1. Introduction

The current trend in modern high-intensity high-energy lasers features more and more complex systems made of several laser amplifiers, followed by a pulse compressor and beam transport over several tens of meters to the target areas, with beams in the range of 0.1 to 0.5 meter in diameter [1]. In such systems, it is advantageous to use image relaying to increase the beam quality similarly to what has been pioneered with nanosecond lasers 30 years ago [2]. However, broadband amplification creates an additional complication for refractive systems and compensation strategies are necessary [3,4]. Moreover, nonlinear effects (B-integral) prevent using refractive telescopes after compression, although the non-perfect compression gratings [5] induce mid-scale phase-to-amplitude spatial modulations [6] in the intermediate field. As a consequence, many systems operate at a reduced energy to avoid damage to the transport mirrors.

Parabolic mirrors are widely used as an alternative to refractive elements over a variety of fields and applications, such as all-reflective astronomical telescopes [7], the focusing of parallel beams or vice versa. Whenever objects in the focal plane shall not obstruct the incident light, an off-axis parabolic mirror (OAP) is used, where the focus is deflected from the optical axis of the incoming light. A pair of these mirrors can also be used as afocal telescope, which features several, often desirable properties, such as mechanical stability, complete achromatism, possibly aberration-free image relaying and a vanishing B-integral.

Still, even though OAPs are used for final beam focusing in high-intensity lasers systems almost exclusively [8,9], there are barely systems that use OAP telescopes for beam transport. Aside from budgetary reasons, one of the main causes for this is the alignment of such a system, which is commonly viewed as overly complicated. A recent work on a similar topic proposes an intricate alignment procedure that aims for interferometric precision [10], which exceeds the requirements for simple beam transport in high-intensity lasers and is only applicable for centimeter-scale beams.

In this work, we revisit OAP telescopes in the context of high-intensity lasers in order to reduce the stigma of complicated alignment. We find that the requirements on beam quality set by these systems can be met with far simpler alignment strategies and we propose a setup that is designed not only to meet these requirements, but to enable fast and repeatable alignment of these telescopes on a regular basis. This makes OAP telescopes more user friendly than all-reflective counterparts composed of multiple spherical mirrors [11] and therefore a viable option for corresponding budgets.

Such systems dramatically reduce the systematic source of spatio-temporal coupling in high-intensity lasers [12] and avoid additional chromatism compensation modules [13]. Also, the imaging properties of such a telescope can be used after pulse compression to efficiently exploit Adaptive Optics (AO) for focal spot optimization and therefore increasing the accessible intensity [14].

To achieve this, we introduce a simple, geometrical model of OAP telescopes in section 2 which can be modified to derive beam propagation and imaging. Using further geometric considerations, we analytically derive the influences of misalignment in section 3. Based on these findings, we propose a setup which enables an alignment procedure that meets the precision demanded by reasonable requirements on beam quality in section 4. We also give a detailed description of how to assemble such a system and how the alignment procedure is performed. In section 5., we derive a set of easy-to-use equations that allow estimating the beam quality that can be expected for any setup using the proposed alignment scheme. Lastly, in section 6., we compare these findings with the alignment precision that can be reached for a single OAP.

2. General mathematical description

In order to find a model to describe aberrations caused by misalignment of an OAP telescope, we need an analytical description of the telescope itself and the rays that pass it. Therefore, we introduce the parametrization of a misaligned OAP surface in section 2.1 and the formal description of the beam propagation in section 2.2.

2.1 Parametrization of a misaligned OAP

An OAP can be characterized in different ways, but the most common one chooses the propagation properties relative to a chief ray. Here, the transversal distance to the OAPs optical axis, called off-axis distance, combined with a reflection angle, called off-axis angle, and the effective focal length, are used. Using the superscript $^{(i)}$ to denote the number of the parabola in a telescope, in this work these properties are written as follows: These properties are visualized in Fig. 1. Note that $\alpha _{oa}$ is written without superscript as this angle has to be the same for all OAPs in a correctly aligned telescope in order to avoid beam distortions. These properties can now be used to derive a simple parametric representation of the parabola surfaces $\vec {O}^{(i)}$ such that the OAPs optical axis matches the z-axis and the focal point equals the origin of the coordinate system, which is shown in detail in Supplement 1 of this publication. The resulting relations are:

For a more common description, alignment errors can be added. As the surface is parametric, an arbitrary translation and rotation of the surface can be written in simplified homogeneous coordinates [15], omitting the last row of the transformation matrix as no further transformations have to be applied:

 

Fig. 1. Sketch of an OAP with according parameters

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2.2 Formal description of beam propagation

Using this representation, the propagation of a ray through the telescope can be calculated. An OAP telescope is composed of two OAPs, while both optical axes ideally overlap and $b^{(1)}$ and $b^{(2)}$ are of opposite signs. Naturally, the beam path is calculated in four steps:

A detailed description on how to do so can be found in Supplement 1 of this publication, along with two examples. Some relevant notations of the beam path are marked in Fig. 2, where $l$ denotes propagation parameters and $\vec {k}$ the normalized wave vectors. $||\cdot ||$ denotes the euclidean norm.

 

Fig. 2. Sketch of an OAP telescope with beam path

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3. Influence of misalignment

In order to derive demands on alignment precision, the influence of different degrees of freedom on the beam has to be known. Specifically, the OAPs can be translated relative to each other, which leads to wavefront (WF) deformations, or rotated around the focal spot, which distorts the near field and changes the local intensity. We examine the translation in section 3.1 and the rotation in section 3.2.

3.1 OAP translation

As a translation of the first OAP could be compensated for by a shift of the incoming ray, the coordinate system is chosen such that $\vec {T}^{(1)} = \vec {0}$. Therefore, only the translation of the second OAP has to be considered. A schematic view of the Optical Path Difference (OPD) for a single ray is depicted in Fig. 3. For small amounts of translation, the displacement of the OAP surface can be reduced to the component $\Delta \vec {n}$ normal to the local surface in a first approximation. This holds true as long as the amount of translation remains small compared to the curvature of the surface, i.e. $\|\vec {T}^{(2)}\| \ll b^{(2)}$. Using the local gradient to estimate the new intersection point of a ray with the surface, one can estimate the additional OPD $P$, depicted in green in Fig. 3:

Here, $\vec {k'}$ describes the direction of propagation between the OAPs, is collinear to $\Delta \vec {d}'$ and is used to get the signed norm of $\Delta \vec {d}'$. On the right side, $d^{(1)} = -a^{(1)}r^2 + b^{(1)}$. A detailed derivation of Eq. (2) can be found in Supplement 1 of this publication, together with considerations about higher order terms and why they can be neglected. This equation indicates two things:

 

Fig. 3. Sketch of the second, translated OAP

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First, the resulting WF error is approximately linear over the amplitude of the displacement. Nonlinear influences arise from the curvature of the surface, which only become relevant for larger displacements.

Second, the WF error depends on the F-number (given by the extent of $x$ and $y$ and the effective focal length) and off-axis angle of the telescope, but not on the magnification $\beta$. Using this property, one can demonstrate that the wavefront error is independent from the beam size too.

In other words, this indicates that a telescope used for the transport of a 20$\,$cm beam has to be aligned with the same absolute precision as a telescope for a beam diameter of 5$\,$mm, as long as the off-axis angle and the F-number stay the same. These results will be used in section 5 to derive the demands on alignment precision in parabola translation.

Figure 4 shows the OPD found in Eq. (2) for the various translation direction components, given for a beam radius of $1.5 b^{(1)}$ around the origin which corresponds to a theoretical on-axis telescope.

 

Fig. 4. OPD given by Eq. (2), plotted over the coordinates of the input rays in units of $b^{(1)}$, relative to the translation of the second parabola in $x$-, $y$- and $z$-direction (left to right). Note that the origin does not correspond any beam center but to the symmetry axis of the undisturbed parabolic surfaces. Any circular beam of a real telescope cannot include the origin.

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Giving the $x$ and $y$ axis in units of the parent focal length $b^{(1)}$ has the advantage of eliminating the scale of the system from the plot and therefore making it applicable to all beam sizes.

The OPD axis (color scale) is scaled with the amplitude of translation in each corresponding direction and therefore has no unit as well. In other words, the actual OPD can be obtained by multiplying the unit-less values from the plot with the amount of translation.

The OPD for any translation can be calculated by superimposing these three components as Eq. (2) is linear in all directions of translation.

The spatial distribution of the OPD in Fig. 4 reveals that the resulting WF errors are not linked to pure aberrations in the Zernike or Seidel sense. Even the defocusing OPD that occurs for translation in $z$ direction (right plot) grows linearly with the distance to the center rather than quadratic.

Figure 5 shows the relevant beam region for an off-axis telescope with $F=3$ and an off-axis angle of $45^\circ$, which corresponds to a set of telescopes that can be built in practice. The OPD corresponds to a small, circular subset of the ones shown in Fig. 4, centered in positive $x$ direction from the origin.

 

Fig. 5. OPD, given by Eq. (2), of a 45$^\circ$ telescope with an F-number of 3, plotted over the coordinates of the input rays in units of $b^{(1)}$, relative to the translation of the second parabola in horizontal, vertical and longitudinal direction (left to right). The piston and slope of the central ray were subtracted for better comprehension.

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For easier comprehension, we decomposed $\vec {T}^{(2)}$ into transversal and longitudinal components relative to the chief ray between the OAPs. We also subtract the OPD and slope at the position of the chief ray in order to give a clearer picture of the resulting WF.

Considering only this subset of the OPD, the resulting aberrations can be described more accurately by aberrations in the Zernike sense, i.e. the WF errors for transversal translation are a compound of mainly astigmatism and coma (left and center), while longitudinal translation primarily features defocus.

3.2 OAP rotation

A similar consideration has to be done for the rotational misalignment of the telescope. Again, as the pointing of the input beam can be well adjusted to the first OAP using turning mirrors, the misalignment of the telescope can be expressed through a single rotation of the second parabola around the origin.

Note that this is an utterly different scenario than the rotational misalignment of a single OAP with respect to the incident beam, which itself causes huge WF defects [16]. In this case, the pointing of the incident beam is assumed to match the optical axis of the OAP, while the term ’rotational misalignment’ purely refers to the orientation of the two OAPs to each other. Framed differently, this rotation is equivalent to the pointing difference between the ingoing and the outgoing beam of the telescope.

While this motion does not naturally correspond to the axes of rotation of a typical optical mount, we will find that this description is inherently elegant in order to split different effects of misalignment mathematically. Also, it integrates nicely with the alignment procedure we propose in section 4.

In order to simplify the calculation we keep the second OAP in place while rotating anything else, i.e. the first OAP and the incoming beam, about the negative angle. This does not change the scenario but enables easier analysis of the outgoing beam by calculating the intersections of individual rays with the second OAP.

As rotation around the z-axis has not to be considered due to the rotational symmetry of the system and only small misalignments are taken into account, we find the output coordinates of each ray by solving

A detailed derivation of this expression is given in Supplement 1 of this publication. The error of these approximations is less than 1% of $b^{(1)}$ if $0.2 b^{(1)} \leq r_{in} \leq 3.7 b^{(1)}$ and $\omega \leq 5^\circ$.

Equations (3) and (4) are straightforward translations of the input coordinates to the output coordinates. For $\omega =0$, $u=\beta x$ and $v=\beta y$. For $\omega \ne 0$, the trigonometric terms on the right side in the brackets represent the regular translation due to the parabola rotation, while the negative magnification $\beta _R(\psi )$, given by Eq. (5), now varies spatially and therefore distorts the beam.

The spatial distortion itself is not really problematic, since it does not affect the quality of the focal spot. The issue lies rather in the fluence distribution, which is changed in the case of beam-transport telescopes or at least wrongly measured if such a telescope is used for beam diagnostics: In regions where the outgoing ray density is contracted, the intensity will be raised in an uncontrolled manner and may reach damage thresholds of subsequent optics. This is especially threatening if amplifier stages follow the OAP telescope as higher intensities will extract more energy from the medium.

The factor of the local intensity can be calculated directly using the local magnification:

However, the approximations used to derive Eq. (5) generate a singularity in Eq. (6). This leads to a deviation from the actual value which grows rapidly when approaching the origin. Alternatively, as an empirical approach, a simple quadratic function in $r = x\sin (\psi ) + y\cos (\psi )$ can be fitted to numerical values:

A numerically calculated example of the local brightness factor is shown in Fig. 6, center, next to the relative error of the approximation in Eq. (7). It can be seen that the error is less than 1% of the local brightness factor for any ray that passes a telescope with $\omega \leq 5^\circ$ within an off-axis distance less than $5\, b^{(1)}$. This should cover most applications and is more precise than the analytical approach as shown in Eq. (6).

 

Fig. 6. Left: Input coordinates (red) are translated to output coordinates (green) for a $8 \times 8b^{(1)}$ region. The big and small circles indicates a region for $0.2 b^{(1)} \leq r_{in} \leq 3.7 b^{(1)}$, while the two medium circles mark the relevant area for a 90$^\circ$, F=1 telescope. Center: Numerically calculated brightness factor at the telescope output for a $12\times 12b^{(1)}$ region. Right: Corresponding deviation of Eq. (7) in percent with contour lines at steps of 0.5%, excluding 0%. The purple line indicates a region for $r \leq 5 b^{(1)}$. For these examples, a telescope with $\beta = -1, \omega = 5^\circ$ and $\psi = 0^\circ$ was used.

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4. Proposed alignment procedure

In this section, we propose a setup that allows easy and repeatable alignment of an OAP telescope. First, we motivate the design of the setup by what we learned about the influence of misalignment above. After that, we present the setup itself. In the last step, we describe the fine alignment procedure that can be easily repeated whenever necessary.

4.1 Design motivation

In section 3., Eq. (2) gives an expression for the WF-errors that arise when translating the OAP with respect to their ideal position and Eq. (7) approximates the relative deviation in local brightness depending on the rotation angle of the OAPs with respect to their ideal orientation.

Considering the translation, Eq. (2) clearly indicates that the translation converts to WF errors in a linear manner with a potentially large factor. This suggests that the alignment precision for translation has to be measured in the order of microns.

However, as the local intensity of a real, large-aperture laser beam naturally suffers from small-scale defects, there is no point in demanding the local intensity factor given by Eq. (7) to be much smaller than the amplitude of these defects, which usually can be measured in single-digit percentages. This indicates that the rotation of the OAPs, given by $\omega$, only has to be aligned with a precision in the order of degrees. Therefore, the rotation can be adjusted during coarse alignment with reasonable results, while the translation has to be fine tuned. The coarse alignment is done while setting up the telescope, which is described in detail in Supplement 1 of this publication.

As translation of an OAP leads to a shift of the focal spot, we therefore conclude that overlapping the focal spots of both parabolas will lead to sufficient alignment precision. We verify this statement in section 5 after we introduced the alignment procedure in the following subsections.

4.2 Setup

A setup to achieve this is shown in Fig. 7. The central component is a focal fiducial mark at the position of the intermediate focal spot. One possibility is a motorized needle with sufficiently small tip of the order of one micron or less. Etched tungsten needles are a good choice for this.

For the alignment, a collimated full aperture beam is injected into the telescope from either side to adjust the corresponding OAP. For the forward direction, the laser beam of the system itself can be used. For the backward direction, a beam can be injected though a leaky mirror. This may seem inefficient as the vast majority of the light gets reflected before entering the main beam path, but as the alignment is checked in the focal plane, all the transmitted energy will be concentrated in one spot, easily delivering enough intensity.

 

Fig. 7. Schematic of the proposed setup for fast, repeatable alignment of an OAP-telescope, discussed in section 4.2

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The setup of an alignment beam is easy while the aperture of the telescope stays small, but bigger apertures require more effort. A good way to save time and material is to combine the alignment beam path with the path of beam diagnostics that are often found after large, critical telescopes in laser systems. This is depicted schematically in Fig. 7.

Having the fiducial and the beams ready, the focal spots can now be observed with microscopes in the forward and backward direction. These should have enough spatial resolution to at least resolve the diffraction limited focal spot to several pixels and a numerical aperture large enough to collect the full beam. For the alignment procedure, the microscope can either be moved into the beam path directly, or the beam can be deflected into the microscope using small, movable turning mirror as depicted in Fig. 7. The latter may be more stable mechanically and is also a good solution for telescopes with space constraints.

One should keep in mind that the whole energy of the beam enters the microscope. While this is likely not a problem for the backwards beam, the laser system itself should be turned down to minimum power and appropriate filtering has to be done within the microscopes.

With the alignment setup in place, the OAPs can now be aligned using motorized mounts. These have to feature a total of five axes - all degrees of freedom except rotation around the z-axis. Having one translational axis parallel to the intermediate beam direction makes the alignment a lot more comfortable as the beam can easily be defocused that way.

As constructing such a mount for big, heavy OAPs may be a challenge, the transversal translation axes can be substituted with motorized tip/tilt axes of an adjacent turning mirror, which should be as close to the OAP as possible. However, this makes the alignment procedure less convenient.

A detailed instruction on how to set up and coarsely align such a system is given in Supplement 1 of this publication. The fine alignment procedure is described in the following.

4.3 Fine alignment procedure

The actual fine alignment procedure is easy to perform and can be done in less than ten minutes. It is similar to the medium alignment procedure during the setup described in Supplement 1, differing mainly in the fact that the microscopes resolve the diffraction limit of the focal spot, enabling a close-to-perfect alignment. The following steps have to be performed for both, the forward and the backward direction using the corresponding beam, OAP, turning mirror and microscope. Some images of this process can be found in section 6., Fig. 11.

After following these steps for both beam directions, the OAP telescope is fully aligned. In the next section, we calculate the precision that one can reach using this procedure.

5. Analysis of alignment precision

If an OAP telescope shall be constructed, there are usually requirements on the beam quality. As seen in section 3., misalignment can heavily influence the WF of the beam, as well as the local brightness. The WF is of particular interest for high-intensity lasers as the focus size and therefore the maximum intensity is limited by aberrations. The brightness on the other hand may lead to damage in subsequent optics, especially if another amplifier follows the telescope.

In this section, we analyze the spatial precision that can be reached with the described method and translate that into quantified beam properties. This allows to quick-check the beam quality that can be expected for a certain setup and if this quality will live up to any external requirements.

5.1 OAP translation

As translational misalignment leads to different WF errors, the quality of the resulting WF needs to be quantified in order to determine the demands on alignment precision. A particularly suitable property is the Root Mean Square (RMS) of the WF. This is due to two reasons: First, the RMS for small WF errors is directly linked to the Strehl ratio of the focal spot [17] which is useful when formulating a criterion on beam quality. Secondly, one can easily show that the RMS grows linearly with the amplitude of the considered WF, and is therefore proportional to the amplitude of the translation vector $\vec {T}'$, as seen in Eq. (2). Therefore, the RMS can be expressed in units of the modulus of the translation vector.

The RMS of a specific setup is calculated by using Eq. (2), subtracting piston and tilt, integrating the remaining squared OPD over the aperture and normalizing with the area of the aperture.

Doing this, we performed a parameter scan over the $F$-number from 0.5 to 100 and the off-axis angle from 1$^\circ$ to 135$^\circ$ for translation in both transversal and longitudinal translation. The result is shown in Fig. 8.

Here, the white areas indicate setups which are not technically realizable as the aperture would include the origin: Rays through the origin would be reflected back onto themselves and therefore cannot hit the second OAP or, considered in another way, would be obscured by the rear side of the second OAP before hitting the first one. We expressed the resulting RMS in units of $T'_i$, as discussed above.

 

Fig. 8. The WF RMS for different setups relative to the translation amplitude of the second OAP. The contour lines indicate orders of magnitude. Note that the RMS-per-displacement axis is dimensionless as this property has to be multiplied with the amount of misalignment in units of length to be applicable.

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From the surface plot, several things become apparent. First of all, the maps for transversal translation (left and center) are nearly identical, which is why we will represent both of them by the set of numerical data for horizontal translation from now on. Second, all the errors decrease with increasing F-number. This is intuitive as increasing the F-number corresponds to decreasing the aperture size, which will naturally decrease the WF errors. Third, the RMS will also increase with increasing off-axis angle, especially in the case of transversal misalignment. However, this parameter is less sensitive.

In order to find a simple formulation of the WF RMS, depending on the alignment precision, we apply an additional fit to the numerical results.

Transversal translation In order to approximate the RMS per translation $\delta$ for transversal misalignment over a sufficiently large area, a function with a relatively large number of parameters has to be used. One should keep in mind that the chosen function over the $F$-number and the off-axis angle $\alpha$ is arbitrary with the only purpose to provide estimations while planning a telescope:

Here, $\alpha$ has to be given in radians. Figure 9 shows the the deviation of the approximations from the numerical values on the left.

 

Fig. 9. Left: The fit error of transversal translation, described by Eq. (8). The black contour lines indicated steps in 2.5e-3, starting at 0. The purple line indicates the threshold according to Eq. (9), above which the fit error may exceed 2.5×10$^{-3}$. Right: The best RMS that is guaranteed to be achievable by the proposed alignment scheme in transversal directions. The black contour lines correspond to the tics of the color palette.

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As one can see, the deviations increase for small $F$-numbers and large off-axis angles. From a practical point of view, however, such configurations are unlikely to be used as they pose some additional challenges. Telescopes based on these parameters would require very large optics compared to the beam diameter and a coating able to manage huge differences in incident angles. Furthermore, they would be rather impractical to use. With that in mind, systems that satisfy

Using Eq. (8), one can approximate the best RMS error that is guaranteed to be achieved for a telescope that is aligned according to the alignment procedure described in section 4. Here, the precision of the translation is naturally limited by the size of the focal spot between the OAPs. The diameter of the focal spot in vacuum for a homogeneously illuminated pupil corresponds to the diameter of the airy disc and can be expressed as $D_f \approx 0.61 \, \lambda / \mathrm {\textit {NA}}$ where $\lambda$ is to the central laser wavelength and NA the numerical aperture of the focusing optic [18]. This holds true as long as Marechals criterion is fulfilled, i.e. the RMS of the aberrations present in the beam is lower than $\lambda /14$ [19]. If not, $D_f$ has to be increased accordingly.

Using this, we can write

Longitudinal translation While longitudinal translation mainly causes beam defocusing which could potentially be compensated by other optics, this is generally unintended and impractical from an imaging point of view. It also may change the beam intensity in extreme cases. Therefore, even though the beam quality criterion may be chosen more loosely here, a consideration analog to the last paragraph is necessary.

The RMS of the aberrations caused by longitudinal translation can be approximated by

Again, $\Delta$ is the best RMS that can be guaranteed to be achieved in longitudinal alignment for a specific setup. Figure 10, right, shows the values given by Eq. (12) for the valid parameter space. Obviously, even the best setups only give a best guaranteed RMS of roughly 50 mλ. This means that traditional methods to align the defocusing of the system have to be employed, e.g. using a shearing interferometer. An example on how to use Eq. (12) is given in Supplement 1 of this publication.

 

Fig. 10. Left: The fit error of longitudinal translation, described by Eq. (11). The black contour lines indicate steps in 2.5e-3, starting at 0. The purple lines indicate the threshold according to Eq. (9), above which the fit error may exceed 2.5×10$^{-3}$. Right: The best RMS that is guaranteed to be achievable by the proposed alignment scheme in the longitudinal direction. The black contour lines correspond to the tics of the color palette.

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5.2 OAP rotation

The local intensity factor is given by Eq. (7). Due to the monotonous behavior of this function, the maximum local intensity boost for a given alignment precision $\omega _p$ can be calculated by evaluating Eq. (7) at the maximum radius $r_{max}$ within the pupil:

Here, $\Delta I_{max}$ is the maximum local intensity boost in the outgoing beam that occurs within the pupil for a given misalignment angle $\omega _p$. A boost of $\Delta I_{max} = 0$ refers to the regular ougoing intensity.

An example on how to use this equation is given in Supplement 1 of this publication.

6. Comparison with the alignment of a single OAP

The precision calculated above only refers to the alignment of the two OAPs relative to each other. Of course, the alignment precision of each individual OAP has to be taken into account as well. For comparison, we briefly analyze the precision that can be reached for a single OAP.

Our procedure minimizes the astigmatism of the beam by defocussing and shortening the resulting line in the focal plane to a minimum by tilting the OAP. The evolution of the focal spot is shown in Fig. 11. Close to the diffraction limit, the elongation of the focal spot becomes minimal and can be observed easiest in the first dark ring of the focal spot while oversaturating the camera and choosing a suitable color palette (see Fig. 12, left and right). Here, we can apply a simple criterion to determine if the elongation is visible or not: If the illumination in the first dark ring along the major semiaxis of the focal spot is more than two times the illumination along the semiminor, the elongation can be observed (see Fig. 12, center). For a homogeneously illuminated, circular pupil, this criterion is fulfilled at 5.6 mλRMS of astigmatism, which therefore is the best guaranteed alignment precision according to this criterion. For certain telescopes, this precision can be lower than the one of the relative alignment (see Fig. 9, on the right). This holds true for larger F-numbers and smaller off-axis angles. Practically speaking though, this precision corresponds to a Strehl ratio of 0.999, which is well above what is realistically reachable in lasersystems today due to other sources of WF aberrations. Therefore, the performance of the system will not be limited by the alignment of the individual OAPs.

 

Fig. 11. Images of a real focal spot while aligning an OAP according to the steps described in section 4.3. Strong astigmatism generates a line-shaped focal spot. 1) The backlit needle after step 2. - 2) Beam focused to a line when starting step 3. - 3) Oblique astigmatism minimized after step 3.1. - 4) Normal astigmatism minimized after step 3.2. - 5) Focal spot after two iterations of alignment

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Fig. 12. Left: The far field of a homogeneously illuminated pupil with 40 mλRMS of defocus, 10 times oversaturated. The two points that are used to calculate the dark-ring contrast are marked. Center: The contrast between the brightest and darkest point in the darkring of the focal spot for over the amount of astigmatism. The black cross marks the point where the visibility criterion is fulfilled. Right: The far field with additional 32 mλRMS of astigmatism.

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7. Conclusion

In this work, we have revisited the topic of OAP telescopes in the context of high-intensity lasers. Using an analytical model, we identified the beam defects caused by misalignment, which is WF errors for translation (Eq. (2)) and beam distortion and therefore local intensity changes for rotation around the focal spot (Eq. (7)). Based on that, we proposed an alignment scheme that aims to overlap the focal spots of both OAPs. Finally, we have shown that this scheme leads to sufficient alignment precision in all terms but defocusing, which has to be done with conventional alignment techniques. We also provided expressions that allow fast estimation of the beam quality that can be expected from a certain setup using the proposed alignment scheme (Eqs. (10), (12) and (14)). By doing this, we have shown that the alignment of an OAP telescope is no game breaker and can even be performed during operation easily. Knowing the advantages of a purely reflective beam transport, one can therefore safely consider to use them in upcoming facilities.