crossing remains unaffected.
3. Extra-cavity experiment
The reflection from an uncoated quarter-wave plate was first investigated in the output beam from a cw Ti:Sapphire laser with the set-up shown in Fig. 1. In this case the etalon, the analyzing waveplate and the polarizer were first aligned with the laser polarization, which was subsequently rotated by ~5° with a half-wave plate. The two photo detector signals I 1 and I 2 were recorded as a function of laser wavelength for a fixed tilt angle. Fig. 3 shows the measured sum and difference signals together with their ratio:
over about three free spectral ranges of the etalon. The steep positive slopes through zero would provide an ideal discriminant for stabilizing the laser wavelength to the maximum etalon transmission. The ratio data show the same general shape as the theoretical prediction based on Eq. (5) although with a slightly lower amplitude. This is most likely due to a less than complete extinction of the reflection of the resonant polarization due to walk-off in the tilted waveplate and the less than ideal optical quality — a waveplate rather than an etalon. This manifests itself as an offset in the sum signal. If we arbitrarily assume that 1/3 of the signal observed on resonance is due to this ‘wrong’ polarization and compensate for this in deriving the experimental values for S, the agreement between theory and experiment is significantly better.
4. Intra-cavity experiment
The comparison of theory and experiment are slightly less straightforward in an intra-cavity configuration. As the etalon is tuned, for instance, by tilting it the laser will jump between successive cavity modes. We would therefore expect a signal of the type given by Eq. (6) to be a sequence of smooth curves through zero separated by a sign changing jumps.
We have demonstrated this using a cw tunable vertical external cavity surface-emitting laser (VECSEL). The details of this system are described elsewhere . For the present discussion it suffices to note that the laser operates with a linear three-mirror cavity formed by a Bragg mirror immediately below the semiconductor gain region, a curved folding mirror and a flat output coupler. The overall cavity length is approximately 0.25 m leaving ample space for intra-cavity elements in the segment between the folding mirror and the output coupler. Single-frequency operation over a tuning range of ~20 nm is obtained by the insertion of a single-plate birefringent filter and a 200 GHz FSR etalon with 20% reflecting coatings.
A crystalline quartz etalon designed as a 17λ/4 plate at the laser wavelength of 975 nm is mounted on a galvanometer with a horizontal rotation axis. The etalon is ‘walked off’ slightly in the horizontal direction as shown in Fig. 1 to enable an aluminum coated pick-off mirror mounted at approximately 45° to separate the reflected light from the path of the cavity. The laser polarization is horizontal and the etalon axes are rotated by ~2° relative to horizontal/vertical. The polarization analyzer is set up as described above. The ratio S of the difference and sum of the detector signals is generated by an analogue multiplier and recorded as the etalon is tilted. Fig. 4 shows an example.
Electronic stabilization is achieved by feeding back to the galvanometer rotation angle a signal derived from the time integral of S. With the addition of a servo system for the cavity length a frequency stability of better than 10 kHz has been achieved with this system .
A new technique has been demonstrated for stabilizing an intra-cavity solid etalon to a cavity mode to ensure long-term single-mode operation. The technique is based on the use of a quarter-wave plate as the etalon. By analyzing the polarization reflected from the etalon a signal can be derived enabling electronic stabilization of the single-mode selection. One of the optic axes of the etalon is at a slight angle with respect to the laser polarization. It is worth noting that the insertion of the birefringent etalon in the laser cavity does not appear to have a detrimental effect on the laser tuning or output polarization.
The technique has been demonstrated on a laser system with a tuning range of ~20 nm, which has allowed us to use a relatively high order waveplate. For a laser system with a wider tuning range, such as that offered by the Ti:Sapphire gain medium, a low-order (preferably zero-order) plate would be required. The required thickness for the operation as an intra-cavity etalon can be achieved either by using a low-birefringence material of by optically contacting together two plates with a λ/4 thickness difference.
This scheme is related to the familiar stabilization scheme developed by Hänsch and Couillaud for stabilizing the frequency of a laser to passive cavity containing a polarizing element . In their case the polarization of the light impinging on the cavity is rotated slightly relative to the plane of the polarizer. This results in a frequency dependent polarization of the light reflected from the cavity, which enables the implementation of an electronic locking scheme.
This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) and by Scottish Enterprise under the Proof of Concept Scheme.
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