1. Introduction

A major issue in the design of a single-longitudinal-mode laser in a standing-wave cavity is longitudinal spatial hole-burning which limits the achievable single-mode output power. To reduce the destabilizing effects of hole-burning in standing-wave cavities, several approaches have been proposed, such as introduction of additional spectral loss modulation with an etalon filter placed inside the cavity [1], use of electro-optic phase modulation [2], cavities with polarisation rotation [3, 4], or employment of short unstable laser cavities with a high-gain laser element [5]. As an alternative to a standing-wave cavity, one can use a travelling-wave ring-cavity containing a non-reciprocal optical element [6]. All these laser designs with exception of short linear unstable cavities contain at least one additional optical element to enforce single-longitudinal-mode oscillation. This is usually not desirable as it eventually increases costs and decreases the reliability of the device. On the other hand, unstable single-longitudinal-mode cavities have been shown to exibit substantial increase in linewidth or excess spontaneous emission noise due to non-power-orthogonality of the transverse modes [5, 7, 8]. Considering all this, the simple two-element linear cavity design would be an attractive approach for miniature single-longitudinal mode lasers.

Due to spatial hole-burning, one single longitudinal mode will not deplete the gain in a linearcavity laser completely, and to prevent other modes from lasing, large loss or low small signal gain for these spectrally adjacent modes is needed. Simply speaking, the bandwidth of loss times gain must be narrower than the free spectral range of the cavity. Zayhowski’s analysis in Ref. [9] confirms this, a strong interdepence is found between the achievable single-mode laser power and the allowed cavity parameter space in terms of cavity length, gain bandwidth and spectral loss. By use of a narrowband spectral filter, the parameter window for laser power and cavity length can be substantially increased, as we will show in this paper.

Here we demonstrate a very simple linear-cavity laser with a spectral filter integrated in one of the cavity mirrors. With only two components in the cavity, we get single-longitudinal-mode lasing for fairly high powers and a comparatively long laser cavity. This means that there is space left in the cavity to introduce another element, e.g. a nonlinear crystal for second harmonic generation. The cavity consists of a relatively broadband Nd:GdVO₄ gain medium with a dielectric mirror coated onto one facet, and a volume Bragg grating in glass [10], working both as an output coupler and a spectral filter. This cavity acts as a combination of a Fabry-Perot cavity and a narrow bandpass filter, with some specific features caused by the properties of the thick Bragg grating. A volume Bragg grating is an ideal filter for single-longitudinal-mode selection, since it can have a very narrow bandwidth compared to ordinary dielectric multilayer mirrors. This is the result of a small refractive index variation in a long device, just as for fibre Bragg gratings.

Previous works on volume Bragg gratings have shown line narrowing and wavelength locking of high power laser diodes [11] and 20 times narrowing of an optical parametric oscillator [12]. Recently, we reported single-longitudinal-mode lasing in a diode-pumped ErYb:glass laser [13], although in a double cavity setup. Longitudinal multimode locking of Ti:sapphire and Cr:LiSAF has also been reported [14]. To the best of our knowledge, no-one has previously demonstrated such a simple technique as ours for a standing-wave single-longitudinal-mode laser, based on only two components, a gain medium and a volume Bragg grating.

2. Theory

A theoretical treatment giving the condition for single-longitudinal-mode operation in a linear cavity laser with spatial hole-burning is given by Zayhowski in Ref. [9]. Here we will build upon these results to analyse our linear cavity containing a volume Bragg grating as output coupler. We recall the main results of Ref. [9]: The ratio of the absorbed pump power, P _a , to the pump at threshold, $P_a^{\mathit{\text{th}}}$ , has to fulfil the inequality

for single-mode operation, i.e. for a first mode (no. 1) to oscillate but not a second one (no. 2). Here β gives the ratio between the logarithmic loss, γ, and the emission cross section, σ, for the first and second mode, β=σ₁ γ ₂/(σ₂ γ ₁), while ψ is the average over the gain medium length l of the cosine of the double phase difference between the two modes, weighted by the unsaturated inversion N ₀, ψ=${\int }_0^l$ N ₀(z)cos(2(k ₁-k ₂)z)dz/${\int }_0^l$ N ₀(z)dz. Here k ₁ and k ₂ represent the wave vectors of the two longitudinal modes. We later compare the theoretical prediction of Eq. 1 with the performance of our laser, see Fig. 6.

The amplitude reflectivty for light incident on a volume Bragg grating in the plane wave approximation, is given by the classic paper by Kogelnik [15]. We assume that the Bragg grating consists of a sinusoidal refractive index variation with amplitude n ₁ of period Λ around a constant index n ₀, n(z)=n ₀+n ₁ sin(2πz/Λ), with a length L in the z-direction. Furthermore, we assume an internal angle between the grating vector and the incident light wave vector of θ. From Fourier theory considerations it can be seen that this (sinusoidal) variation will give a first order reflection (only) at the Bragg wavelength λ _B =2n ₀Λcosθ. The electric field amplitude reflectivity, r=√Rexp(iφ), around the Bragg peak for a wave vector k=k _B +δk=2πn ₀/λ _B +δk, is given by

with α=(κ ²-(δ k cos²θ)²)^1/2, the grating strength given by κ=n ₁ k/(2n ₀)=n ₁ π/λ and m an integer defined to give the phase continuity. Here we have chosen to let zero phase correspond to zero detuning. For normal incidence, θ=0, the peak reflectivity is R ₀=tanh² κL, the spectral bandwidth, defined as the spectral distance between the two zeros closest to the peak, is Δλ=λ_B ($n_1^2$/$n_0^2$+4Λ²/L ²)^1/2, and conversely the angular bandwidth between the two first zeros for a constant wavelength λ_B is Δθ=2(Δλ/λ_B )^1/2. An example of Eq. 2 and Eq. 3 for the grating we use is given in Fig. 2.

In practice the beam in our experiments is not a plane wave but has a gaussian transverse distribution. Nevertheless, we still want to use the rather simple plane-wave equations, which in the case of normal incidence are valid as long as the divergence of the gaussian beam over the length of the grating is essentially less than the grating’s angular bandwidth.

It should be noted that the phase variation around the peak is approximately linear with the detuning, a Taylor expansion in the detuning of Eq. 3 for normal incidence gives φ(δk/κ)=-$R_0^{1/2}$ δk/κ+O((δk/κ)³). By comparison with the phase change for a light field of wave vector k_B +δ k over a distance l, φ=(k_B +δk)l, and remembering that with a reflection we get a change of sign, it is natural to assign a round-trip effective optical cavity distance 2L _cav for the light reflected in the grating by 2L _cav≡l=±φ/δk, which yields 2L _cav=$R_0^{1/2}$/(2arctanh $R_0^{1/2}$)L. The same result has been derived elsewhere for fibre Bragg gratings [16], using a slightly different method. With this result in mind, it is possible to treat a Fabry-Perot cavity consisting of one ordinary broadband dielectric mirror and one thick Bragg grating mirror as an ordinary Fabry-Perot cavity, with the length of the Bragg grating given by the effective cavity length. This works fine close to the Bragg peak where the laser is most likely to lase, however farther away one needs to take into accout the full expression for the phase variation in Eq. 3, see Fig. 2. The frequency of the peak transmission of such a combined Bragg grating Fabry-Perot, as a function of the cavity length, is shown by solid lines in Fig. 3.

 

Fig. 1. Laser setup consisting of a Nd:GdVO₄ crystal and a volume Bragg grating.

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3. Experimental setup

The laser cavity was flat-flat, composed of a Nd:GdVO₄-crystal and the volume Bragg grating and pumped longitudinally, see Fig. 1. The 1.5 mm long Nd:GdVO₄-crystal was a-cut and doped with 1 at.% Nd³⁺. Both surfaces were flat, with the second one wedged 7° rotated around the c-axis, to minimize parasitic reflections. The first surface was HR-coated for the laser wavelength at 1060 nm and HT-coated at the pump wavelength of 808 nm, whereas the second surface was AR-coated at 1060 nm. The laser crystal was placed in a temperature-controlled holder kept at 20 °C. The Nd:GdVO₄ host crystal has similar performance as the more well-known Nd:YVO₄, and was chosen primarily because the fluorescence spectrum for the σ-polarization in Nd:GdVO₄ overlaps reasonably well with the Bragg grating we had available. The peak of the gain spectrum in the σ-polarisation in our crystal was measured to be at around 1065.5 nm with a FWHM bandwidth of 1 nm. In addition, there is also another peak at 1063.4 nm, with higher emission cross section for the π-polarisation and a FWHM of 0.6 nm. Thanks to the wedge and the birefringence in GdVO₄, the σ-polarisation is easily chosen by aligning the Bragg grating properly. However, since the emission cross section for the σ-polarization is larger than the one for π-polarization at the Bragg peak, σ-polarisation is preferred also for an unwedged crystal.

The volume Bragg grating (Optigrate, Florida, USA) was 5.0 mm thick with a clear aperture of 6 mm by 5 mm. In order to reduce parasitic reflections, the facets were tilted by 2° with respect to the grating direction, as well as AR-coated at 1060 nm. The reflectivity was measured by a narrowband Ti:sapphire laser. To avoid errors from the fact that the mode size in the laser cavity is rather small, of the order 50 µm to 90 µm 1/e²-radius, and thus not strictly a plane wave, we used an equally small focus size for the reflectivity measurements. In Fig. 2 we show the experimental data, where the data points above 50% is from a transmission measurement and the ones below from a reflection one, in order to increase the accuracy. Here a lossless grating is assumed, a point which we will discuss later. We measured a peak reflectivity of 98.8% at 1066.06 nm and a FWHM bandwidth of 0.22 nm. This gives a full angular bandwidth of about 30 mrad, to be compared with a maximal 4 mrad divergence of a 50 µm beam in the grating, justifying our plane wave assumption in Section 2. In Fig. 2 we also show theoretical fits to Eq. 2 and Eq. 3, where the grating thickness is the fit parameter. The fit yields a reflection effective thickness of 4.5 mm, slightly less than the physical length, and a refractive index modulation amplitude of n ₁=2.2×10^-4. From the peak reflectivity and the reflection effective thickness we deduce an effective cavity length of L _cav=0.77mm. The Bragg gratingwas placed on a motorized, high-precision translation stage so that the distance between the grating and the gain crystal could be varied from 1 mm upwards in steps of 50 nm.

 

Fig. 2. Bragg grating reflectivity (blue), experimental points and solid line fit, as well as phase change (solid red line) with linear approximation (dashed red line).

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The laser was pumped by an 808 nm laser diode in the π-polarisation, delivering up to 3.3 W at the laser. For convenience, the laser diode was kept at a fixed temperature of 20 °C, resulting in a wavelength shift with pump current, and thus a varying absorption in the Nd:GdVO ₄ between 40% for low powers up to 60% at maximum pumping. To compensate for this, we give the absorbed pump power henceforth. The pump was imaged to form a circular 50 µm 1/e ²-radius beam waist in the laser crystal, however with different divergence in the π- and σ-directions due to different beam propagation factors of M ²~30 and M ²~70, respectively.

4. Results and discussion

The cavity was made geometrically stable by the thermal lens formed by the absorbed pump and the quantum defect of 24%, causing both thermal bulging and a positive lens due to a positive dn/dT in GdVO₄. The location of the pump focus along the propagation direction was chosen to optimise the laser action near the threshold. We used the above-stated pump focus radius to get a strong enough thermal lens to ensure a TEM₀₀ mode, though no serious attempt for optimisation was made. Indeed, an M ₂ of 1 was measured up to absorbed pump powers of 1W. However, the M ₂ value increased up to 1.5/2 in the π/σ -direction for 2.1 W absorbed pump. We believe that with a somewhat different pump or laser design this point could be improved.

To evaluate the laser performance and to tune the laser to an appropriate setting, the cavity length was fine-tuned by the translation stage. The laser’s characteristics where simultaneously monitored by measuring the laser output power with a power meter, the absolute laser wavelength with an optical spectrum analyser and, more in detail, the relative frequency detuning with a scanning Fabry-Perot interferometer. From the Fabry-Perot traces it could also be seen if the laser was lasing with one or more longitudinal modes. In Fig. 3, an example of the recorded frequency, ν, from the Fabry-Perot, calibrated by the spectrum analyser data, is shown versus cavity length detuning for an absorbed pump power of 0.14 W with the Bragg grating placed 1 mm from the gain crystal. With a refractive index of 1.97 for GdVO₄ in the σ-polarisation direction and using the effective cavity length of the Bragg grating, the optical length of the cavity is 4.7 mm. Since dν/ν=-dL/L for an ordinary Fabry-Perot cavity, the cavity length, L, can also be deduced from the slope in Fig. 3. This yields a value of 4.7 mm, confirming the theoretical prediction of an effective cavity length. Here it should also be pointed out, that the tuning we see in between the mode jumps is truly continuous.

 

Fig. 3. Frequency detuning with changing cavity length, theoretical prediction (line), and experimental laser wavelength, dots for single mode and crosses for multimode.

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In Fig. 4 we show four examples for different pump powers of how the wavelength can be tuned by changing cavity length over a distance exceeding the laser wavelength. The figure shows the single-mode-operation tuning ranges for different absorbed pump powers. It can be seen that the single-mode range narrows down from approximately 25 GHz for 0.14 W to about 15 GHz at 2.1 W of the absorbed power. This behaviour stems from the spatial hole-burning and will be discussed below in more detail. At the same time, the centre of the tuning range also shifts to longer wavelengths at higher pump powers. We attribute this shift to local heating of the grating in the laser channel. Since there was no thermal contact between the grating and the laser crystal, this indicates that indeed there is some absorption in the grating. It has previously been reported that the temperature tuning of the Bragg peak is ~0.01 nm/K [11, 12, 14], in our case this would correspond to a local heating in the grating of 15 K at full power. The assymetry in laser power for higher pump powers is probably due to higher gain for shorter wavelengths since the Bragg grating’s reflection peak is on the side of the peak of the spectral gain. We use the data in the middle of the single-mode tuning ranges in Fig. 4 to extract the laser power dependence on the absorbed power, shown in Fig. 5. The laser shows a threshold of 0.07 W absorbed power and initially a slope of 37%.

 

Fig. 4. Wavelength and frequency detuning versus laser power with changed cavity length for four absorbed powers: 0.14 W(blue), 0.69 W(green), 1.5 W (red), 2.1 W(black). Dots is for a single longitudinal mode and crosses for multimode lasing.

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Fig. 5. Absorbed pump power versus laser output power.

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We evaluated the losses of the Bragg-locked laser by comparing the laser threshold to that of a laser with an ordinary dielectric mirror output coupler, in a Findlay-Clay analysis. Seven different ordinary mirrors with reflectivities ranging from 60% to 98% were used and all cavities had a length of 4.7 mm. By adjusting the alignment of the output coupler properly, as explained in Section 3, the σ-polarisation was chosen. As expected, the laser with an ordinary output coupler lased longitudinally multimode at the peak of the fluorescence spectrum at 1065.5 nm. Here, the emission cross section has twice the value compared to at the Bragg grating wavelength of 1066.06 nm, so we compensated the loss-evaluation data by dividing the Bragg grating-locked laser threshold by a factor of 2 to get a fair comparison. The analysis gave a single-pass loss of the cavity with conventional mirrors of 1.2%, and 1.7% for the Bragg grating cavity, i.e. only a minor loss introduced by the grating according to these data.

The most important subject of this investigation is to determine over what range of pump powers and cavity lengths the volume Bragg grating-locked laser operates in a single longitudinal mode. To this end, measurements were done separately for both varying pump power and fixed cavity length, as well as for varying cavity length and fixed pump power. First, the data presented in Fig. 4 with varying pump power were used to measure the single-mode operation range for a fixed cavity length of 4.7 mm (triangles in Fig. 6). Second, we made another measurement for varying cavity lengths, while keeping a fixed absorbed pump power of 0.69 W and 2.1 W, respectively, represented by circles and squares in Fig. 6. The data confirm that higher pump powers and a longer cavity (i.e a shorter free spectral range), both make single-mode lasing more difficult, as should be expected. For the maximum absorbed pump power available of 2.1 W, the laser is found to lase single-mode for cavity lengths up to about 8 mm. It should be taken into account, when we discuss a single longitudinal mode, that the laser for absorbed pump powers above 1 W is multimode in the transversal direction, which in fact shows up as a small extra peak in the Fabry-Perot trace about 1 GHz above the main peak.

 

Fig. 6. Single-mode tuning range versus absorbed pump power for 4.7 mm cavity length (red triangles), and versus cavity length at 0.69 W (blue dots) and 2.1 W (green squares) absorbed power. The theoretical prediction from Eq. 1 is given as dashed lines in corresponding colours.

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The experimental data for the single-mode tuning span is also compared to the predictions given by Eq. 1, shown by dashed lines in Fig. 6. Here we assume that the unsaturated inversion is exponentially decaying in the z-direction, i.e. proportional to the absorbed pump, with a decay length given by the total amount of absorbed pump. Furthermore, we use the wavelength-dependent reflectivity of the Bragg grating (see Fig. 2), while the emission cross section wavelength dependence is found from a direct measurement of the fluorescence. We also take into account a slight variation with pump power of these input data. To fit the theory to the experimental data, we use a wavelength independent loss as a fit parameter, and minimize the error for the data with varying absorbed pump. The best fit is found for a loss of 5% single pass. We then get a fair agreement between theory and experiment for changing absorbed power, as well as for shorter cavity lengths. Nevertheless, the model fails to predict single-mode lasing for as long cavity lengths as we measure, probably due to more complicated effects than the ones included in the model, e.g. spatial cross relaxation [9], which tend to reduce spatial hole-burning.

For comparison, we use the theoretical model to estimate the limits of the cavity length for single-longitudinal-mode operation with ordinary broadband dielectric mirrors, coated directly onto the laser crystal, where the dominant filter is the gain bandwidth. We assume pumping 30 times above threshold (corresponding to the case of 2.1 W of absorbed pump power in the Bragg grating laser) and a pump absorption of 0.6mm^–1. Since the fluorescence in Nd:GdVO₄ at 1066 nm has a double peak, we simplify the analysis and assume a single Lorentzian emission cross section with a FWHM of 1 nm. With these parameters, the model gives a maximum cavity optical length of 0.6 mm for single-mode operation. Indeed, as mentioned earlier, with an ordinary mirror as output coupler, our laser was multimode for a cavity optical length of 4.7 mm.

Regarding the losses in the Bragg grating, our measurements clearly show that there is loss, though different methods give slightly different values, between 0.5% and 4%, if we assume a 1% single-pass loss for other reasons. At least some of the loss is in the form of absorption, as shown by the tuning of the Bragg peak with laser power. For comparison, in Ref. [10] a loss of less than 0.01 cm^–1 is found, which in our 5 mm long grating would be less than 0.5% loss.

5. Conclusions

We demonstrate a standing-wave laser design which increases the parameter range in terms of power and cavity length over which single-longitudinal-mode lasing is possible. The laser comprises only two components, the Nd:GdVO₄ gain medium with a dielectric mirror coated onto it, and a volume Bragg grating working both as a filter and a cavity mirror. We get up to 0.85 W of power at 1066 nm in a single-longitudinal-mode for a cavity optical length of up to 8 mm. We also show that for shorter cavities and lower powers, the laser wavelength can be tuned continuously over 25 GHz.