Modelling and characterisation of continuous wave resonantly pumped diamond Raman lasers

Muye Li; Ondrej Kitzler; Richard P. Mildren; David J. Spence

doi:10.1364/OE.426067

1. Introduction

Stimulated Raman scattering (SRS) is a third-order nonlinear laser-material interaction, and lasers based on SRS can efficiently generate new Stokes-shifted wavelengths spaced by a resonant phonon frequency [1,2]. SRS is an attractive method to shift conventional lasers to different wavelength bands for applications such as atom cooling, sensing, fluorescence imaging, etc. [3–5]; SRS has no requirement for phase matching and so relaxes constraints on thermal management, and the Raman-beam-clean-up can improve the spatial beam characteristics of the generated beam compared to the pump [6].

Modelling of continuous-wave (CW) crystalline Raman lasers is well developed for intra-cavity systems where the Raman and inversion gain share a laser cavity [7] and external-cavity lasers pumped by a non-resonated pump laser [8], where only the Raman field is resonated. Modelling of cavity-enhanced Raman lasers has been aimed at describing Raman cascading in micro-spheres [9] and waveguide Ring resonators [10].

In this paper, we report new experimental results and associated modelling results of resonantly-pumped Raman lasers generating first or second Stokes output. We have previously reported a first Stokes diamond laser generating 1 W at 883 nm [11] and a second Stokes diamond laser generating 360 mW at 1101 nm [12] pumped by a continuous-wave Ti:sapphire laser (SolsTiS, M Squared Lasers Ltd.) at 890 nm and 851.5 nm, respectively. Here, we present a deeper investigation of a first Stokes laser, and extend the output power of the second Stokes Raman laser to 1.5 W. We develop an analytical model of cascaded Stokes conversion in bulk lasers with cavities of lower finesse, and model the effects of imperfect mode-matching. The model is used to understand the performance of the lasers and to guide the design of new lasers.

2. First Stokes model

We first derive an analytical model for a first Stokes Raman laser, with the Raman crystal inside a cavity resonantly-pumped by a single-longitudinal-mode (SLM) pump laser.

We define an input/output coupler (IC/OC) mirror with reflectivities ${R_P},{R_S}{\; }$and ${R_{SS}}$ for the pump, first Stokes and second Stokes field, respectively, grouping all other losses (including coating imperfections, absorption and scattering) into a single loss term $L\; $for each field. For a ring enhancement cavity, the incident pump power is resonantly enhanced inside the cavity by a factor [13]

(1)$$\alpha = \frac{{{P_{Pintra}}}}{{{P_{Pinci}}}} = {\left\{ {\frac{{{{({1 - {R_P}} )}^{0.5}}}}{{1 - R_P^{0.5}{G^{0.5}}}}} \right\}^2},$$

where ${P_{Pintra}}$ and ${P_{Pinci}}$ are the intra-cavity pump power and incident pump power, respectively, G is the round-trip change in the pump power due to intracavity gain and passive loss; in our case this can be less than one to model pump loss due to both passive losses and depletion by conversion to Stokes power. The fraction of the incident pump power reflected from the input mirror is

(2)$$\beta = \frac{{{P_{Prefl}}}}{{{P_{Pinci}}}} = {\left\{ {\frac{{R_P^{0.5} - {G^{0.5}}}}{{1 - R_P^{0.5}{G^{0.5}}}}} \right\}^2},$$

in which $\beta $ goes to zero for perfect impedance matching when the cavity losses equal the input mirror transmission.

In practice, perfect coupling of the pump into the TEM₀₀ cavity mode is difficult, owing to mode mismatch and polarization error. The unmatched fraction will always be reflected from the input mirror when the cavity is locked for coupling of the TEM₀₀ mode. We account for this by defining [13]

(3)$${P_{Pinci}} = \gamma {P_{Pinci0}},$$

where ${P_{Pinci0}}$ is the full pump power incident on the cavity, $\gamma $ is a mode matching factor, and ${P_{Pinci}}$ is the pump power that can be correctly coupled into the cavity. The total power reflected from the cavity ${P_{Prefl0}}$ comprises the unmatched fraction as well as the fraction of the matched power rejected because of imperfect impedance matching$\; {P_{Prefl}}$:

(4)$${P_{Prefl0}} = ({1 - \gamma } ){P_{Pinci0}} + {P_{Prefl}} = ({1 - \gamma } ){P_{Pinci0}} + \beta \gamma {P_{Pinci0}}.$$

We will see that this reflected power is an important way of experimentally diagnosing the laser performance.

Assuming the laser has reached steady state, the loss and gain for the first Stokes must be equal over a round trip, thus

(5)$${R_S}({1 - {L_S}} ){e^{\frac{{{g_s}l{P_{Pintra}}}}{{{A_1}}}}} = {R_S}({1 - {L_S}} ){e^{\frac{{{g_s}l\gamma \alpha {P_{Pinci0}}}}{{{A_1}}}}} = 1,$$

where ${R_S}$ is the reflectivity of first Stokes, ${L_S}$ is Stokes passive loss, ${g_s}$ is the gain coefficient for Stokes, l is the crystal length, ${A_1}$ is the overlap area of pump and Stokes fields. ${A_1}$ is generally defined as [1]

(6)$$\frac{1}{{{A_1}}}\; \; \; = \frac{1}{l}{\smallint\!\!\!\smallint }\overline {{I_p}({r,z} )} \; \overline {{I_s}({r,z} )} dAdz,$$

where the z integral is over the crystal length, and $\overline {I({r,z} )} = \; I({r,z} )/\smallint I({r,z} )dA$ is a normalized intensity profile. If the confocal parameters of the fields are much greater than the crystal length, this simplifies to ${A_1} = \pi ({\omega_P^2 + \omega_S^2} )/2$, where ω_P,S is the waist radius of pump and Stokes field.

The relationship between the Stokes output power and its intra-cavity power is

(7)$${P_{Sout}} = ({1 - {R_S}} ){P_{Sintra}}.$$

Considering the round trip loss experienced by the pump for a particular Stokes power and assuming small roundtrip change in the intracavity fields as in [1], we can write the round-trip fractional change G for the pump as

(8)$$G = ({1 - {L_P}} ){\textrm{e}^{ - \frac{{{g_P}{P_{Sintra}}l}}{{{A_1}\; }}}} = ({1 - {L_P}} ){\textrm{e}^{ - \frac{{{g_s}{P_{Sout}}l}}{{{A_1}{\eta _s}({1 - {R_S}} )\; }}}},$$

where ${g_s} = {\eta _s}{g_P}$ and ${\eta _s} = \frac{{{\lambda _p}}}{{{\lambda _s}}}$ is the quantum defect of the Raman process, where ${\lambda _p}$ is the pump and ${\lambda _s}$ the Stokes wavelength. When Eqs. (1), (5) and (8) are combined to eliminate $\alpha $ and G the first Stokes output power can be expressed as

(9)$${P_{Sout}} = \frac{{{\eta _s}{A_1}({1 - {R_s}} )}}{{{g_s}l}}\left\{ {\ln [{{R_P}({1 - {L_P}} )} ]- 2\ln \left[ {1 - \sqrt {\frac{{\; {P_{Pinci0}}}}{{{P_{Sth}}}}} \left( {1 - \sqrt {{R_P}\; ({1 - {L_P}} )} } \right)} \right]} \right\},$$

where the incident threshold power ${P_{Sth}}\; $is found by setting ${P_{Sout}} = 0$ to be

(10)$${P_{St\textrm{h}}} ={-} \frac{{{A_1}\ln [{{R_s}({1 - {L_S}} )} ]{{\left[ {1 - \sqrt {{R_P}({1 - {L_P}} )} } \right]}^2}}}{{{g_s}l\gamma ({1 - {R_P}} )}}.$$

As $\alpha $ in Eq. (5) is inversely proportional to incident pump power, the intra-cavity power defined in Eq. (1) remains constant above Stokes threshold. The reflected power changes however, and reaches minimum for the impedance matched condition $G = {R_P}$ at

(11)$$P_{Pinci0}^{\prime} = \frac{{{A_1}({1 - {R_P}} )}}{{{g_s}l\gamma }}\ln \left( {\frac{1}{{{R_s}({1 - {L_S}} )}}} \right),$$

with Stokes output power of

(12)$$P_{Sout}^{\prime} = \frac{{{A_1}{\eta _s}({1 - {R_S}} )}}{{{g_s}l}}\ln \left( {\frac{{1 - {L_P}}}{{{\textrm{R}_p}}}} \right).$$

The results can be simplified with some approximations. Using

(13)$$R_P^{0.5}{G^{0.5}} \approx 0.5({1 + {R_P}G} ),$$

which is valid for strong resonant enhancement and low pump power (${\textrm{R}_P}G \to 1$), Eq. (1) simplifies as

(14)$$\alpha = \frac{{4({1 - {R_P}} )}}{{{{({1 - {R_P}G} )}^2}}}.$$

Furthermore assuming highly reflective mirrors and low losses (${R_P},\; {R_S}$ ≈ 1 and ${L_P},\; {L_S}$ << 1), the natural logarithms in Eq. (9) are approximated according to $\ln ({1/x} )\approx 1 - x$, when $x \to 1$, which yields Eq. (9) as

(15)$${P_{Sout}} = {\mathrm{\sigma }_S}\left( {\frac{{\sqrt {{P_{Pinci0}}} }}{{\sqrt {{P_{Sth}}} }} - 1} \right)$$

with threshold pump power

(16)$${P_{Sth}} = \frac{{({1 - {R_S} + {L_S}} ){A_1}}}{{4{g_s}l\gamma ({1 - {R_P}} )}}{({1 - {R_P} + {L_P}{R_P}} )^2}$$

and slope efficiency

(17)$${\mathrm{\sigma }_S} = \frac{{{A_1}{\eta _s}({1 - {R_S}} )\left( {\frac{1}{{{R_P}}}{\; } - {\; }1 + {L_P}} \right)}}{{{g_s}l}}.$$

This approximate result agrees with the equations presented by Min et al. [9].

3. Experimental setup and results from a first Stokes laser

For the model verification, we built cavity-enhanced ring resonators optimized for first and second Stokes output. For the first Stokes system, we used a SLM tunable CW Ti:sapphire laser (SolsTiS, M Squared Lasers Ltd.) as the pump source with wavelength tunable from 700 nm to 1000 nm. For the second Stokes system shown in Fig. 1, we used a 1064 nm CW fiber laser (SLM Moglabs laser seed and Precilasers amplifier) which offered more pump power.

Fig. 1. Schematic layout of the experimental setup for second Stokes generation. In the first Stokes setup, M3 was the IC/OC mirror with R_P = R_S = 98%, while other mirrors were HR. Inset shows the reflectivity curve of M1 designed for the second Stokes setup.

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The ring resonator was a bow-tie cavity comprising of a mirror providing input and output coupling (IC/OC) and three highly reflective (HR) mirrors. In the first Stokes setup the IC/OC mirror (M3 in this case) was partially reflective (PR = 98%) for pump and Stokes wavelengths. In the second Stokes setup, M1 was the IC/OC mirror with the reflectivity optimized to efficiently generate at second Stokes (reflectivity is shown in the inset of Fig. 1). Dichroic mirrors DM1 and DM2 in combination with HR and dichroic mirror DM3 were used to retro-reflect the particular Stokes field into the cavity to induce its unidirectional oscillation, as we showed previously in [11]. The Raman gain medium was a 5 × 2×1 mm³ CVD-grown Brewster-cut diamond crystal (Element Six Ltd.), installed on a copper holder and placed between M1 and M2 in the pump beam waist of radius of 20 μm. The diamond was cut for propagation along the [110] direction and polarization parallel to the [111] direction to access the highest Raman gain. The resonator was locked for resonant-enhancement of the pump using the Hansch-Couillaud locking method [14].

Figure 2 shows the behavior of the first Stokes laser, pumped at ${\lambda _P}$ = 851 nm, with corresponding Stokes wavelength at ${\lambda _s} = $ 960 nm. The maximum, unidirectional, Stokes power reached 610 mW for a pump power of 3.4 W. The first Stokes was confirmed to be single-longitudinal mode using a scanning Fabry-Perot interferometer (Thorlabs, 10 GHz free spectral range and 67 MHz resolution).

Fig. 2. Measured (symbols) and modelled (lines) output power and reflected pump power for the first Stokes laser.

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We also collected data for the pump power reflected from the input mirror, which turns out to be important information to correctly model the cavity behavior. To fit the model to the Stokes threshold, slope efficiency, and measured reflected pump power we used a gain coefficient of ${g_S}\; $=$\; 14.5$ cm/GW (at 960 nm) [15], although we note that there is considerable uncertainly around this value [16]. The M² of the fields was set to 1, pump waist radius was measured and compared with a ABCD cavity simulation to be 20 μm (pump areas calculation takes Brewster cut diamond into account). We assumed that losses were equal for both fields, and that the pump and Stokes beams had identical confocal parameter, which is valid in a doubly resonant cavity and determines the Stokes waist size. The IC/OC mirror parameters were measured as ${R_P}$ = ${R_S}$ = 98%. With these measured quantities, passive loss of 1.2% for both fields and a pump mode-matching factor $\gamma = 0.81$ were found to best match the measured data. The solid gray curve shows the reflected power from just the matched fraction of the pump power ${P_{Prefl}}$, indicating, that the cavity would become impedance matched for 3.75 W of pump power, which is just above the maximum pump power available. It is notable that only for 100% mode-matched incident pump would the reflected power reach zero. This is in practice hardly achievable and would be impractical as the reflected pump power ${P_{Prefl0}}$ also facilitates the locking signal to the cavity.

4. Model for second Stokes and above

In the second Stokes model derivation, the accumulated first Stokes field serves as a pump for the second Stokes field. The base equations (Eqs. 1–4, 6) for resonant enhancement and fields overlap still apply. After the second Stokes threshold the steady state Stokes field in Eq. (5) is modified to include loss due to the conversion to second Stokes as

(18)$${R_S}({1 - {L_S}} ){e^{ - \frac{{{g_s}l{P_{SSintra}}}}{{{A_2}}}}}{e^{\frac{{{g_s}l{P_{Pintra}}}}{{{A_1}}}}} = 1,$$

where ${A_1}$ and ${A_2}\; $are the overlap areas of the pump and first Stokes field and first Stokes and second Stokes field, respectively, ${P_{SSintra}}$ is the second Stokes intracavity power. Expressing the second Stokes output power using second Stokes reflectivity ${R_{SS}}$ of the IC/OC as ${P_{SSout}} = ({1 - {R_{SS}}} ){P_{SSintra}}$ and ${P_{Pintra}} = \alpha \gamma {P_{Pinci0}}$ (from Eqs. (1) and (3)) we find

(19)$${P_{SSout}} = ({1 - {R_{SS}}} )\left( {\frac{{{A_2}\alpha \gamma {P_{Pinci0}}}}{{{A_1}}} + \frac{{{A_2}\textrm{ln}({{R_s}({1 - {L_S}} )} )}}{{{g_s}l}}} \right).$$

The gain has to equal loss for the second Stokes field in the steady-state, therefore

(20)$${R_{SS}}({1 - {L_{SS}}} ){e^{\frac{{{g_{ss}}l{P_{Sintra}}}}{{{A_2}}}}} = {R_{SS}}({1 - {L_{SS}}} ){e^{\frac{{{\eta _{ss}}{g_s}l{P_{Sintra}}}}{{{A_2}}}}} = 1,$$

where ${g_{ss}} = {\eta _{ss}}{g_s}\; $is the second Stokes gain coefficient and ${\eta _{ss}} = \frac{{{\lambda _s}}}{{{\lambda _{ss}}}}$, where ${\lambda _{ss}}$ is the second Stokes wavelength. Combining Eqs. (20) and (8) we find

(21)$$G = ({1 - {L_P}} ){[{R_{SS}}({1 - {L_{SS}}} )]^{\frac{{{A_2}}}{{{A_1}{\eta _S}\; {\eta _{SS}}}}}}.$$

Substituting Eq. (21) into (1) and Eq. (1) into Eq. (19), the second Stokes output power takes a simple form

(22)$${P_{SSout}} = {\sigma _{SS}}({{P_{inci0}} - {P_{SSth}}} ),$$

where

(23)$${\sigma _{SS}} = \gamma \frac{{{A_2}}}{{{A_1}}}\frac{{({1 - {R_{SS}}} )({1 - {R_P}} )}}{{{{\left( {1 - \sqrt {{R_P}({1 - {L_P}} ){{({{R_{SS}}({1 - {L_{SS}}} )} )}^{\frac{{A2}}{{A1{\; }{\eta_S}{\; }{\eta_{SS}}}}}}} } \right)}^2}}}$$

represents linear slope efficiency and ${P_{SSth}}$the pump threshold for second Stokes

(24)$${P_{SSth}} = \frac{{ - {A_1}}}{{{g_s}l\gamma ({1 - {R_P}} )}}{\left( {\sqrt {{R_P}({1 - {L_P}} ){{[{R_{SS}}({1 - {L_{SS}}} )]}^{\frac{{{A_2}}}{{{A_1}{\eta_S}\; {\eta_{SS}}}}}}} - 1} \right)^2}\ln [{{R_s}({1 - {L_S}} )} ],$$

These final results can be simplified under analogous assumptions to first Stokes model, i.e., ${R_P}$,$\; {R_S}$, ${R_{SS}}$ ≈ 1 and ${L_P}$,$\; {L_S}$, ${L_{SS}}\; $<< 1 to obtain

(25)$${P_{SSth}} \approx \frac{{{A_1}({{L_S} + 1 - {R_s}} )}}{{{g_s}l\gamma }}\frac{{{{\left\{ {1 - {R_P}\left[ {({1 - {L_P}} )- \frac{{{A_2}({1 - {R_{SS}}\; + {L_{SS}}} )}}{{{A_1}{\eta_S}\; {\eta_{SS}}}}} \right]} \right\}}^2}}}{{4({1 - {R_P}} )}}.$$

According to Eq. (20), the first Stokes intra-cavity power becomes constant above second Stokes threshold, pinned at the intensity required to make the gain and loss equal for the second Stokes field. Since it is the first Stokes intensity, through its influence on G, that determines whether the cavity is impedance matched, we can design the laser such that the laser is always impedance matched for all second Stokes output powers. To do this, we set $G = {R_P}$, and find that we should choose

(26)$${R_{SS}} = \frac{1}{{({1 - {L_{SS}}} )}}{\left( {\frac{{{R_P}}}{{1 - {L_P}}}} \right)^{\frac{{{A_1}\; {\eta _S}\; {\eta _{SS}}}}{{{A_2}\; }}}}.$$

This impedance match condition, as expected, does not depend on pump power, and a second Stokes laser can be in theory impedance matched for all output powers with reflected power ${P_{Prefl}}$ remaining zero. Practically, imperfect mode-matching leads to a linear increase in reflected power ${P_{Prefl0}}$.

Using the same derivation method and induction, we can write equations for the behavior of laser configured to generate an arbitrary Stokes order. The behavior of lasers designed to output odd-order Stokes fields is different from those for even-order Stokes output. Figure 3(a) shows the behavior of a third Stokes laser, and Fig. 3(b) a fourth Stokes laser. The mirrors and other input parameters are similar to the experiments above, only resonating the pump and intermediate fields, and outputting the desired order. The output of odd-order Stokes lasers grows nonlinearly as the square root of power above threshold (this includes first Stokes output) while the even-order Stokes laser output grows linearly (this includes second Stokes output). Lasers designed to output odd Stokes orders can only be impedance matched at one pump power, whereas lasers designed to output even Stokes fields can be impedance matched for all output powers. When a new Stokes field is generated, the Stokes fields with the opposite parity to the new Stokes field are pinned: For example, when the fourth Stokes is generated, the power of first and third Stokes is constant.

Fig. 3. (a) The third Stokes and (b) the fourth Stokes output as a function of pump power. The model parameters are: passive loss of all fields = 1%, pump waist radius = 20 μm (and Brewster crystal), ${\lambda _P}$ = 851 nm, $\gamma \; $ = 1, ${R_P}\; $ = 96.7%, ${R_1}\; $= ${R_2}$ = 99.8%, and ${R_3}$ = 98.5% in (a) and ${R_3}$ = 99.8%, ${R_4}$ = 98.5% in (b).

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5. Results for a second Stokes laser

We built a second Stokes laser, as described in Fig. 1. For this laser we used a 1064 nm pump laser, resulting in ${\lambda _s}$ = 1240 nm and ${\lambda _{ss}}$ = 1485 nm. The output second Stokes power, the first Stokes power, and the reflected pump power are shown in Fig. 4. The maximum second Stokes power was 1.55 W for 17.1 W of pump power. As in the first Stokes system, retro mirrors (DM2 and DM3 in Fig. 1) were used to counter-propagate the first and second Stokes fields. In this wavelength range, we currently do not have an instrument to resolve the mode structure of the second Stokes output, and we leave for future work the analysis of whether the second-Stokes laser operated on a single-longitudinal mode. We note that in our previous work with a second-Stokes laser [12], the output was only SLM for lower output powers, and there remains an open question as to whether this can be maintained further above threshold.

Fig. 4. Measured (symbols) and modelled (lines) output power and reflected pump power for the second Stokes laser. The dashed lines are fitted model results with a fixed $\gamma $ value, whereas the solid lines use the variable $\gamma ^{\prime}$.

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The first Stokes power remains close to 1.07 W above the second Stokes threshold, reflecting the fact that it must provide gain to balance the constant second Stokes loss. Using ${g_s} = $ 11.2 cm/GW${\; }$ [15], we use the model to fit the matching parameter$\; \gamma = 0.81$, and fitting passive losses (the same for all fields) as ${L_P} = {L_S} = {L_{SS}}{\; } = 1.02\%$. The dashed modelling curves in Fig. 4 fit the measured data reasonably well below second Stokes threshold, but diverge significantly above second Stokes threshold where the predicted second Stokes slope efficiency is too high and the predicted reflected pump is too low. Using the model to investigate possible causes, it appears that changing the mode matching parameter $\gamma $ is the only reasonable way to explain the discrepancy. Using a simple heuristic formula to explore this idea, we set $\gamma $ above second Stokes threshold to be a function of pump power added above the second Stokes threshold as

(27)$$\gamma ^{\prime} = \frac{\gamma }{{1 + k({{P_{Pinci0}} - {P_{SSth}}} )}},$$

where k is a proportionality factor. As the solid line in Fig. 4 shows, using k = 0.038 simultaneously improves the fit of second Stokes slope and the growth of the reflected pump power, supporting this explanation. With this k, this models a change in $\gamma $ from 0.81 at second Stokes threshold to 0.57 at full pump power. To confirm experimentally why the mode matching might change needs more investigation: one possibility is small changes to the cavity transverse mode due to onset of thermal lensing and/or distortion.

We also considered what influence other changes to cavity parameters might have. For example, we measured a change in reflectivity of the IC/OC mirror as a function of temperature: a change of about 80°C shifted ${R_P}\; $and ${R_{SS}}$ by about -0.4% and -0.1%, respectively. (Such a local change in temperature of the mirror coating in the vicinity of the laser spot might be feasible.) However, such changes do not significantly affect the model output and are not able to explain the strong increase in the reflected pump power.

6. Optimization

To optimize a first or second Stokes laser, the main design feature is the IC/OC mirror. Figure 5(a) shows the modelled Stokes output power for 3.4 W of pump power, as a function of the reflectivity of the IC/OC mirror for the pump and Stokes fields (other experimental parameters are as in the first Stokes laser modelling presented above). The red lines are reflectivity value pairs for which the laser is impedance matched, and the red dot represents the output coupler reflectivities used in the experiments above. The mirrors used in the experiment generated somewhat lower power than the maximum of 620 mW that could have been achieved using a mirror with higher first-Stokes reflectivity.

Fig. 5. The output Stokes Power as a function of pump and Stokes reflectivity. (a) The first Stokes laser, (b) the second Stokes laser. The parameters are identical to the experiments, the red line shows the impedance matching condition. The red spots are the output coupler reflectivities used in the experiment.

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We can analytically find the optimum reflectivities for the mirror by finding the maximum of Eq. (9) with respect to ${\textrm{R}_S}$ and ${\textrm{R}_P}.$ We find

(28)$${R_{S - optimum}} = 1 + {L_S} - \sqrt {\frac{{{L_s}gl\gamma {P_{Pinci0}}}}{{{L_P}{A_1}}}} ,$$

(29)$${R_{P - optimum}} = 1 - \frac{{{L_P}}}{{{L_S}}}\sqrt {\frac{{{L_S}gl\gamma {P_{Pinci0}}}}{{{L_P}{A_1}}}} .$$

Note that here we use the approximation $\ln ({1/x} )\approx 1 - x$ for simplicity, thus the optimum equations are valid in a relatively low pump power range.

Figure 5(b) shows the optimization of the IC/OC mirror reflectivities for the pump and second Stokes fields (the reflectivity for the first Stokes is assumed 100%, with any transmission accounted in the Stokes passive loss). We use $\gamma ^{\prime} = 0.57$, corresponding to its fitted value at maximum pump.

For the second Stokes field the optimal reflectivities are

(30)$${R_{SS - optimum}} = 1 + {L_{SS}} + \frac{{{A_1}{L_P}}}{{{A_2}\lambda }} - \frac{{gl({{A_1}{L_P} + {A_2}\lambda {L_{SS}}} )\gamma {P_{Pinci0}}}}{{\sqrt {{A_2}^2gl{\lambda ^2}({{A_1}{L_P} + {A_2}\lambda {L_{SS}}} )\gamma {P_{Pinci0}}({1 + {L_S} - {R_S}} )} }},$$

(31)$${R_{P - optimum}} = 1 - \frac{{{A_2}gl\lambda ({{A_1}{L_P} + {A_2}\lambda {L_{SS}}} )\gamma {P_{Pinci0}}}}{{{A_1}\sqrt {{A_2}^2gl{\lambda ^2}({{A_1}{L_P} + {A_2}\lambda {L_{SS}}} )\gamma {P_{Pinci0}}({1 + {L_S} - {R_S}} )} }},$$

where $\lambda = \frac{{{\lambda _{ss}}}}{{{\lambda _P}}}$. The mirrors used in the experiment were close to optimal, generating 1.55 W compared to the 1.67 W that might have been achieved with an optimized mirror.

In both cases, for higher loss than assumed in Fig. 5 the Stokes output range will shrink toward higher mirror reflectivities. Increasing pump intensity by increasing pump power or decreasing spot size has the opposite effect, as expected. In both cases the Stokes output power is less sensitive on the reflectivity of the pump. As shown by the red lines, impedance matching is a necessary but not sufficient condition to maximize the Stokes power; the optimum impedance matched pair of ${R_P}$ and ${R_S}$ (${R_{SS}}$) have to be found according to Eqs. (28)–(31).

Using the optimum mirror specification, we can plot the maximum achievable output power as a function of the available pump power, as shown in Fig. 6. Here we use the losses measured in our experiments, but assume ideal mode matching $\gamma = 1$ to represent the best that can be achieved. The conclusion from Fig. 6 is that an ideal resonantly-pumped first Stokes laser with output power above 1 W requires as little as 3 W of pump power; a second Stokes output of a watt requires just a 6 W pump laser.

Fig. 6. The maximum Stokes output power as a function of pump power for optimum mirror design according to Eqs. (28)–(31). The input parameters are the same as for our experiments.

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7. Conclusion

In this paper, we presented experimental demonstration and modeling of a cavity-enhanced Raman ring lasers generating the first and Second Stokes output. The model considers all experimental parameters including mode matching and corresponds to the experimental measurements reasonably well, with a divergence for the second Stokes model suggesting that the mode matching deteriorates above the second Stokes threshold. The optimization analysis will enable the design practical CW Raman lasers pumped with relatively small pump lasers at the few-watt level.

Funding

Australian Research Council (LP110200545).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Modelling and characterisation of continuous wave resonantly pumped diamond Raman lasers

Abstract

1. Introduction

2. First Stokes model

3. Experimental setup and results from a first Stokes laser

4. Model for second Stokes and above

5. Results for a second Stokes laser

6. Optimization

7. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (31)

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