1. INTRODUCTION

Within the context of the crucially increasing energy needs, the International Thermonuclear Experimental Reactor (ITER) project [1] was launched in 1985 and is under construction in the south of France. ITER is an experimental reactor aiming at the development of fusion energy corresponding to a power of $\simeq 500\mathrm{MW}$ during about 1000 s. This naturally raises the question of the next step toward a demonstration power plant (DEMO) [2], which shall be the first real fusion power plant connected to the electric network to demonstrate electricity production over long periods. For a long term operation, during which the fusion reactions have to provide 1.5 GW of thermal power, additional plasma heating of 150 MW is required. Two heating systems are, so far, being considered: (i) antennae injecting high-power microwaves, which transfer energy to the plasma charged particles by resonant interactions. Located near the plasma, these antennae undergo high thermal loads. (ii) The neutral beam injection (NBI) system, which injects high-energy neutral atoms into the plasma core, the energy of which is transferred to the plasma particles by collisions. This system has the advantage to be far from the plasma core, and it is foreseen to be the main heating system in DEMO.

As a matter of fact, unlike ITER, the overall efficiency of the DEMO NBI system is a key parameter. It directly impacts on the net electrical power produced by the reactor and the electricity cost. In fact, in ITER neutrals, ${\mathrm{D}}^0$ are generated by accelerating negative ions ${\mathrm{D}}^{-}$ up to 1 Mev before being neutralized in a ${\mathrm{D}}_2$ gaseous cell. While diffusing along the NBI system, injected gas causes important losses through different mechanisms, thus limiting the injector efficiency to less than 30% [3]. Photodetachment was proposed as an alternative to gaseous neutralization [4] and represents a possible breakthrough to the ITER low NBI system efficiency, since it offers a potentially high beam neutralization rate ($\lesssim 100\%$), limits losses, and allows the recovery of non-neutralized negative ions [5,6]. Hence, 60% would be foreseen as the efficiency of the future DEMO NBI photo-neutralization-based system [7].

However, photo-neutralization has an obvious drawback: the weakness of the photodetachment cross section [8] combined with the high velocity of accelerated ions push the required optical power for an interesting neutralization rate up to the megawatt range. Meanwhile, the actual power consumed by the photodetachment process remains in the watt range for a ${\mathrm{D}}^{-}$ beam of tens of amperes. Optical cavities clearly appear as adapted to this issue, offering the possibility to load high optical power in a well-defined optical mode [9]. A suitable configuration representing a compromise among optical stability [10], reduction of optical power, and overlapping with the ion beam was identified to be a four times refolded half-symmetric resonator ($g\simeq 0.9$) (see top part of Fig. 1) with an intra-cavity beam radius $\simeq 1\mathrm{cm}$ to less than 10% for $\lambda \simeq 1064\mathrm{nm}$ in order to guarantee the homogeneity of the neutral beam. The cavity total length is $L\simeq 100\mathrm{m}$, and the stored optical power is 3 MW [11], reached thanks to a finesse of 10,000 and an injected laser in the kilowatt (kW) range.

 

Fig. 1. Top: negative ions are provided by a source then electrostatically accelerated to 1 Mev. Photo-neutralization occurs within a four times refolded cavity focusing 3 MW laser power. The cavity whole length is 100 m. Bottom: inserting an adequate telescope reduces the cavity length down to 30 m while keeping the beam width unchanged in the laser–ions interaction area.

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DEMO is expected to be a complex machine, in which important overload issues might be encountered, and 25 m per cavity arm might not be available. Keeping the same intra-cavity beam radius, i.e., maximizing the laser–ion beam overlap, while reducing the arm’s length, pushes the cavity toward the marginal stability area ($g\lesssim 1$). This is a well-known issue encountered in second generation gravitational wave detectors, for which power and signal recycling cavities accommodate a large beam radius ($\simeq 1\mathrm{cm}$) on a short distance ($\simeq$ few meters). The stability of these cavities is critical, since it impacts on the detection efficiency [12]. It was proved that adding an adequate telescope is likely to stabilize these cavities without significantly increasing their lengths, while providing the same beam radius to the Michelson interferometer [13,14]. This issue was also encountered in pulsed dye laser cavities accommodating a combination of a telescope and a grating [15]. The aim of this paper is to show that the DEMO NBI cavity can be shortened to $\simeq 30\mathrm{m}$ thanks to the same procedure (see bottom part of Fig. 1). In the general cavity configuration, the stability can be characterized by the round trip Gouy phase [10]. In the second section, we explain how a telescope can keep the cavity away from the marginal stability area, and three specific configurations are analyzed. In the third section, the optimum configuration is inferred from a calculation based on the $ABCD$ matrix formalism [10]. Several criteria required by the NBI system are considered, and the target of a 30 m long cavity is reached. Based on an obvious scaling law, the same stability condition can be brought to an optical bread board cavity. An experimental setup is implemented, which is described in the fourth section. The cavity round trip Gouy phase is measured with 16 mrad accuracy. This will be the subject of the fifth and final section.

2. INTRA-CAVITY TELESCOPE: A SIMPLE APPROACH

The stability of a two mirror cavity is a textbook case: the condition $0<g<1$ expresses that a mathematical paraxial Gaussian mode that is superimposed on itself after a round trip mathematically exists [10]. It is assumed that the areas $g\lesssim 1$ and $g\gtrsim 0$ correspond to marginal stable cavities, i.e., configurations that are very sensitive to misalignment like effects (mismatching, mirror surface defects). These stability areas can roughly be bounded, since misalignment like effects closely depend on the cavity configuration and not only on the $g$ factor. However, it is experimentally agreed that configurations corresponding to $0.02<g<0.98$ allow optical power matching close to 100%.

In the general case, the stability of a cavity configuration is measured by the round trip Gouy phase [16,17]. In fact, the stability of the cavity is directly related to its ability to lift the degeneracy between the fundamental mode and the first higher-order modes. Thus, for a non-degenerate cavity, misalignment like effects do not couple energy into intra-cavity first higher-order modes [beam shift and tilt into (${\mathrm{HG}}_{10}$, ${\mathrm{HG}}_{01}$) Hermite–Gauss mode, longitudinal mismatching into (${\mathrm{HG}}_{20}$, ${\mathrm{HG}}_{02}$, ${\mathrm{HG}}_{11}$) [18]]. Let ${\varphi }_G$ be the fundamental ${\mathrm{HG}}_{00}$ cavity mode Gouy phase and the resonance frequencies spacing $\mathrm{\Delta }\nu$ between the fundamental mode ${\mathrm{HG}}_{00}$ and ${\mathrm{HG}}_{10}$. The resonance condition gives [10]

The cavity considered here is represented on Fig. 2(a). The intra-cavity beam expands along the telescope $({\mathrm{M}}_1-{\mathrm{M}}_2)$, where the Gouy phase is mainly accumulated, thus stabilizing the cavity with respect to the hypothetical cavity made of the $({\mathrm{M}}_2-{\mathrm{M}}_3)$ section. Before going on to the rigorous calculation of the Gouy phase based on the $ABCD$ matrix formalism [10], we propose to give some Ref. [19] like estimations on the cavity stability. First, we neglect the astigmatism effect related to angle $\theta$ between the incident and reflected beam. Thus, the system is equivalent to the cavity represented on Fig. 2(b), for which the mirror ${\mathrm{M}}_2$ is replaced by a lens the focal length of which is half of the radius of curvature of ${\mathrm{M}}_2$, i.e., $f=R_2/2$. In the $({\mathrm{M}}_2-{\mathrm{M}}_3)$ section, the waist is located on the flat mirror ${\mathrm{M}}_3$. In the paraxial approximation, its image through the lens represents the beam waist in the $({\mathrm{M}}_1-{\mathrm{M}}_2)$ section. According to [19], we have with the notation used in Fig. 2(b):

 

Fig. 2. (a) Laser beam is injected by the mirror ${\mathrm{M}}_1$ and naturally expands through the telescope before being reflected to become subparallel between ${\mathrm{M}}_2$ and the flat mirror ${\mathrm{M}}_3$. The $({\mathrm{M}}_2-{\mathrm{M}}_3)$ section can be folded several times to cover the whole ion beam. (b) Aside from the angle $\theta$, the mirror can be simulated by a lens of focal length $f=R_2/2$.

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We start with the particular configuration $L_2=L_1=f$, which implies $l=0$. This means that the two waists on both sides of the lens are located on the focal planes and coupled regardless of the beam radius $w_0^{\prime }$. This case requires ${\mathrm{M}}_1$ to be flat, i.e., $R_1=\infty$ and is similar to the two mirrors confocal cavity [20]. Obviously, for a concave mirror ${\mathrm{M}}_1$, i.e., $R_1>0$, $w_0$ vanishes to have a Gaussian beam waist on a curved mirror. According to Eq. (3), $w_0^{\prime }$ goes to infinity. No cavity can close up if ${\mathrm{M}}_1$ is slightly moved to the left hand side. Whereas, slightly increasing $l$ brings $w_0^{\prime }$ to a finite value, whilst keeping an interesting magnification factor $w_0^{\prime }/w_0$. Finally, for a subparallel beam in the $({\mathrm{M}}_2-{\mathrm{M}}_3)$ section, increasing $L_2$ does not change the cavity configuration for $z_R^{\prime }\gg L_2>f$. Thus, at the first order in $f/z_R^{\prime }$, Eqs. (2) and (3) give $L_1-l\simeq f$ and $w_0\simeq \lambda f/\pi w_0^{\prime }$, which simply represents the Fraunhöfer diffraction limit [21], while considering the beam width unchanged in the $({\mathrm{M}}_2-{\mathrm{M}}_3)$ section. We then infer that the beam just after the right side of the lens represents a diffracted far field with respect to the beam waist position $“w_0”$, and that the Gouy phase accumulated between $“w_0”$ and the mirror ${\mathrm{M}}_2$ is ${\varphi }_{“w_0”-M_2}\simeq \pi /2$. Finally, since $L_2\ll z_R^{\prime }$, ${\varphi }_{{\mathrm{M}}_2-{\mathrm{M}}_3}\simeq 0$, the round trip Gouy phase is

It then appears that the whole cavity stability is related to the $g$ factor of the hypothetical half-symmetric cavity $(M_1-“w_0”)$ given by $g_h=1-l/R_1$ through its Gouy phase ${\varphi }_{M_1-“w_0”}=\arccos (\sqrt{g_h})$ [10]. With the same sense of purpose, we can associate an equivalent $g$ factor to the whole cavity, namely, $g_e$, related to its Gouy phase ${\varphi }_G$, and Eq. (4) gives $g_e=1-g_h$.

Considering the hypothetical cavity $(M_1-“w_0”)$, the beam waist $w_0$ is given by

During the former proof, $L_2$ was asymptotically removed, whereas $L_1$ is shown to be close to $R_2/2$. Hence, the cavity configuration depends only on three parameters: $R_1$, $R_2$ and $g_e$ their acceptable values. Note that shortening the subparallel beam cavity section down to 30 m requires us to keep the telescope section very short, i.e., $L_2\simeq 30\mathrm{m}$, and $R_2/2\simeq L_1\ll L_2$. We then set the maximum value of $R_2/2$ to 10% of the whole cavity length, i.e., $R_{2,\max }=6,\mathrm{m}$. With some algebra, one gets

The NBI requires $w_0^{\prime }\simeq 1\mathrm{cm}$ and a stable cavity, i.e., $g_e\simeq 0.5$, we obtain $R_{1,\max }<1.5\mathrm{cm}$, which is hardly available for standard high-quality mirrors. Therefore, $g_e$ is to be brought as close as possible to zero or one, although $R_1$ evolves slowly with $g_e$. We arbitrarily choose the value $g_e\simeq 0.04$, since it is small enough to increase $R_1$ to acceptable values (for $R_2=6\mathrm{m}$, we have $R_1\simeq 15\mathrm{cm}$) and offering some latitude with respect to the marginally stable area $0<g_e<0.02$. The case $g_e$ when close to 1 is symmetric within this approximation and appears not to be in the full calculation analysis developed in the next section.

It is worth taking a look at how $g_e$ depends on $(R_1,R_2)$ in the $g_e\simeq 0.04$ region. Since $g_e\ll 1$, we have $l\ll R_1$. So, combining Eqs. (3) and (5) gives to the first order in $l/R_1$:

In Fig. 3, Eq. (8) is displayed for different values of $R_2$, corresponding to the condition $L_1\lesssim L_2/10$. Interestingly, higher values of $g_e$ imply a higher sensitivity to a slight variation of $R_1$; this sensitivity increases when $R_2$ decreases. This provides confirmation on the validity of $g_e\simeq 0.04$. In the next section, we conduct the rigorous calculation based on the $ABCD$ matrix formalism.

 

Fig. 3. Values $R_1=15,7,1.5\mathrm{cm}$ correspond, respectively, to $R_2=6,4,2\mathrm{m}$ to have $g_e\simeq 0.04$. $R_1=1.5\mathrm{cm}$ excludes the $R_2$ as small as 2 m.

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3. CAVITY CONFIGURATION BASED ON THE ABCD MATRIX FORMALISM

The $ABCD$ matrix formalism was initially invented for ray propagation in paraxial optical systems before being generalized to the propagation of Gaussian beams [19,22]. So far, several interpretations have been given to this formalism and its relationship with the Fresnel diffraction theory (see, for example, Refs. [23,24]). It has been extended to lossy optical systems and also applied to atomic matter waves [25].

The $ABCD$ matrix formalism corresponds to a linear optical element to a $2\times 2$ matrix of unity determinant, which describes how it transforms a Gaussian beam. Let an optical element be described by the matrix:

Obviously, a series of optical elements is described by the corresponding products of matrices. In the following, we apply this formalism to the lens-based cavity represented in Fig. 2(b), for which no astigmatism is considered.

A. Lens-Based Cavity

Choosing the mirror ${\mathrm{M}}_1$ as the reference plane, the round trip matrix ${\mathrm{M}}_{\mathrm{rt}}$ is calculated (see the supplemental material 8.A for explicit results). The cavity eigenmode $q_1$ is obtained using Eq. (10) to be

The index “1” stands for the mirror ${\mathrm{M}}_1$ on which cavity Gaussian mode is defined. The last equation admits a single physical solution given by

$m_{\mathrm{rt}}=(A_{\mathrm{rt}}+D_{\mathrm{rt}})/2$ can be interpreted as the cavity stability factor since a real solution exists provided the condition $0<m_{\mathrm{rt}}^2<1$. More specifically, Eq. (12) implies

In addition to the propagation phase, the Gouy phase appears as the argument of the wave complex amplitude ratio after a single round trip. With the notation of Ref. [10], the latter is given by

In order to establish the value of $w_0^{\prime }$, $q_1$ is propagated using the matrix $M^{\prime }$ from the mirror ${\mathrm{M}}_1$ to the mirror ${\mathrm{M}}_3$ (see 8.A Supplemental Material for explicit results). With some algebra, we get

B. Effect of Angle $\mathsf{\theta }$

We now consider the mirror-based cavity represented on Fig. 2(a). $\theta$ is the angle between the incident beam on ${\mathrm{M}}_2$ and its reflection. Hence, the impinging beam on ${\mathrm{M}}_2$ sees two different radius of curvatures in the tangential and the sagittal planes, i.e., $R_{2t}=R_2\times \cos \theta$, and $R_{2s}=R_2/\cos \theta$ [26]. Therefore, the cavity admits for each plane a different round trip cavity matrix ${\mathrm{M}}_{\mathrm{rt},t}$ and ${\mathrm{M}}_{\mathrm{rt},s}$ simply obtained by replacing the corresponding radius of curvature in Eq. (A1). To each matrix corresponds a “half-” cavity mode, which implies in the general case both astigmatism (the two modes have different waist position) and ellipticity (the two modes have different waist values). In the following, we get particularly interested in the effect of angle $\theta$ on the cavity stability. To this aim, we consider the round trip amplitude ratio which becomes

For small angle $\theta$, the radius of curvature seen by the sagittal “half-” mode changes by $\delta R_{2,s}=R_2\times {\theta }^2$, whereas for the tangential “half-” mode $R_2$ changes by $\delta R_{2,t}=-\delta R_{2,s}$. Based on the approximate Eq. (8), the factor $m$ undergoes a variation $\delta m_{t,s}=\pm 4g_e{\theta }^2$. With some algebra, we obtain the first order,

Obviously, $|\delta m_t|=|\delta m_s|=\delta m\simeq {\theta }^2$, and we infer from Eq. (15) that $\delta g_e\simeq {\theta }^4$. Angle $\theta$ is estimated to be $\theta \simeq 0.05\mathrm{rd}$, and $\delta g\simeq {10}^{-6}$. The effect of angle $\theta$ is then negligible.

C. Numerical Investigation

In the following, $L_2=30\mathrm{m}$, and we study the three specific cases ($R_1=15\mathrm{cm}$, $R_2=6\mathrm{m}$), ($R_1=7\mathrm{cm}$, $R_2=4\mathrm{m}$), and ($R_1=1.5\mathrm{cm}$, $R_2=2\mathrm{m}$) previously identified. On Fig. 4, $w_0^{\prime }$, $g_e$, and $w_1$ (beam radius on mirror ${\mathrm{M}}_1$) are displayed as functions of the parameter $l$.

 

Fig. 4. Variation of $w_0^{\prime }$, $g_e$, and $w_1$ with respect to $l$ for the three cases ($R_1=15\mathrm{cm}$, $R_2=6\mathrm{m}$), ($R_1=7\mathrm{cm}$, $R_2=4\mathrm{m}$), and ($R_1=1.5\mathrm{cm}$, $R_2=2\mathrm{m}$). The case ($R_1=15\mathrm{cm}$, $R_2=6\mathrm{m}$) offers more latitude to choose an optically stable configuration responding to the injector requirements.

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The condition $w_0^{\prime }=1\mathrm{cm}$ appears to have two solutions for each case: a $g_e\gtrsim 0$ solution corresponding to the simple calculation conducted in Section 2, and a $g_e\lesssim 1$ solution, which simply corresponds to the half-near-concentric hypothetical cavity, i.e., $l\simeq R_1$. The second solution is particularly interesting, since it offers a $\simeq 5$ times larger beam radius on mirror ${\mathrm{M}}_1$ than the first solution, i.e., less thermal load [27]. However, in case the condition, $w_0^{\prime }=1\mathrm{cm}$ is considered as strict, and the second solution corresponds to the $g_e>0.98$ in the three considered cases. But, if a slightly smaller beam is acceptable, $g_e$ can be brought below the critical value of 0.98. This is only possible for the case $R_2=6\mathrm{m}$, for which the slope $w_0^{\prime }$ with respect to $l$ is low enough to offer the required latitude, allowing a significant variation of $g_e$. Hence, the configuration ($R_1=15\mathrm{cm}$, $R_2=6\mathrm{m}$) appears to fit the best with the system requirements.

The effect of angle $\theta$ has been investigated by slightly varying the value of $R_2$ in the interval $[R_2\times \cos \theta ,R_2/\cos \theta ]$. The $g_e$ variation is kept below $2\times {10}^{-8}$, and, as expected, the effect on the cavity stability is negligible. Moreover, the ellipticity along the cavity is kept close to 1 within less than 1%, which is delicately measurable. Therefore, we can neglect the effect of angle $\theta$ for the considered configurations. However, it is worthwhile to outline that some configurations show important astigmatism, which needs to be compensated [28].

4. EXPERIMENTAL SETUP AND PRELIMINARY CHARACTERIZATION

A. Setup

In this section, we propose to set up a table-top-telescope-based cavity. In order to bring the $\simeq 30\mathrm{m}$ to a meter scale experiment, while keeping the same stability property, we use the following rule: for a length scaling factor $\alpha$, the radii of curvature follow the same scaling factor $\alpha$, while beam radii are multiplied by $\sqrt{\alpha }$ in order to keep the ratio cavity length to the Rayleigh range unchanged. We choose the injector configuration ($L_2=30\mathrm{m}$, $R_1=15\mathrm{cm}$, $R_2=6\mathrm{m}$) as a starting point, and it is given in the second column of Table 1. The third column applies the scaling law to bring the cavity length to $\simeq 1\mathrm{m}$. $w_0\simeq 18\mathrm{\mu m}$, thus obtained is at the limit of the paraxial approximation [10]. Therefore, we would rather use more common values for $R_1$ and $w_0$, while adjusting $L_1$ to keep the same beam waist $w_0^{\prime }$. The obtained cavity is closer to the instability threshold, and the condition $L_1\ll L_2$ is lost (see the fourth column of Table 1).

Table 1. Geometrical Features of Considered Cavities and Corresponding Intra-cavity Beam Characteristics

View Table

Displayed in Fig. 5, the experimental setup to test the cavity configuration is described above. It accommodates a common injection system made of (i) a single frequency Yb-doped fiber laser providing a single mode beam at $\lambda =1064\mathrm{nm}$, (ii) a Faraday isolator allowing the detection of the cavity reflected beam, (iii) a lens-based mode adaptation system, and (iv) alignments mirrors. As for the injection mirror ${\mathrm{M}}_1$, the mirror ${\mathrm{M}}_2$ was chosen to be partially transmissive in order to monitor the intra-cavity beam in both arms. The mirrors’ power reflectivity was then measured to be $r_1\simeq 0.9938$, $r_2\simeq 0.9942$, and $r_3\simeq 1$ (measurement uncertainties are not important in the following). Hence, the cavity finesse and its power reflectivity at resonance differ from the two mirrors’ cavity case and read

 

Fig. 5. Meter scale telescope-based cavity and its beam injection system. The laser frequency is modulated in order to scan the cavity resonances. Two output beams through the mirror ${\mathrm{M}}_2$ are monitored using a CCD camera.

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B. Measurement of the Cavity Contrast

The cavity contrast is measured by monitoring the reflected beam while modulating the laser frequency. The best result was obtained by optimizing both the positions of the matching lenses and the injection mirrors, and the optimum was $C=77\%$. The power dispatched on non-resonant higher-order modes, which are due to residual misalignment and mismatching or simply to the initial non-Gaussianity of the laser beam, explains 8% of the contrast defect. The remaining 7% could be explained by intra-cavity losses of a few 100 ppm due to the decay of the mirrors’ coatings under a non-purified environment.

C. Output Beams

Output beam diameters were measured through mirror ${\mathrm{M}}_2$ by using a WinCamD Series beam profiling camera [29]. A simultaneous caption of these beams passing through a diminishing lens is displayed on Fig. 6(a). Taking into account the mirror width, one can calculate the expected beam diameters using ABCD propagation laws and find ${\mathrm{\varphi }}_1\simeq 4400\mathrm{\mu m}$ for beam (1), and ${\mathrm{\varphi }}_2\simeq 3685\mathrm{\mu m}$ for beam (2). In order to measure beam radii, the cameras were placed separately at 10 cm from the output mirror ${\mathrm{M}}_2$ on each beam. Based on a fit procedure provided by the camera software [see the lower part of Fig. 6(b)], ${\mathrm{\varphi }}_1$ and ${\mathrm{\varphi }}_2$ were determined with an accuracy estimated at $\simeq 5\%$, which is much lower than the accuracy on the theoretical estimation. We obtain ${\mathrm{\varphi }}_1\simeq 4398\pm 220\mathrm{\mu m}$, and ${\mathrm{\varphi }}_2\simeq 3665\pm 183\mathrm{\mu m}$, which is compatible with the theoretical calculations. With so low accuracy, no conclusion can be inferred from these measurements. However, the authors thought that this is worth being reported.

 

Fig. 6. (a) Simultaneous capture of the output beams going through a diminishing lens, which inverts the spots positions. (b) Measurements of the output beams’ radii using the camera software fit procedure. According to the fit, beams are Gaussian at $\simeq 91\%$.

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5. CAVITY STABILITY

According to Eq. (15), the equivalent $g_e$ factor is related to the cavity Gouy phase through

In the following, we show that estimating the Gouy phase, known to be a standard experimental procedure, turns out to be complicated when the accuracy represents an important issue.

A. Principle of the Measurement

The standard procedure to measure the cavity Gouy phase is based on the measurement of the following:

The Gouy phase is then given by

B. Astigmatic Cavity

As described in Section 4.A, the Gouy phase is the angle of the amplitude ratio on a round trip ${\alpha }_2/{\alpha }_1$. In order to take into account the cavity refolding, we generalize Eq. (17) to an arbitrary Hermite–Gauss mode ${\mathrm{HG}}_{i,j}$:

Thus, Eqs. (24) and (25) are generalized to

With some algebra, we show that the Gouy phase is given by

C. Measurement Procedure

Experimentally, the laser frequency is scanned while observing the coupling efficiency through, for example, the cavity reflection. In Fig. 7, the reflection signal of the cavity is displayed, while modulating the laser frequency by applying a voltage on a piezoelectric transducer (PZT) actuator provided for such a purpose. For want of a frequency reference, measuring the frequency differences involved in Eq. (26) consists of measuring the applied voltage on the PZT actuator. There is no need for an absolute measurement of the laser frequency or the cavity length in the perfect case. However, two issues could potentially arise:

 

Fig. 7. Laser frequency is modulated by applying a triangular voltage (orange curve) ten times amplified (purple curve). Resonances are identified on the reflection signal (cyan curve). Resonance peaks are asymmetric due to the cavity photon lifetime, and potential related systematic errors are canceled by differential measurement procedure. It is impossible to relate the ${\mathrm{HG}}_{10}$ to either surrounding ${\mathrm{HG}}_{00}$ peaks. This uncertainty is expressed by the modulo $2\pi$ in Eq. (31).

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In Appendix A.2, we show that if the modulation frequency is correctly chosen, the Gouy phase is given by

D. Determination of the Function ${\mathsf{D}}_{\mathsf{\nu }}(\mathsf{f},\mathsf{V})$

One should note that by changing the time $t_{q,00}$, at which the measurement is done, the voltage $V(t_{q,i0})$ [which is a solution of the implicit Eq. (A6)] changes. For lack of having a complete scan of the function $D_{\nu }(f,V)$, this allows us to have several realizations of

For a given frequency $f$, $n=5$ sets of data were taken at different times $t_{q,00}$ made of $n^{\prime }=3{\mathrm{HG}}_{00}$ peaks, corresponding to the voltages ${(V_{f,i,j})}_{1\le i\le 5;1\le j\le 3}$, surrounding two ${\mathrm{HG}}_{10}$ peaks, corresponding to the voltages ${(V_{f,i,j}^{\prime })}_{1\le i\le 5;1\le j\le 2}$.

Let ${(Y_{f,i^{\prime }})}_{1\le i^{\prime }\le 2\times 5}$ be a column vector defined by

To determine $D_{\nu }(f,V)$, we assume it is a polynomial function of order $N$:

The solution shall be noted as ${\overrightarrow{\alpha }}_0$. The probability distribution ${\chi }^2$ has a mean value equal to one with a standard deviation ${\sigma }_{{\chi }^2}=1/\sqrt{n(n^{\prime }-1)-N}\simeq 0.378$. Hence, the fit is considered as valid when

Whereas the case ${\chi }_f^2>1.38$ is simply rejected, the case ${\chi }_f^2<0.62$ means that the uncertainties on the measurements are overestimated.

In practice, for $f\lesssim 100\mathrm{Hz}$, the cavity is perturbed by air turbulence, and its resonances are difficult to acquire. For $f\ge 100\mathrm{Hz}$, the uncertainty on the voltage measurements is roughly estimated by considering each peak width. For each modulation frequency $f$, the uncertainty on the voltage measurement ${\sigma }_{f,V}$ is uniform for the corresponding set of measurement points. The blue marks on Fig. 8 represent ${\sigma }_{f,V}$ with respect to $f$ and the corresponding ${\chi }_f^2$. For $f>700\mathrm{Hz}$, ${\chi }_f^2$ is higher than the upper limit of the confidence range (represented by the dashed lines) and is simply rejected. This might be explained by an important distortion of the modulation signal for frequencies close to the system mechanical resonance. For $f=200$, 400, 500, 600, and 700 Hz, ${\chi }_f^2$ is beneath the lower limit and needs to be re-evaluated. In fact, since the cavity is considered as a length reference, ${\sigma }_{f,V}$ is the unique uncertainty that operates for a given modulation frequency $f$. We choose to decrease ${\sigma }_{f,V}$ until ${\chi }_f^2$ reaches the lower limit of the confidence range (red marks on Fig. 8). Henceforth, all sets $f\le 700\mathrm{Hz}$ are considered as valid.

 

Fig. 8. In the upper figure, uncertainties on voltage measurement experimentally estimated (blue marks) and reevaluated (red marks) are displayed in order to include corresponding ${\chi }^2$ values into the confidence range (dashed lines on the lower figure). Measurements data for $f\ge 800\mathrm{Hz}$ are simply rejected.

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Thanks to the fit procedure, $\alpha (f)$, $\beta (f)$, and $\gamma (f)$ are determined and are represented in Fig. 9. Whereas uncertainties in $\alpha$ are too important to reveal a distinguishable behavior, $\beta$ and $\gamma$ clearly decrease (in absolute value) with respect to $f$. The function $D_{\nu }(f,V)$ is displayed on Fig. 10 and clearly shows a nonlinear behavior with respect to $V$ decreasing with respect to $f$. Moreover, the modulation efficiencies decreases with $f$ with an increase in the nonlinear behavior. The system acts as a nonlinear low-frequency filter.

 

Fig. 9. Evolution of $\alpha$, $\beta$, and $\gamma$ with respect to $f$. The relative uncertainties are $\simeq 40\%$ for $\alpha$, $\simeq 10\%$ for $\beta$, and $\simeq 1\%$ for $\gamma$. $\beta$ and $\gamma$ show a clear behavior with respect to $f$. The red mark on the lower figure corresponds to ${\gamma }_c$ and was arbitrarily placed at $f=0\mathrm{Hz}$.

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Fig. 10. Frequency response of the laser with respect to the voltage in the [0, 30] V range for different modulation frequencies $f$. Uncertainties on $\alpha$, $\beta$, and $\gamma$ are not considered. The modulation efficiency decreases with increasing $f$.

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With the function $D_{\nu }(f,V)$ now computed, we use $V_{f,i,j}$ and $V_{f,i,j}^{\prime }$ to calculate the set of Gouy phases $({\varphi }_{f,i,j})$. From Eq. (A10), it becomes

In order to estimate the effect of correlations between different ${\varphi }_{f,i,j}$, we apply Eq. (A12) to ${\varphi }_{f,v}$. The variance ${\sigma }_{{\varphi }_{f,c}}^2$ thus obtained is compared to ${\sigma }_{{\varphi }_{f,v}}^2$, and the relative difference $\delta {\sigma }_f^2\lesssim 0.08$ indicates the correction to be made on the weight of each term ${\varphi }_{f,i,j}$. Then, the average ${\varphi }_f$ is likely to scan a supplemental range of $\simeq \pm \sqrt{(2\times 5)\times \delta {\sigma }_f^2\times {\sigma }_{{\varphi }_{f,v}}^2}$. At the end, we obtain the average ${\varphi }_f={\varphi }_{f,v}$ with a variance ${\sigma }_f^2={\sigma }_{{\varphi }_{f,v}}^2(1+10\delta {\sigma }_f^2)$. Results are displayed with blue marks on Fig. 11.

 

Fig. 11. Distribution of ${\varphi }_f$ with respect to the modulation frequency $f$ (blue marks) and its average value (red mark) arbitrarily placed. Considering their uncertainties, data points ${\varphi }_f$ are coincident.

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Finally, since all measurements for different modulation frequency $f$ are independent, the average is simply given by

Numerically, we obtain ${\varphi }_{G0}=2.765\pm 0.014\mathrm{rad}$ within $1\sigma$, which the red mark corresponds to in Fig. 11. As can be seen on this figure, ${\varphi }_f$ are sharply distributed. The spread of the measurements dots is quantified by

The calculation of the systematic error $\mathrm{\Delta }\varphi$ is straightforward. Its uncertainty is estimated using Eq. (A12) with respect to $(L_1,L_2,R_1,R_2,\theta )$, where $\theta =0.05\pm 0.01\mathrm{rad}$. It turns out that the uncertainty on $\theta$ is dominating, and we obtain $\mathrm{\Delta }{\varphi }_G=-0.015\pm 6\times {10}^{-3}\mathrm{rad}$. At the end, the Gouy phase of the refolded cavity and the corresponding equivalent $g$ factor are

The accuracy is then increased by a factor of 20 with respect to the theoretical estimation in Section 5, and we conclude that the cavity is certainly stable.

6. THERMAL EFFECTS ISSUE

Thermal effects in the photo-neutralization cavity for future NBI was partially addressed in Ref. [11]. The surface distortion of a mirror undergoing an impinging 3 MW beam was considered, and the thermal compensation system was proposed. Except for the input mirror, this compensation can be used for both configurations displayed in Fig. 1. However, these configurations have an important drawback if used on a NBI: each folding mirror sees $2\times 3\mathrm{MW}$ incident beams, which increases the impinging power up to 6 MW instead of 3 MW. The ring cavity (top of) configuration is the alternative to be adopted. It can also be extended to the telescope-based cavity (bottom of Fig. 12). New astigmatism issues related to the injection mirror are expected, but should not affect the cavity stability.

 

Fig. 12. Refolded cavity in the ring configuration for both simple and telescope-accommodating cases.

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7. CONCLUSION

Fusion reactors are a concentration of technology covering different areas. One of the main issues is to make all these systems coexist in a reduced volume. As metrological systems, optical cavities for photo-neutralization of negative ions in NBI are to be adapted to the harsh environment of fusion reactors. In this paper, we demonstrated that a telescope-based cavity in a neutral beam injector is likely to reduce space issues in the environment of a future fusion reactor. A key point in this configuration is the stability of the cavity despite the fact that its length is much shorter than the Rayleigh range within the useful part. This stability is characterized by the round trip Gouy phase, which must be kept far from $0\equiv [2\pi ]$. A meter scale telescope-based cavity was settled. Its Gouy phase was measured to be 2.750 rad with a standard deviation of 16 mrad, thus confirming its stability.