## Abstract

We present an architecture for a multipass amplifier based on a succession of optical Fourier transforms and short propagations that shows a superior stability for variations of the thermal lens compared to state-of-the-art $4f$-based amplifiers. We found that the proposed multipass amplifier is robust to variations of the active medium dioptric power. The superiority of the proposed architecture is demonstrated by analyzing the variations of the size and divergence of the output beam in the form of a Taylor expansion around the design value for variations of the thermal lens in the active medium. The dependence of the output beam divergence and size is investigated also for variations of the number of passes, for aperture effects in the active medium, and as a function of the size of the beam on the active medium. This architecture makes efficient use of the transverse beam filtering inherent in the active medium to deliver a beam with excellent quality (TEM00).

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Multipass amplifiers are used for the generation of the currently highest laser pulse energies and average powers [1–6]. Oscillators, regenerative amplifiers, and multipass amplifiers are limited in pulse energy by the damage of optical components induced by the high intensity. Multipass amplifiers possess a higher damage threshold compared to laser oscillators and regenerative amplifiers due to the absence of intracavity enhancement and nonlinear crystals [7,8]. Optical damage can be avoided by increasing the laser beam size, but this typically increases the sensitivity of the beam propagation and output beam characteristics (size and divergence) to the active medium thermal lens. The active medium design and material choice can be optimized to decrease the thermal lens, e.g., in thin-disk lasers [9–12]. Still, the higher pump powers and the increased beam size needed for energy and power scaling make the thermal lens in the active medium the most severe limitation to overcome [6,13,14]. For these reasons an optical layout has to be chosen that minimizes the changes of the beam propagation for variations of the thermal lens of the active medium.

Commonly, relay imaging ($4f$-imaging) from active medium to active medium is used to realize a multipass amplifier where the output power is independent of the thermal lens of the active medium because the beam is imaged from pass to pass [15]. The $4f$-design exhibits various advantages: it sustains equal beam size at each pass on the active medium independent of the dioptric power of the active medium and independent of the beam size at the active medium. Moreover, numerous passes can be realized with only a few optical elements [16–26]. Yet, this design does not represent the optimal solution for a multipass amplifier because the phase front curvature of the beam after propagation through this optical system depends strongly on variations of the active medium dioptric power.

In this paper, we present a novel multipass architecture whose propagation from pass to pass is given by a succession of Fourier transforms and short propagations. The working principle, the practical implementation, and the advantages in terms of stability to variations of dioptric power (thermal lens) and aperture effects of the active medium are detailed.

## 2. THERMAL LENS AND APERTURE EFFECTS

In this section, we briefly review well-known facts and various definitions needed in the following sections.

When a laser beam crosses the pumped active medium, it is amplified and it experiences a position-dependent optical phase delay (OPD) that distorts its phase front curvature. When simulating the propagation of the laser beam, this distortion can be mostly accounted for by approximating the active medium with a spherical lens [27,28] that can be easily described within the ABCD-matrix formalism [29].

The position-dependent gain (and absorption in the unpumped region) can be approximated by an average gain per pass and a position-dependent transmission function that shows a gradual transition from maximal transmission (including gain) on the optical axis to zero transmission far from the axis. This position-dependent gain is commonly referred to as “soft aperture.” In this study we consider only Gaussian beams in the fundamental mode and we approximate the super-Gaussian aperture of the pumped active medium as the Gaussian aperture. This approximation is justified because the Fourier transform (that underlies the here-presented designs) converts the high-frequency spatial distortions of the beam caused by the super-Gaussian aperture into a halo of the fundamental mode that is eliminated in the next pass at the active medium. Therefore, the beam propagating in the amplifier shows only minor higher-order distortions from the fundamental mode so that the description of the super-Gaussian aperture can be effectively accomplished using a Gaussian aperture [30]. In fact, a Gaussian aperture transforms a Gaussian input beam into a Gaussian output beam [31]. The relation between the $1/{e}^{2}$-radii of the input (${w}_{\mathrm{in}}$) and output (${w}_{\mathrm{out}}$) beams takes the simple form

where ${W}^{2}$ is the $1/{e}^{2}$-radius of the intensity transmission function of the Gaussian aperture [31,32]. This simple behavior of a Gaussian aperture acting on a Gaussian beam (TEM00-mode) can be captured into an ABCD-matrix for Gaussian optics as [30,31,33–36]An active medium (AM) usually exhibits both a lens effect (position-dependent OPD) and an aperture effect (position-dependent gain). These can be merged into a single ABCD-matrix describing the active medium ${M}_{\mathrm{AM}}={M}_{\mathrm{lens}}\xb7{M}_{\mathrm{aperture}}$, where ${M}_{\mathrm{lens}}$ is the ABCD-matrix of a thin spherical lens. The product of these two ABCD-matrices becomes

The similarity of ${M}_{\mathrm{AM}}$ and ${M}_{\mathrm{lens}}$ allows us to define a complex dioptric power $\tilde{V}$ ($\tilde{V}\in \mathbb{C}$) of the active medium as

Its real part $V$ is the “standard” dioptric power (focusing–defocusing effects), while the imaginary part accounts for the Gaussian aperture. The real part can be articulated as a sum of various contributionsIn Sections 3–7, for simplicity, but without loss of generality, we assume that ${V}_{\mathrm{unpumped}}+{V}_{\mathrm{thermal}}^{\mathrm{avg}}=0$. In Section 8, when realistic implementations of multipass amplifiers are presented, we relax this assumption. Moreover, in the various plots presented in this paper we have fixed the value of the aperture size to be $W=4{w}_{\mathrm{in}}$. For comparison, in some figures we also present results for $W=\infty $ to highlight the aperture effects.

A laser mode size of ${w}_{\mathrm{in}}\approx 0.7{R}_{p}$, where ${R}_{p}$ is the radius of the super-Gaussian pump spot, is typically used in the thin-disk laser community for efficient laser operation in the fundamental mode. The above choice of a Gaussian aperture with $W=4{w}_{\mathrm{in}}$ rests upon the fact that it produces a similar reduction of the fundamental mode size and a similar reduction of the fundamental mode transmission compared to a super-Gaussian aperture fulfilling the relation ${w}_{\mathrm{in}}\approx 0.7{R}_{p}$ [30].

## 3. STATE-OF-THE-ART MULTIPASS AMPLIFIERS

The commonly used state-of-the-art multipass amplifiers are based on $4f$-relay imaging ($4f$) from active medium to active medium [15–25] so that in these amplifiers the beam proceeds as a succession of propagations

The geometrical layout and the corresponding propagation of a laser beam through a $4f$-segment is given in Fig. 1(a). By concatenating, for example, six of such propagations, the 6-pass amplifier of Fig. 1(b) can be realized.It can be demonstrated that the $4f$-imaging warrants the relation ${w}_{\mathrm{in},N}={w}_{\mathrm{out},N-1}$ [15]. Here, we defined ${w}_{\mathrm{in},N}$ as the size of the beam incident on the $N$th pass at the active medium and ${w}_{\mathrm{out},N}$ as the size of the beam departing from the $N$th pass at the active medium after amplification. In the absence of aperture effects ($W=\infty $) in the active medium ${w}_{\mathrm{out},N}={w}_{\mathrm{in},N}$. Hence, in the absence of aperture effects the beam size is identically reproduced at each pass on the active medium independent of the size of the laser beam. This property, together with the simplicity of the implementation (many passes using few optical elements), are the main advantages of the $4f$-based multipass architecture. However, for nonvanishing aperture effects ($W\ne \infty $) the beam size is decreased when passing the active medium ${w}_{\mathrm{out},N}<{w}_{\mathrm{in},N}$ [see Eq. (1)]. Because the $4f$-imaging reproduces the beam ${w}_{\mathrm{in},N+1}={w}_{\mathrm{out},N}$ the beam size is continuously decreased at each pass ${w}_{\mathrm{out},N}<{w}_{\mathrm{out},N+1}$ as is visible in Fig. 1(b).

By considering the various curves presented in Fig. 1(b) another, even more crucial, drawback of the $4f$-based architecture emerges, namely, that the divergence of the output beam strongly depends on variations of the dioptric power $\mathrm{\Delta}V$ of the active medium.

The output beam characteristics and their dependence on variations of $\mathrm{\Delta}V$ can be derived by making use of the ABCD-matrix and complex beam parameter formalism [33]. The complex beam parameter $q$ is defined as

where $R$ is the radius of curvature of the phase front and $w$ is the $1/{e}^{2}$-radius of the TEM00 Gaussian beam. In this framework the complex beam parameter at the output of an optical system (${q}_{\mathrm{out}}$) can be computed knowing the ABCD-matrix of the optical system and the complex beam parameter at its input (${q}_{\mathrm{in}}$) using the relation [33] The ABCD-matrix describing the Gaussian beam propagation from the first pass to the last pass of an $N$-pass amplifier having ($N-1$) $4f$-imaging propagations readsFrom Eq. (9) it is possible to extract the phase front curvature and size of the output beam. When aperture effects are neglected ($W=\infty $) they are given by

While the output beam size does not depend on variations of the dioptric power $\mathrm{\Delta}V$, the phase front curvature of the output beam shows the above-mentioned accumulation of the dioptric power deviation $\mathrm{\Delta}V$, leading to an increasing deviation from the design value of ${R}_{\mathrm{out},N}^{-1}=0$ with an increasing number of passes. This implies a strong dependence of the output beam divergence to variations of the operational conditions.When aperture effects are included, the beam size right after the $N$th pass on the active medium shrinks with increasing $N$ as

Imaging properties of the $4f$-scheme, also distortions of the OPD beyond what is accounted for by the spherical lens, as well as distortions caused by the deviation from a purely Gaussian aperture, are accumulated at each pass. Thus, while propagating in the $4f$-based multipass amplifier the beam shows increasing mode deformation (beyond the TEM00-mode) resulting in an output beam with decreased beam quality and increased dependency on running conditions [1,37].

For $4f$-based amplifiers there are two methods to reduce the accumulation of phase front distortions. The first is by changing the distance between the two lenses used for the imaging [see Fig. 1(a)] [18,38] so that their separation departs from $2f$. To put it in another way, the distance between the focusing elements of the $4f$-propagation can be adjusted to preserve the $q$-parameter also for a curved active medium [39]. The second approach is by realizing the $N$-pass $4f$-amplifier as a succession of $2N$ $4f$-segments so that between each pass on the active medium there is an intermediate image of the active medium. At the position of this intermediate image, a deformable mirror can be placed adjusted to produce an opposite phase front distortion compared to the active medium [1]. These methods are particularly suited to compensate for nonzero values of the sum ${V}_{\mathrm{unpumped}}+{V}_{\mathrm{avg}}$. For time-dependent variations of $\mathrm{\Delta}V$, a time-dependent correction of the distance or of the dioptric power of the deformable mirror would be necessary, making the system significantly more complex. Especially difficult to be compensated are fast variations that might occur during the pulsed amplification process [38]. Contrarily, the compensation achieved in the Fourier-based amplifier architecture presented in this work is passive and thus well suited also for fast variations of $\mathrm{\Delta}V$.

## 4. FROM $\mathsf{4}\mathsf{F}$-IMAGING TO A DOUBLE FOURIER TRANSFORM WITH FILTERING

In some realizations of multipass amplifiers based on $4f$-imaging [15,23,40–43], the problematic accumulations of beam distortions at each pass have been suppressed by placing tight apertures (pinholes) in the focus (center) of the $4f$-imaging segments as shown in Fig. 2(a). Mode filtering is thus achieved at the cost of output efficiency. In these designs the $4f$-propagation is split into a $2f$-propagation, an aperture, and a second $2f$-propagation. The $2f$-propagations act as optical Fourier transforms, while the pinhole filters the beam components beyond the fundamental Gaussian mode.

These schemes indicate a way of how to modify the $4f$-imaging propagation to reduce the accumulation of beam deformations, namely, by using optical Fourier-transform propagations to transport the beam from pass to pass and by using the aperture available in the active medium itself as a mode cleaner. In this new scheme, the pinhole of Fig. 2(a) is replaced by the active medium (more precisely two passes in the active media) as shown in Fig. 2(b) that exhibits an intrinsic soft aperture. Hence, the active medium acts not only as gain medium but also as mode cleaner, maximizing the gain for the fundamental mode. For comparison, a $4f$-based 4-pass amplifier has similar aperture losses for the fundamental mode as the multipass amplifier of Fig. 2(b), but it does not provide mode cleaning. Multipass amplifiers based on this succession of Fourier propagations and passes at the gain medium featuring a soft aperture will be evaluated in the following sections.

## 5. 2-PASS AMPLIFIER WITH FOURIER TRANSFORM PROPAGATIONS

In this section, we investigate the properties of a Fourier-based 2-pass amplifier as sketched in Fig. 3(b). Here, the beam crosses first the active medium, propagates through an optical segment acting as an optical Fourier transform, and then passes the active medium a second time. The properties of this system are compared to the 2-pass amplifier shown in Fig. 3(a), which is based on a $4f$-imaging propagation.

To quantify the sensitivity to variations of $\mathrm{\Delta}\tilde{V}$ of the beam leaving the 2-pass amplifier we first evaluate the corresponding ABCD-matrix. The ABCD-matrix of the 2-pass amplifier ${M}_{\mathrm{Fourier},2}$ based on the Fourier transform results from the product

By combining Eq. (14) with Eq. (7) we can deduce the complex beam parameter that is replicated from the input plane (just before the first pass) to the output plane (just after the second pass) ${q}_{\mathrm{out}}={q}_{\mathrm{in}}$. The solution reads

Therefore, unlike the $4f$-imaging that can reproduce any beam size for any value of $f$, the Fourier transform can reproduce the size of the input TEM00 beam at its output only provided that the complex parameter of the input beam fulfills the condition ${q}_{\mathrm{in}}={q}_{\mathrm{stable}}$. Note that in this case the output beam size equals the input beam size only when the number of passes in the active medium is even.When designing multipass amplifiers the pragmatic choice for the input beam is to assume that the active medium has no dioptric power and no aperture effects ($V=\mathrm{\Delta}V=0$ and $W=\infty $), i.e., $\mathrm{\Delta}\tilde{V}=0$. In this case, the stability condition given in Eq. (15) simplifies to ${q}_{\mathrm{stable}}=iF$ so that our practical choice for the input beam parameter is

Using Eq. (6), this condition can be re-expressed as It is evident from this equation that the focal length $F$ needed to realize the Fourier transform reproducing the size of the beam from active medium to active medium depends on the desired beam size at the active medium.The benefits of the Fourier transform can be evinced by following the beam propagation shown in Fig. 3(b). The beam size of the input beam ${w}_{\mathrm{in},1}$ is reduced by the soft aperture with $W\ne \infty $ of the active medium so that ${w}_{\mathrm{out},1}<{w}_{\mathrm{in},1}$. The smaller beam is, however, converted by the Fourier transform into a larger beam ${w}_{\mathrm{in},2}>{w}_{\mathrm{in},1}$ [see Fig. 3(b)]. At the second pass through the active medium, this larger beam is reduced by the aperture ${w}_{\mathrm{out},2}<{w}_{\mathrm{in},2}$ so that the resulting beam has roughly the size of the input beam ${w}_{\mathrm{out},2}\approx {w}_{\mathrm{in},1}$. There is thus a counteracting interplay between the reduction of the beam size caused by the aperture, and the increase of the beam size related with the Fourier transform that stabilizes the beam size close to the value expressed by Eq. (17) so that similar beam sizes are reproduced at each pass. Because of this compensation, variations of the width $W$ of the soft aperture do not significantly affect the output beam size so that a precise knowledge of the aperture effect is not needed when designing Fourier-based multipass systems.

For an analogous compensation mechanism, the Fourier-based 2-pass amplifiers exhibit output beam sizes and divergences that are insensitive to variations of $\mathrm{\Delta}V$ [see Fig. 3(b)]. Indeed, an active medium with focusing dioptric power turns the entering collimated beam into a converging beam, but the following Fourier transform converts this converging beam into a diverging beam. At the second pass, the focusing dioptric power of the active medium essentially cancels the divergence of the impinging beam. In such a way, the beam leaving the 2-pass amplifier has similar size and divergence as the collimated input beam. An analogous compensation occurs for an active medium with defocussing dioptric power.

Summarizing, there is a counteraction between the Fourier transform and the active medium that results in output beam characteristics only weakly dependent on variations of $\mathrm{\Delta}\tilde{V}$. The reduced sensitivity to variations of $\mathrm{\Delta}V$ of the Fourier-based compared to the $4f$-based amplifier can be clearly appreciated by comparing the divergences of the output beams in Figs. 3(a) and 3(b). The $4f$-imaging leads to an accumulation of the lens effects from pass to pass, while the Fourier transform leads to a substantial compensation of these lens effects. This compensation applies also for other deformations of the phase front occurring at the active medium beyond the spherical distortion already accounted in $V$.

The complex parameter of the outgoing beam after propagation along the 2-pass amplifier can be calculated from Eq. (7) and the ABCD-matrix elements of Eq. (14). By assuming an input beam parameter as given in Eq. (16), we find that the complex parameter of the output beam for small variations of $\mathrm{\Delta}\tilde{V}$ takes the form

Remarkably, in this case ${q}_{\mathrm{out},2}$ depends only quadratically on variations of the complex-valued dioptric power $\mathrm{\Delta}\tilde{V}$. This is in sharp contrast to the linear dependence observed in Eq. (9) for the amplifier based on $4f$-imaging. In other words, the output beam for the 2-pass Fourier-based amplifier in first order does not depend on $\mathrm{\Delta}\tilde{V}$ so that the output beam characteristics such as phase front curvature (given by radius ${R}_{\mathrm{out},2}$) and beam size ${w}_{\mathrm{out},2}$ are very insensitive to small variations of the dioptric power and aperture effects of the active medium.

The beam size ${w}_{\mathrm{out},2}$ and the phase front curvature ${R}_{\mathrm{out},2}$ obtained from ${q}_{\mathrm{out},2}$ are shown in blue in Figs. 4(a) and 4(b), respectively, for variations of the dioptric power $\mathrm{\Delta}V$ and aperture size of $W=4{w}_{\mathrm{in}}$. Similar plots are given in Figs. 4(c) and 4(d) for $W=\infty $ to highlight the role of the aperture. When comparing these blue curves with the red dashed curves, which represent the output beam characteristics for the $4f$-based 2-pass amplifier, it becomes evident that for a $4f$-based amplifier the output beam size strongly depends on $W$ and that the output divergence strongly depends on $\mathrm{\Delta}V$. Oppositely, the Fourier-based multipass amplifier has output characteristics insensitive to small variations of $\mathrm{\Delta}V$ and $W$.

Assuming no aperture effects ($W=\infty $), the Taylor expansions of the output beam characteristics take the form

It is also interesting to investigate how the Gaussian aperture of the active medium affects the output beam size. Assuming that $V=\mathrm{\Delta}V=0$, we find that the output beam size takes the simple form

The dependence of the phase front radius ${R}_{\mathrm{out},2}^{-1}$ on aperture effects has also been investigated. Similar to the $4f$-based amplifier, also in this case the phase front radius (divergence) of the beam leaving the amplifier, for $V=0$, does not depend on the aperture size $W$.

## 6. 4-PASS AMPLIFIERS WITH FOURIER TRANSFORM PROPAGATIONS

In this section, we consider the 4-pass amplifier based on optical Fourier transforms obtained by concatenating two of the optical segments of Fig. 3(b). The resulting propagation follow the sequence

as sketched in Fig. 5(a). We assumed here that the propagation between the second and the third pass on the active medium is so short that it can be neglected. This can either be realized with a propagation that is physically short compared to the Rayleigh length of the beam [see Fig. 5(a)] or as sketched in Fig. 5(b) by inserting $4f$-imaging between the second and the third pass that per definition has a vanishing effective optical length ($B=0$ for the ABCD-matrix of a $4f$-imaging propagation). In this case, the propagation in the 4-pass amplifier follows the sequence The complex parameter of the output beam can be obtained following a similar procedure as in the previous section and can be approximated by a Taylor expansion in $\mathrm{\Delta}\tilde{V}$ asThe size and divergence of the output beam for $\mathrm{\Delta}V$ variations can be obtained by combining Eq. (22) with Eq. (6). Assuming for simplicity that $W=\infty $ we find

A plot of ${w}_{\mathrm{out},4}$ and ${R}_{\mathrm{out},4}^{-1}$ for variations of $\mathrm{\Delta}V$ is shown in blue in Fig. 6 and compared with the results given in dashed red for a $4f$-based amplifier with 4 passes and same beam size. For the Fourier-based amplifier the “stability region,” where the impact of $\mathrm{\Delta}V$ variations to the output beam characteristics is small, increases with the number of passes while it decreases for $4f$-based amplifiers (compare Fig. 6 with Fig. 4).

## 7. MULTIPASS AMPLIFIER WITH A LARGE NUMBER OF PASSES

Following the reasoning of the above sections, an amplifier with a large number of passes ($N=8,12,16,\dots $) inheriting the same insensitivity to variations of the active medium lens and aperture effects can be realized by concatenating several of the 4-pass segments presented in Fig. 5.

An example of an 8-pass amplifier modeled according to this scheme is shown in Fig. 7. Size and phase front curvature of the output beam for an 8-pass and a 16-pass amplifier as a function of $\mathrm{\Delta}V$ are shown in Figs. 8 and 9, respectively. The accumulation of lens and aperture effects in the $4f$-based amplifiers becomes striking when considering the red dashed curves in these figures. In such multipass designs, the size of the output beam ${w}_{\mathrm{out}}$ becomes significantly smaller than the input beam size ${w}_{\mathrm{in}}$, while the phase front curvature $1/{R}_{\mathrm{out}}$ of the output beam becomes increasingly sensitive to variations of $\mathrm{\Delta}V$ (large slope of the red curves).

Contrarily, in the Fourier-based amplifiers the output beam characteristics show an oscillatory behavior around the design value for variations of $\mathrm{\Delta}V$. For vanishing aperture effects this oscillation is bound (see black dashed lines in Figs. 8 and 9) by the following relations:

The Taylor expansion of the complex beam parameter after a propagation through a Fourier-based 8-pass amplifier is

Alike previous sections we have investigated the dependence of the output beam size on the aperture size $W$. Assuming for simplicity that we are in the center of the “stability region” ($V=\mathrm{\Delta}V=0$) we find that

## 8. PRACTICAL REALIZATION

The Fourier-based multipass amplifiers sketched in the previous sections show excellent output beam stability for variations of $\tilde{V}$ but their practical implementation as such leads to long propagation lengths. In fact, the large mode size at the active medium needed for high-power and high-energy operation requires the use of large focal lengths $F$:

This result was obtained by solving Eq. (17) for $F$. For example, a beam size at the active medium of 1 mm requires $F=3\text{\hspace{0.17em}}\mathrm{m}$ for $\lambda =1030\text{\hspace{0.17em}}\mathrm{nm}$ so that each Fourier transform is 6 m long.At the same time, it is challenging to realize an active medium with vanishing dioptric power $V={V}_{\mathrm{unpumped}}+{V}_{\mathrm{thermal}}^{\mathrm{avg}}=0$ at the desired operational conditions as assumed in previous sections.

The realization of the Fourier transform propagation using two pairs of Galilean telescopes solves both these issues: it shortens the physical propagation length and allows for the use of active media with nonvanishing dioptric power. However, the focal lengths and the positions of the mirrors forming the telescope pairs must be chosen so that the propagations from active medium to active medium correspond approximatively to a Fourier transform, i.e., their ABCD-matrices must have the form $A=D=0$ and $B=1/C=-F$.

In the thin-disk lasers sector it is customary to use thin disks (active media) with a focusing focal length (${f}_{AM}$). Therefore, as shown in Fig. 10(a), the active medium itself can be used as curved mirror to realize the Galilean telescopes. To fully define the layout of the Fourier transform, we need to further assume the position of the defocusing optical elements (lens or mirrors) forming the telescope pairs. In the layout shown in Fig. 10(a), it was assumed that these defocusing elements were placed at a distance $0.25L$ after and before the active media, where $L$ is the distance from active medium to active medium. Under these assumptions, we find that the telescopic pair act as a Fourier transform when the focal length ${f}^{\prime}$ of these defocusing elements fulfill the relation

In the practical implementation, however, there is a limit in the minimal value of the focal length ${f}_{\mathrm{AM}}$ given by the optical damage at the defocusing elements and the laser-induced breakdown in air. Indeed the beam sizes at the defocusing elements and in the focus of the telescopic system decrease with decreasing ${f}_{\mathrm{AM}}$.

We conclude by sketching the optical layout and the corresponding beam size evolution for two 8-pass Fourier-based amplifiers realized with the above-described Galilean telescopes. In these designs the active medium that has a focusing dioptric power $V$ is also utilized as the focusing element of the Galilean telescope. In Fig. 11(a), the propagation from active medium to active medium follows the sequence

The shortening of the propagation due to the insertion of telescopes can be better appreciated with a numerical example. A large focal length of $F=90\text{\hspace{0.17em}}\mathrm{m}$ [see Eq. (35)] is needed to realize a multipass amplifier based on the scheme of Fig. 7 (without telescopes), having a beam size ${w}_{\mathrm{in}}=5.4\text{\hspace{0.17em}}\mathrm{mm}$ and a wavelength of $\lambda =1030\text{\hspace{0.17em}}\mathrm{nm}$. Assuming the short propagation length is ${L}^{\prime}=1\text{\hspace{0.17em}}\mathrm{m}$ the total length of the 8-pass amplifier would be ${L}_{\mathrm{total}}=720\text{\hspace{0.17em}}\mathrm{m}$. Using the layout of Fig. 11(a), the same beam size at the active medium can be obtained for an active medium with focal length ${f}_{\mathrm{AM}}=1\text{\hspace{0.17em}}\mathrm{m}$ and a defocusing element with focal length ${f}^{\prime}=-150\text{\hspace{0.17em}}\mathrm{mm}$ placed at a distance of 861 mm. In this way the distance AM-Fourier-AM is reduced to 3.445 m (instead of 180 m). Assuming that ${L}^{\prime}=1\text{\hspace{0.17em}}\mathrm{m}$ the total length of the 8-pass amplifier is reduced to ${L}_{\mathrm{total}}=16.8\text{\hspace{0.17em}}\mathrm{m}$ (instead of 720 m). Because this telescope reduces the beam size in the midplane of the Fourier transform to $w=0.57\text{\hspace{0.17em}}\mathrm{mm}$, optical damage issues have to be considered. Yet, this beam size is significantly larger than the size of $w=0.06\text{\hspace{0.17em}}\mathrm{mm}$ in the focal plane of a $4f$-imaging with $f=1000\text{\hspace{0.17em}}\mathrm{mm}$.

## 9. CONCLUSION

Numerous studies have investigated the influence of the active medium thermal lens on resonator (oscillator) eigenmodes [33,45]. Oscillator designs have been developed that mitigate this influence, reducing the dependence of the oscillator performance and output beam characteristics to variations of the thermal lens of the active medium. However, similar studies have not (or insufficiently) been undertaken for multipass amplifiers. This might be associated with the fact that the $4f$-relay imaging, that is commonly used when realizing multipass amplifiers, offers two obvious advantages. The first is that its imaging properties reproduce the same transverse intensity profile (size) from pass to pass independent of the size and shape of the beam and independent of the active medium dioptric power. The second is that several passes can be implemented with only few optical elements. Less obvious and known are the disadvantages. The first is that the beam size is reproduced only for a vanishing aperture effect at the active medium. Second, there is a pile up of phase front distortions at each pass on the active medium that strongly affects the output beam divergence and the beam quality. Imaging-based amplifiers are thus well suited for low-power lasers where the thermal lens is small but become less apt for high-power lasers with good output beam quality.

Here, we have presented a multipass amplifier architecture based on a succession of optical Fourier propagations. Contrary to the $4f$-imaging the layout of this Fourier transform propagation needs to be designed to match the desired beam size at the active medium. Moreover, the implementation of this architecture requires one additional mirror for every pass, making it more complex compared to the $4f$-based layouts. However, the implementation of these minor additional complexities is rewarded with the compensation of the thermal lens effects occurring in this architecture, which significantly decreases the sensitivity of size and divergence of the output beam to variations of the dioptric power and aperture effects of the active medium. Therefore, this architecture overcomes present energy- and power-scaling limitations related with the thermal lens effect in the active medium. Furthermore, the mode filtering that naturally occurs from the interplay between the Fourier transform propagations and the soft aperture of the active medium provides excellent beam quality with high efficiency.

We found that the output beam size and phase front radius of a Fourier-based multipass system are less sensitive to variations of the active medium thermal lens than a single-pass amplifier. Hence, from the point of view of the stability to thermal lens effects, it is advantageous to realize amplifiers as multipass amplifiers where the propagating beam is redirected several times to the same active medium. Various active media can be utilized provided they have identical or similar dioptric power. The multipass architecture presented here is thus particularly suited for thin-disk lasers where the gain per pass is small and multiple passes are required. An example of such a multipass amplifier realized according to this architecture is presented in Refs. [46,47], where the disk is acting as active medium, soft aperture, and focusing mirror of the Galilean telescopes used to realize the Fourier transform.

We want to point out that the here-presented architecture can be used in other contexts as well. For example, a recent development in Faraday insulators reduced the thermal lens effect resulting in TEM00 up to 400 W average power [48,49]. However, commercially available laser systems already provide output powers exceeding 10 kW [50]. In the laser system at advanced LIGO the focusing thermal lens effect of terbium gallium granate used in the Faraday rotator was compensated using a plate of deuterated potassium dihydrogen phosphate exhibiting a defocusing thermal lens [51]. Following our design [as presented in Fig. 3(b)] the effective thermal lens of the Faraday rotator can be almost canceled (see Fig. 4) without additional transmissive elements simply by splitting the rotator in two halves and by implementing a Fourier transform between them.

## Funding

Swiss National Science Foundation (SNF 200021_165854); H2020 European Research Council (ERC) (ERC CoG. #725039, ERC StG. # 279765); ETH Research Grant (ETH-35 14-1); ETH Femtosecond and Attosecond Science and Technology (ETH-FAST) (NCCR MUST).

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