Peak-power enhancement of a cavity-dumped cesium-vapor laser by using dual longitudinal-mode oscillations

Masamori Endo

doi:10.1364/OE.409022

1. Introduction

Diode-pumped alkali lasers (DPALs) are optically pumped, near-infrared, continuous-wave (CW) gas lasers. A vaporized alkali metal—such as potassium, rubidium, or cesium—mixed with helium buffer gas at nearly atmospheric pressure serves as the active medium. A DPAL is optically pumped by a narrow-band laser diode at the D₂ transition, and it lases at the D₁ transition. Figure 1 shows an energy diagram and the transition paths of a Cs DPAL. In a Cs DPAL, methane or ethane is added to the medium to facilitate upper-state mixing [1].

Fig. 1. Energy diagram of a Cs DPAL.

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DPALs have been extensively studied in the past decade as a potential source for extremely high-average-power applications [2,3]. Because of their high efficiency and exceptional beam quality, they have potential industrial applications as well [4]. High-power industrial lasers are often operated in a pulsed mode, since this allows—for example—better coupling with materials and a reduced heat-affected zone. Because DPALs are scalable to megawatts, they are expected to be useful also as high-energy, ground-based laser drivers for the removal of space debris from orbit [5,6]. For such an application, periodic pulses are clearly more advantageous than CW for a given average power, since the momentum-coupling coefficient (the momentum produced per unit energy of the laser) is inversely proportional to the square root of the pulse width [7].

The most common pulsing technique for CW lasers is Q-switching. However, this technique is not applicable to DPALs, because the radiative lifetime of the lasing transition is too short (on the order of 10 ns) [2] to store energy in the upper lasing state. Instead, we have proposed the use of the cavity-dumping (CD) technique for repetitive, enhanced pulse operation of a DPAL [8]. We have developed a one-dimensional, time-dependent code to simulate a CD-DPAL and have shown that an enhancement of a factor of 20–60 is expected if it is applied to our 3-W-class DPAL [9].

In response to the positive outcome of the simulation, we also conducted an experimental investigation of this technique [10], achieving a 14 ns pulse with 77 W peak power from the aforementioned apparatus. Unexpectedly, the waveform of the pulsed output was not a trapezoid characterized by the resonator length, but instead it had periodic spikes at 220 MHz or 440 MHz. The period of the spike strongly suggests interference between longitudinal modes. Although a homogeneously broadened CW laser like a DPAL in principle oscillates in a single longitudinal mode, spatial hole burning (SHB) can enable the excitation of multiple longitudinal modes [11,12].

This phenomenon is actually beneficial for our purposes, because it results in higher peak power than the average power circulating inside the optical resonator. In other words, it adds additional modulation to the periodic pulses of a DPAL. However, the pulse shape was not always spiky. Sometimes it exhibited a smooth trapezoidal shape, as if there were only one longitudinal mode present. To fully exploit this effect for enhancing the output power of the CD-DPAL, we were therefore motivated to investigate the cause of the spikes and to find a way to maximize the peak power.

In the present study, we have investigated multiple longitudinal-mode oscillations of a DPAL. In Section 2, we describe the basic idea of multiple longitudinal-mode oscillations resulting from SHB. In Section 3, we describe the experimental setup and the results obtained by varying the resonator length. In Section 4, we introduce the one-dimensional numerical simulation code and present some results that explain the experimental observations. In Section 5, we discuss guidelines for obtaining sharp peaks from dual-mode oscillations. In Section 6, we summarize our conclusions about enhancing the peak power of a CD-DPAL.

2. Theory

In principle, a laser with a homogeneously broadened gain medium oscillates in a single longitudinal mode, because the gain profile is homogeneously saturated to the level where only the principal mode can maintain steady-state oscillation. However, stable dual-mode—or multimode—oscillations are often observed due to SHB. In their pioneering works, Hertel et al. [13,14] observed SHB in a CW dye laser. They derived an analytical formula for establishing a stable dual-mode oscillation. In the present work, we revisit their explanation but cast it into a more intuitive form. For convenience, we follow their nomenclature.

Figure 2 shows a standing-wave optical resonator consisting of mirrors M1 and M2 separated by a distance L from each other. A short gain module is placed between the two mirrors, and its positions measured from M1 and M2 are b and a, respectively. The spatial electric-field distribution of an oscillating mode at a certain moment may be written as

(1)$${E_\beta }(z) = 2{E_0}\sin \left( {\frac{{\beta \pi }}{L}z} \right),$$

where E₀ is the electric-field amplitude, β is any large rational integer, and z is the distance measured from M1. We call this the βth mode. At the position of the gain module (z = b), the phase of the (β+j)th mode relative to the βth mode is advanced by

(2)$$\Delta \varphi = j\pi \frac{b}{L},$$

as shown in Fig. 2. It follows that the interference between two longitudinal modes due to local hole burning is at a minimum when the phases of the two longitudinal modes differ by

(3)$$\Delta \varphi = \frac{{m\pi }}{2}\,\,\,\,\,({m = \textrm{ }1,\textrm{ }3,\textrm{ }5 \ldots } ).$$

In other words, when the intensity maxima of one longitudinal mode coincide the intensity minima of the second mode, the two modes receive equal gain with minimal interference. Equating Eqs. (2) and (3) gives the condition for dual-mode oscillation:

(4)$$\frac{b}{L} = \frac{m}{{2n}} \,\,\,\,\,({n = \textrm{ }1,\textrm{ }2,\textrm{ }3 \ldots ,\,\,m = \textrm{ }1,\textrm{ }3,\textrm{ }5 \ldots } ).$$

Fig. 2. Schematic diagram of a standing-wave optical resonator and the relative phases of neighboring modes.

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For m = 1, the optimal position of the gain module for dual-mode oscillation thus occurs at 1/2, 1/4, 1/6… of the full resonator length, and the order of the second mode is j = 1, 2, 3…, respectively.

3. Experimental

3.1 Experimental setup

Figure 3 shows a schematic diagram of the experimental apparatus. It is the same as in our previous publication [10], but has been modified as follows: First, the cubic polarization beam splitter (PBS) has been replaced by a Brewster-type PBS (Sigma Koki PBS-25, 4C03-10-895) in order to reduce the insertion loss. Second, the length of the resonator is varied as shown in Table 1.

Fig. 3. Schematic diagram of the experimental setup. PC = Pockels cell; PD = photodetector; PM = power meter; ND = neutral-density filter; DM = dichroic mirror; PBS = polarization beam splitter.

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Table 1. Specifications of the optical resonators^a

View Table

We used four different lengths for the optical resonator. In each case, the length b was fixed and L was varied. For the resonators with L = 0.69 m and 2.0 m, the dual-mode condition is fulfilled for n = 1 and n = 3, respectively, although this is not exact for L = 0.69 m. We selected L = 1.0 m so that dual-mode oscillation was not likely to occur for this resonator, and we used L = 5.0 m to see the effect of small Δφ.

The gain module is made of stainless steel and is capped by anti-reflection-coated glass (BK7) windows at both ends. The gain length is 5 cm. A mechanical chopper, operating at 10 Hz with an 8% duty cycle, is inserted in the resonator. The optical switch is composed of a Pockels cell (PC) and the PBS. The PC (Leysop BBO-TT-4-25-AR895) is operated in λ/4 mode and is optimized for the high-speed driver (FIT FDS4-1M), which provides a 3 kV pulse to the PC with a 0.5 ns (10–90%) rise time and a 20 ns duration. The optical resonator oscillates in s-polarization with respect to the PBS, due to its high polarization selectivity. When a high-voltage (HV) pulse is applied to the PC, the polarization of light that passes twice through the PC is rotated by 90°, and the light reflected from M1 passes through the PBS as an out-coupled beam from the resonator.

The output power is measured with a high-speed Si photodetector (NewFocus 1621) through an OD = 3 neutral-density filter and a condenser lens. The amplified voltage signal from the detector is recorded by a high-speed oscilloscope (Teledyne Lecroy WaveSurfer 10). The nominal bandwidth of the photodetector is 500 MHz, and the sampling rate and bandwidth of the oscilloscope are 10 Gs/s and 1 GHz, respectively.

In all the experiments, the operating conditions were as follows: the gas components were 60 Torr of ethane and 700 Torr of helium, the pump power was 33 W, and the spot size at the center of the gain cell was 0.3 × 0.8 mm FWHM [15]. The operating temperature of the gain cell was set at 88 °C, which we found to be optimal for generating the highest CW output power.

We operated the apparatus in three different ways. First, we operated it in “CW mode.” In this mode, we fixed the mechanical chopper in the open state, the PC was removed, and a 70% partial reflector was used for M1. The output power was measured by power meter PM1 (ThorLabs S425C), and the unabsorbed pump power was measured by PM0 (Ophir FL250A). The measured output power in CW mode was 6.6 W, and the optical-optical conversion efficiency with respect to the absorbed power (gross efficiency) was 60%. The transverse mode of the oscillation was TEM₀₀, judging from the spot size far from the output coupler.

Next, we operated the apparatus in “chopped mode.” In this mode, we rotated the chopper. We also inserted the PC, but we did not operate it. The use of the chopper is intended to prevent optical damage to the PC. The circulating power inside the resonator was measured by PM2 (ThorLabs S121C), where a small fraction (less than 0.1%) of the circulating power spills out from the mirror. We determined the factor for converting the signal level detected by PM2 to circulating power by comparing the readouts of PM1 and PM2 in CW mode. In addition, we measured the internal-loss factor of the optical resonator by varying the reflectivity of M1 and measuring the internal power. The result is discussed in the next subsection.

Finally, we used a highly reflective mirror for M1 and operated both the chopper and the PC, which we refer to as “CD mode” operation. Due to the limitations of our HV power supply, we could only apply HV to the PC at 100 Hz. Therefore, most of the optical power stored inside the resonator is not extracted; this is the reason why we use the chopper in CD mode.

3.2 Experimental results

We first set the resonator length at 2.0 m and operated the apparatus in chopped mode while varying the transmittance of M1 from <0.1% to 85%. For this series of experiments, the pump power was reduced to 12 W. The result is shown in Fig. 4. The internal power P_int in the resonator is plotted against the output coupling coefficient.

Fig. 4. Internal power of the optical resonator measured by PM2 as a function of the output coupling coefficient.

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From the data presented in Fig. 4, one can estimate the internal loss of the resonator by Rigrod analysis [16]. Assuming that all the loss (L_i/pass) occurs between the gain cell and M1—and that the small-signal gain coefficient g₀, saturation intensity I_s, and beam area A are constant across the gain cell—the value of P_int measured at the position of PM2 is given by

(5)$${P_{{\mathop{\rm int}} }} = \frac{{{g_0}l + \ln \sqrt {R - 2{L_\textrm{i}}} }}{{\left( {1 + \frac{1}{{\sqrt {R - 2{L_\textrm{i}}} }}} \right)\left( {1 - \sqrt {R - 2{L_\textrm{i}}} } \right)}}{I_\textrm{s}}A,$$

where l is the gain length, and R is the power reflectivity of M1. By fitting the results in Fig. 4 numerically, we obtained g₀ = 0.61 cm⁻¹ and L_i = 0.014/pass. The fit obtained using Eq. (5) is shown as the dashed line in Fig. 4. In our previous apparatus with the cubic PBS, L_i was 0.034/pass. The magnitude of this reduction cannot be explained solely by the substitution of the PBS. The improved optical alignment of the PC may also have contributed to it.

Next, we operated the apparatus in CD mode with the various resonator lengths shown in Table 1. Figure 5 shows typical waveforms at the four different resonator lengths. The output-pulse waveform was recorded by PD1, while the internal intensity of the resonator was recorded by PD2, as shown in Fig. 3. The voltage signal from PD1 was calibrated absolutely in CW mode by measuring the laser power alternately with the photodetector and with the power meter (PM2). On the other hand, the voltage signal from PD2 was not calibrated, and we estimated its magnitude from the average internal intensity recorded by PM2. The blue traces represent the internal power, and the red traces represent the output power. The optical delay was compensated to make the presentation clearer. The insert in the upper-right corner of each panel is the Fourier power spectrum of the internal power. Note that the recorded waveform was not always like those shown here; for this figure we selected waveforms that show spectrum peaks at integer multiples of the cavity-mode spacing (Table 1).

Fig. 5. Waveforms of the internal power in the resonator (blue) and of the output pulse (red). The insert is the Fourier power spectrum of the internal power.

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Except for L = 1.0 m, the highest peak in the Fourier power spectrum, near 227 MHz, strongly suggests that there are two modes whose spacing coincides with the spatial hole-burning frequency (SHBF) [17], c/4b, where c is the speed of light. However, the smaller peaks that coincide with integer multiples of the cavity-mode spacing suggest that more than two longitudinal modes are oscillating simultaneously. On the other hand, for L = 1.0 m, the observed waveforms were mostly like the one shown in Fig. 5, although they occasionally showed peaks at 150 MHz (the cavity-mode spacing) and its integer multiples.

These observations suggest the following characteristics of the longitudinal modes in a DPAL:

1. When the condition given by Eq. (4) is satisfied, SHB promotes dual-mode oscillation spaced by the SHBF (Δφ = 90°). However, more than two modes were found to be oscillating simultaneously most of the time.
2. When more than two modes are oscillating, complete constructive interference of these modes is not expected, and thus the modulation depth is shallower than for dual-mode oscillation.
3. When L = 1 m, where (b/L) becomes an odd fraction, a beat frequency at the SHBF was not observed, although a high-frequency beat was seen. Beat frequencies near 500 MHz and 900 MHz suggest that the oscillating modes share the same intensity distribution along the optical axis inside the gain module.
4. The state of the longitudinal-mode distribution is unstable. Presumably, it is sensitive to the slightest change in the optical path, for example due to vibration of the resonator mirrors.

4. Numerical simulation

4.1 Modeling

To explain how the oscillating modes are chosen for a given set of conditions, we constructed a one-dimensional numerical simulation that illustrates the essential features of multiple longitudinal-mode oscillations as simply as possible.

Figure 6 shows a schematic illustration of the model. The effective zone for the model calculation is the volume w × w × λ shown in the figure, where w represents the beam width, and λ represents the wavelength of the D₁ transition. The calculation zone is divided into 100 unit cells, and the rate equations and photon-matter interactions are calculated in each cell. The active volume of length l is represented by periodic boundary conditions that repeat the calculation zone. The pump light is assumed to illuminate the side of the active volume in order to avoid the complexity of pump-light depletion. The cavity modes are assumed to be uniform in the transverse dimensions, whereas a standing-wave sinusoidal intensity distribution is assumed in the optical-axis direction.

Fig. 6. Schematic illustration of the numerical simulation model.

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In the calculation zone, each longitudinal mode is assumed to have the following intensity distribution:

(6)$${I_j}(z) = {I_{0j}}{\sin ^2}\left( {\frac{{2\pi z}}{\lambda } + {\varphi_j}} \right)\,\,\,\,\,\,\,({j = \textrm{ }0,\textrm{ }1,\textrm{ }2 \ldots ,\,\,J - \textrm{ }1} ),$$

where j represents the index of the mode order starting from zero, I_0j is the peak intensity of the jth mode, and φ_j is the phase constant given by Eq. (2). Because pressure broadening is very large in a DPAL (typically 20 GHz) compared to the cavity-mode spacing (100 MHz or less), a set of 200 modes was initially assumed. However, we found that the calculation result was not affected by reducing the initial number of modes to 20, so J was assumed to be 19 or 20.

In each unit cell, the following set of atomic rate equations is solved:

(7)$$\frac{{\partial {N_3}}}{{\partial t}} ={-} {\sigma _{31}}({N_3} - 2{N_1}){n_2}c - ({k_\textrm{1}}{N_3} - {k_\textrm{2}}{N_2}){N_{\textrm{C2H6}}} - {A_{31}}{N_3}\, - {k_{\textrm{q2}}}{N_3}{N_{\textrm{C2H6}}} + D\frac{{{\partial ^2}{N_3}}}{{\partial {z^2}}},$$

(8)$$\frac{{\partial {N_2}}}{{\partial t}} ={-} \sum\limits_j {\sigma _{21}^j({{N_2} - {N_1}} )\frac{{{I_j}}}{{h\nu }}} + ({k_\textrm{1}}{N_3} - {k_\textrm{2}}{N_2}){N_{\textrm{C2H6}}} - {A_{21}}{N_2} - {k_{\textrm{q}1}}{N_2}{N_{\textrm{C2H6}}} + D\frac{{{\partial ^2}{N_2}}}{{\partial {z^2}}},$$

(9)$$\frac{{\partial {N_1}}}{{\partial t}} = {\sigma _{31}}({N_3} - 2{N_1}){n_2}c + \sum\limits_j {\sigma _{21}^j({{N_2} - {N_1}} )\frac{{{I_j}}}{{h\nu }}} + {A_{31}}{N_3} + {A_{21}}{N_2}\\ + ({k_{\textrm{q}2}}{N_3} + {k_{\textrm{q}1}}{N_2}){N_{\textrm{C2H6}}} + D\frac{{{\partial ^2}{N_1}}}{{\partial {z^2}}}, $$

where N_i (i = 1, 2, 3) represents the number density of Cs(6²S_1/2), Cs(6²P_1/2), and Cs(6²P_3/2), respectively, and N_C2H6 is the number density of ethane, A_ji represents the Einstein A-coefficient of the transition j → i, hν is the energy of a D₁ photon, n₂ is the number density of D₂ photons, and σ₂₁^j and σ₃₁ are the stimulated-emission cross-sections for the D₁ (jth mode) and D₂ transitions. The rate constants k₁, k₂, k_q1, and k_q2 are same as in our previous publication [15]. This is a simplified reaction model of a DPAL, but it does represent pumping, stimulated emission, upper-state mixing, spontaneous emission, and quenching of the Cs atoms by ethane. The effect of leveling the local Cs atomic-concentration distribution by diffusion is important and is also considered in the model. The diffusion constant D was taken from the literature [18] and is assumed to have the temperature and pressure dependence given by D = D₀(T/T₀)^1.7(p₀/p) [19].

For optical amplification, we employed a simple photon-rate-equation model. Although it has been pointed out that effects such as four-wave mixing [17] and population grating [20] contribute to the evolution of the multiple longitudinal modes in homogeneously broadened CW lasers, our simplified treatment nevertheless reproduces many of the observed phenomena. In our model, the individual oscillation modes are separately calculated from

(10)$$\frac{{\partial {I_{0j}}}}{{\partial t}} = \frac{{c\sigma _{21}^j}}{\lambda }\int\limits_0^\lambda {({{N_2} - {N_1}} ){I_j}\textrm{d}z} - \frac{{1 - R}}{{2L/c}}{I_{0j}} + \frac{{ch\nu \Omega {N_2}{A_{21}}}}{{F({{{\Delta {\nu_{21}}} / {\Delta {\nu_\textrm{R}}}}} )}}\delta ,$$

where R is the reflectivity of the output mirror, Ω is the solid angle of the TEM₀₀ transverse mode of the resonator in question, F is the finesse of the resonator, Δν₂₁ is the spectral broadening of the D₁ transition, Δν_R is the cavity-mode spacing, and δ is a random number from zero to 1, which is calculated every time Eq. (10) is evaluated. The last term in Eq. (10) is a stochastic-noise term; it has also been applied in other photon-rate-equation models [21,22] to represent the chaotic nature of mode growth [17].

The stimulated-emission cross-section of the D₁ transition was assumed to be homogeneously broadened and to have a Lorentzian shape with the broadening constant Δν₂₁; it is given by the following [19]

(11)$$\sigma _{21}^j = \frac{{{\lambda ^2}}}{{8\pi }}{A_{21}}\frac{{{{\Delta {\nu _{21}}} / {2\pi }}}}{{{{[{{\nu_0} - \{{j - (J - 1)/2} \}\Delta {\nu_\textrm{R}}} ]}^2} + {{({{{\Delta {\nu_{21}}} / 2}} )}^2}}}\,\,\,\,\,\,({j = \textrm{ }0,\textrm{ }1,\textrm{ }2 \ldots ,\,\,J - \textrm{ }1} ), $$

where ν₀ is the line center, and Δν₂₁ is calculated as a function of operating temperature, using the method described in Ref. [23].

We have found that the final state of the mode distribution depends strongly on the locations of the resonator modes with respect to the line center of the D₁ transition, even though Δν₂₁ is two orders of magnitude larger than Δν_R. Taking this into account, we next considered the results for two distinct cases. In one series of cases, one of the longitudinal modes coincides with the line center. In these cases, the total number J of longitudinal modes assumed in the model is odd; these are referred to hereafter as “odd cases.” On the other hand, for cases with even J, the line center is in between the two longitudinal modes; these are referred to as “even cases.”

The calculation starts with zero oscillation intensity in the optical resonator. A part of the spontaneous emission is randomized and added to each oscillation mode. It is then amplified by the stimulated emission. The calculation ends when a stable oscillation state is established. Although we performed calculations for a wide range of operating conditions, all the calculations ended with a steady state. No competitive oscillations between two modes [14] were observed. It is interesting to note that the spatial inhomogeneity of the population inversion leveled out to less than 0.1%, the same as in the numerical calculations of Ref. [17].

In all the calculations, w and l were fixed at 1 mm and 50 mm, respectively. The internal intensity of the optical resonator was varied by changing the reflectivity of the output coupler. The operating pressure and temperature were varied to see the effects of these variables on the state of the oscillating modes. The pump power was fixed at 30 W (60 W/cm²), and its spectral width was assumed to be 50 pm. In a typical case, approximately 30% of the pump power was absorbed by the medium, and the typical output power was 6.4 W.

4.2 Results of the calculations

We set the operating conditions to be the same as those in the experiments and performed a series of calculations. The reflectivity of M1 was chosen to be 0.98 so that the internal intensity was consistent with the observations. Figure 7 shows the results. The graphs show the longitudinal-mode intensity distributions for the four different resonator lengths listed in Table 1. Each bar shows the intensity, and its position shows the phase. The line center of the D₁ transition is defined to be φ = 0. Because we set b = 0.33 m, the conversion factor from Δν_R to Δφ is 2.5 MHz/degree. In all cases, both the odd case and the even case are displayed.

Fig. 7. Calculated longitudinal-mode intensities. The conditions are same as in the experiments.

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As can be seen from the figure, there is no case with a single longitudinal-mode oscillation. That is true independent of the calculation conditions: the M1 reflectivity ranged from 0.3 to 0.99 (three orders of internal intensity), the pressure ranged from 360 Torr to 1520 Torr, and the temperature ranged from 353 K to 373 K. From these calculations, it appears to be extremely difficult for a DPAL to oscillate in a single longitudinal mode without employing a mode-selection mechanism.

Next, we investigated the individual results shown in Fig. 7. The number of oscillating modes and their relative intensities differ considerably between the odd and even cases, and this explains why the experimentally observed waveform changes over time. Because only a 0.2 µm change in the resonator length results in a switch from odd to even, the state of the longitudinal modes can easily be switched from odd to even and vice versa. However, by seeding two longitudinal modes with frequencies separated by the SHBF (Δφ = 90°) at the beginning of the calculation, we have also confirmed that a stable dual-mode oscillation can be established wherever the longitudinal modes sit.

Figure 8 shows the intensity of the third mode relative to that of the second mode as a function of the seed intensity. For 0.69 m and 2.0 m, we chose the odd case, in which a dual-mode oscillation is not naturally established. For 5.0 m, we selected the even case. The figure shows that above a certain seed level the third mode is suddenly suppressed to virtually zero. The required intensity for dual-mode operation is less than 100 W/cm², which is on the order of 1% of the typical internal intensity when the resonator is in CD operation. This implies that a stable dual-mode oscillation can be obtained by self-seeding of the resonator; that is, by not dumping the circulating power completely.

Fig. 8. Intensity of the third mode relative to that of the second mode as a function of the seed intensity.

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The only case with dual-mode oscillation is L = 2.0 m, even (see Fig. 7). This is because Eq. (4) is exactly fulfilled only for L = 2.0 m; for L = 0.69 m, there is only 4° of mismatch, but this results in multimode oscillation. The effect of phase mismatch is discussed in Section 5.

For L = 5.0 m, two groups of longitudinal modes with phases nearly 90° apart oscillated simultaneously. This is because the distance between adjacent modes is too close for SHB to discriminate between them, and multiple modes with nearly identical intensity distributions can coexist. This result implies that the common method for producing single longitudinal-mode oscillation—bringing the gain module close to one of the resonator mirrors [24]—does not work for a DPAL. As will be discussed in Section 5, dual-mode oscillation is possible for the L = 5.0 m resonator if Eq. (4) is exactly satisfied, but that occurs only in the lower range of intensities, I < 10 kW/cm².

For L = 1.0 m, in which the phase distance between adjacent modes is 60°, three longitudinal modes are oscillating simultaneously with equal intensity. Note that the phase distance between weaker modes in the even case is 180°, and their intensities are half the central value, exciting three modes oscillating with virtually equal intensities. Interestingly, the sum of three squared sinusoidal functions with phases 60° apart is constant; that is

(12)$${\sin ^2}(\frac{{2\pi z}}{\lambda }) + {\sin ^2}(\frac{{2\pi z}}{\lambda } + \frac{1}{3}\pi ) + {\sin ^2}(\frac{{2\pi z}}{\lambda } + \frac{2}{3}\pi ) = \frac{3}{2}.$$

Therefore, the laser medium is uniformly saturated over the distance z by a set of three modes with phases 60° apart. In reality, this condition seemed less stable than dual-mode oscillation, so beat frequencies with Δφ = 60° (150 MHz) and its integer multiples were not frequently observed.

In summary, our simulation model reproduced many of the observed results:

1. Longitudinal modes that oscillate simultaneously are strongly affected by SHB. A phase difference of 90° is preferred if it is possible.
2. If Eq. (4) is not exactly satisfied, more than two modes oscillate simultaneously. This explains the beat frequencies at integer multiples of the cavity-mode spacing.
3. The status of a longitudinal mode is strongly affected by the relative position of the longitudinal modes with respect to the line center of the D₁ transition. This fact, together with the sensitivity of the mode frequencies to the slightest change in the resonator length, explains the instability of the observed beats.
4. The weaker beats in the case with L = 5 m are explained by multimode oscillations, due to the small spacings between adjacent modes and the mismatch from the condition given by Eq. (4).
5. For L = 1 m, the calculations also explain the reason why dual-mode oscillation was not preferred. The possibility of three-mode SHB was suggested by the calculations, and it was experimentally observed. However, the instability of such an SHB state is not explained by the current model.

5. Discussion

In this section, we discuss the conditions in which a DPAL preferentially operates in dual-mode oscillation. We compared the advantages and disadvantages of the following three resonators: Δφ = 15° (b = 0.333 m, L = 4 m), 30° (b = 0.333 m, L = 2 m), and 90° (b = 0.333 m, L = 0.667 m). We considered the favorable mode arrangement for dual-mode oscillation in these resonators; namely, odd for Δφ = 15° and even for Δφ = 30° and 90°. The figure-of-merit is defined by the purity of the dual-mode oscillation; that is, by the relative amplitude of the third mode with respect to the first and second modes, which we define as the signal-to-noise ratio (SNR).

First, we evaluated how the dual-mode oscillation is affected by the resonator length, leaving the position of the gain module unchanged and the operating pressure set to 760 Torr. The result is shown in Fig. 9, where the SNR is plotted as a function of the internal intensity. The relative phase of adjacent modes, Δφ, is the varied parameter. For Δφ = 90°, dual-mode oscillation is assured over the whole intensity range considered. On the other hand, for Δφ = 15° and 30°, the strength of the third (and fourth) mode increases as the internal intensity is increased. In particular, it is difficult to expect dual-mode oscillation when Δφ = 15°, especially during cavity dumping with a strong internal intensity. The gain module is better placed at the center of the resonator (Δφ = 90°), or at least at 1/6 of the total length (Δφ = 30°).

Fig. 9. SNR plotted against the resonator internal intensity. The relative phase Δφ is the varied parameter.

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Second, we investigated the variation of the SNR with operating pressure. The result is shown in Fig. 10. We chose the resonator length L=2 m (Δφ = 30°) and varied the operating pressure as a parameter. As the figure shows, the operating pressure has a strong effect on dual-mode oscillation. At 1520 Torr, dual-mode oscillation is not expected for the intensity range where cavity dumping is employed. The difference between 360 Torr and 760 Torr also are large. The reason for this is straightforward: discrimination between adjacent longitudinal modes is stronger when the bandwidth of the D₁ line is narrower. This result thus provides a quantitative guideline for obtaining dual-mode oscillation.

Fig. 10. SNR plotted against the internal intensity of the resonator. The operating pressure is the varied parameter.

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In Fig. 7, dual-mode oscillation was not observed for L = 0.69 m, despite the condition expressed by Eq. (4) being approximately satisfied (see Table 1). We therefore calculated how the phase error affects dual-mode oscillation, fixing L = 0.69 m and varying the relative phase from 85° to 95° by changing the gain-module position. The result, shown in Fig. 11, demonstrates that the tolerance to phase error is strikingly small. Since one degree of phase error corresponds to a 3.8 mm shift along the optical axis for a 0.69 m resonator, the 5-cm-long gain module does not fulfill this condition throughout its entire length. This restriction becomes looser when the resonator is longer. Because the length of a DPAL gain module is no shorter than a few centimeters, it is desirable for the resonator length to be several meters.

Fig. 11. SNR as a function of the phase error.

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After determining the above guidelines for obtaining resonator conditions that facilitate dual-mode oscillation, we fine-tuned the optical resonator at the L = 2 m setting and operated the laser again in CD mode. The resulting internal power and output power waveforms shown in Fig. 12 were then obtained. The Fourier transform of the internal intensity suggests that dual-mode oscillation has been achieved. The highest peak power was 250 W, which is higher than the time-averaged internal power, 200 W. The peak power also was 38 times the output power of the CW mode (6.6 W). The internal intensity is not completely modulated, and one possible reason may be insufficient bandwidth of the PD. Judging from the internal-intensity waveform, the peak of the output pulse should be more deeply modulated, and the peak should be higher than that observed. However, the measurement setup did not allow us to capture the full-bandwidth waveform of the output pulse.

Fig. 12. Waveforms of the internal power of the resonator (blue) and of the output pulse (red). The insert is the Fourier power spectrum of the internal power.

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6. Conclusion

We have studied cavity dumping in a Cs-diode-pumped alkali laser both experimentally and theoretically. In a previous study, we observed a saw-like structure in the pulse waveform, presumably due to multimode oscillation resulting from SHB. In this study, we varied the length of the optical resonator and observed the output-pulse waveform as well as the internal intensity of the optical resonator. The observed beat frequency confirmed that the optical resonator was oscillating in more than two longitudinal modes simultaneously. In addition, the beat was observed not to be stable in time. Unless the resonator is equipped with a phase-lock device, it is desirable for the resonator to be oscillating in dual modes, because the amplitude of the beat signal is then strongest. To explain the observed phenomena and derive guidelines for stable dual-mode oscillation, we developed a simple one-dimensional numerical model. The calculations reproduced the observed results well. We found that a suitable condition for dual-mode oscillation is L = 2m, with the gain module placed precisely at the center of the resonator (i.e., within 1/90 of the full length) or at 1/4 or 1/6 of full length. Lower gas pressure is preferable, but even the typical 760 Torr operating condition of a DPAL leads to dual-mode oscillation. After fine-tuning the resonator length, we observed a pulse with 250 W peak power, which is 38 times the CW output of the same apparatus and higher than the time-averaged internal intensity of the optical resonator.

Acknowledgements

The author is grateful to Dr. Fumio Wani and Mr. Hiroki Nagaoka of Kawasaki Heavy Industries Ltd., for their technical and equipment supports of this work. The experimental work was supported by Mr. Taro Yamamoto of Numazu Oxygen Industries Co. Ltd.

Disclosures

The authors declare no conflicts of interest.

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Peak-power enhancement of a cavity-dumped cesium-vapor laser by using dual longitudinal-mode oscillations

Abstract

1. Introduction

2. Theory

3. Experimental

3.1 Experimental setup

3.2 Experimental results

4. Numerical simulation

4.1 Modeling

4.2 Results of the calculations

5. Discussion

6. Conclusion

Acknowledgements

Disclosures

References

Cited By

Figures (12)

Tables (1)

Equations (12)

Optics Express

Curvature of M2 (m)	b (m)	L (m)	b/L	Δφ (deg.)	90/Δφ	Cavity-mode spacing (MHz)
2.0	0.33	0.69	0.48	86	1.05	217
2.0	0.33	1.0	0.33	60	1.5	150
2.0	0.33	2.0	0.17	30	3.0	75
5.0	0.33	5.0	0.066	12	7.5	30