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Abstract
We present an innovative automatic control of angular dispersion for high-power laser systems. A novel, to the best of our knowledge, diagnostic has been developed to visualize angular dispersion in ultrashort near-infrared laser pulses for on-shot analysis. The output of a commercial ultrabroadband oscillator was prepared with an arbitrary chromatic dispersion and sent through a compensation system composed of 4° glass wedges in motorized mounts. These wedges were rotationally controlled in discrete steps about the beam axis in accordance with the diagnostic, via an automated feedback loop, to successfully eliminate angular dispersion to a precision of 5 nrad/nm. The system can be implemented to maintain a zero or nonzero target dispersion for experiments.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. INTRODUCTION
Ultrashort pulse laser science faces a multitude of challenges in maintaining and measuring high-quality pulses, often due to the high time-bandwidth-product associated with femtosecond pulses possessing very broad (${\gt}{100}\;{\rm nm}$) optical bandwidths [1]. Dispersive properties of optics become quickly detrimental over such large bandwidths, compelling careful control and diagnosis of short-pulse lasers [2]. One common complication is angular dispersion (AD), where the transmission through optical media in the laser chain causes divergence between longer and shorter wavelengths, i.e., chromatic aberration. This is most regularly caused by: imperfect alignment of the stretcher and compressor in chirped pulse amplification laser systems [3,4]; noncollinear optical parametric amplification [5]; and transmission through wedged optics (required to mitigate backreflections) [6].
The presence of AD has been shown to lead to pulse-front tilt (PFT) [7]: the laser pulse front lies at an angle to its direction of propagation, resulting in suboptimal experimental outcomes in applications that require the laser’s PFT to be minimized to maximize the pulse intensity on target. More specifically, it has been reported that the presence of PFT in high-powered laser systems causes a wakefield asymmetry, which leads to a deflection of the electron beam [8,9]. Furthermore, high-power laser technology is now developing towards higher repetition rates to provide high average power experiments [10]. This poses a threat to the pulse front stability due to the thermal loading on the amplifiers and compressor gratings [11].
Experiments seeking to manipulate AD commonly do this by intentional misalignment of the compressor [9,12–14]. For large-scale laser systems (PW, fs) [15], this is detrimental to the pulse and system quality [16]; the gratings are mounted in vacuum on susceptible motorized mounts and should be optimally aligned and left untouched, minimizing risk. Furthermore, examples of manipulating AD indicate the experimental benefit of fine-tuning toward a specified value: in order to steer the deflection of protons from target normal sheath acceleration (TNSA) [13], or specify deviation of electron-beam pointing from laser wakefield acceleration (LWFA) [14].
In this context, it is clearly experimentally desirable to possess a method to automate the control of AD within a beamline with minimal impact on the original system and in a fast and repeatable manner. Such an approach would be beneficial in a continuous active loop for high-repetition-rate systems to counter thermal loading, or used occasionally to readjust after dynamic activities such as realignment or alterations to a beamline. Here we present a system capable of simultaneously characterizing and correcting for the presence of AD. We demonstrate it on a commercial oscillator of modest (4 µrad/nm) dispersion, showing automated compression in a matter of minutes using an iterative optimization strategy and electronic feedback to the azimuthal angle of two BK7 wedges.
2. METHODOLOGY
In this study, a diffraction diagnostic method proposed by Galimberti et al. [17] has been used as a way of providing real-time feedback of the AD. Other current techniques to measure angular dispersion in ultrashort pulses can be more complex to implement, requiring alignment, interferometry, and cross-correlation and thus were ruled out for this study [18–20]. Additionally, for high-power—and hence, usually, lower repetition rate—lasers, it is not practical to use conventional far-field image analysis to deduce AD without assuredly eliminating the possibility of astigmatism. Our chosen diffractive diagnostic provides a simple and inexpensive method of visualising AD by diffracting the beam from a polka-dot beam splitter (transmissive orthogonal grating) and evaluating the first-order far-field diffraction patterns. Then, for broadband pulses, these diffraction patterns will appear as lines, as the different frequencies are spread by the grating. It follows that, when aligned properly, a pulse with no AD will show these lines perfectly parallel to the axes of the grating; AD is readily seen by a tilt away from the $x$ and $y$ axes on the camera. These tilt angles can be converted to directional AD values from knowing only the polka-dot grating density and the image size. Example demonstrations of the diagnostic images are given in Fig. 1, obtained through Opticstudio, where an ultrabroadband (800–960 nm) source was simulated to have a large AD caused by a prism compressor with an AD of ${\sim}5\;\unicode{x00B5} {\rm rad/nm}$.
Fig. 1. (a) Example diagnostic images obtained from Opticstudio: a pulse with 5 µrad/nm angular dispersion caused by a misaligned fused silica prism pair; (b) above pulse compensated with BK7 wedges. Associated far-field diffraction patterns are expanded in each case.
The experimental arrangement used to test the correction system is shown in Fig. 2: an ultrabroadband (800–960 nm) femtosecond laser source (LS) was dispersed by an arbitrarily misaligned fused silica prism compressor (CMP) and directed through two motorized compensating wedges (W1, W2), followed by a polka-dot beam splitter acting as a two-dimensional transmissive grating (G), and a focusing lens (L) to the diagnostic camera (C1). Each iteration (calculated wedge movement(s)) inevitably disrupts the pointing of the beam; as such, an automatic alignment loop (mirror M and camera C2) is necessary to keep the light on the diagnostic and maintain measurement [21].
Fig. 2. Schematic layout of the experimental arrangement: a broadband pulse from a commercial oscillator (LS) is arbitrarily dispersed by a misaligned prism compressor (CMP), and then directed by an automatic alignment loop (M, S, C1) through two correcting wedged optics (W1, W2), a polka-dot beam splitter (G) and a focusing lens (L).
To compensate for the AD, two BK7 4° wedged windows were rotated in motorized mounts about the beam axis. These were simulated in Opticstudio to compensate for the dispersed far field in Fig. 1(a) to the best-corrected focus in Fig. 1(b).
The angular deviation of light refracted by a shallow wedge of apex angle $\alpha$ is
In the case of two identical wedges, rotated about the $z$ axis by angles ${\theta _1}$ and ${\theta _2}$, it is useful to introduce the two-dimensional rotation matrix $R({\theta _z})$ to visualize the vectorized deviation angles in both $x$ and $y$ axes, projected into the coordinate frame of wedge 2,
Equation (2) can be expanded and simplified to present the AD, ${\boldsymbol \Phi}$, caused by two wedges, rotated by ${\theta _1}$ and ${\theta _2}$, as a two-dimensional vector,
We define a diagnostic measuring residual vector AD of ${\boldsymbol \delta} = (\sigma ,\rho)$, consisting of the current wedge configuration and the existing pulse AD. We seek small (for stability) wedge movements to converge the system until ${\boldsymbol \delta} = 0$, i.e., ${\boldsymbol \Phi}({S^{i + 1}},{T^{i + 1}}) = {\boldsymbol \Phi}({S^i} + dS,{T^i} + dT) = {\boldsymbol \Phi}({S^i},{T^i}) + {\boldsymbol \delta}$. This is achieved when
Clearly the denominator in Eq. (4) could cause instability. A correction factor, $\varepsilon$, is therefore substituted in place of $\cos (T)$ when |$\cos (T)$| < $\varepsilon$, preserving sign; a similar approach is applied for $\sin (T)$. The final step sizes are then limited to an input gain ($\gamma$) to account for the assumptions made previously and to ensure stability of the system.
Figure 3 shows a flow diagram of the algorithm used to control the prism movements as per the diagnostic readout. The diffracted orders on the camera are translated as directional AD values. Step sizes for each wedge are then calculated using the current positions and the remaining error. The laser pointing is then necessarily corrected to rectify the misalignment by refraction. The loop can be left running in a continuous operation mode for systems where there is reason to suspect fluctuations in AD (such as thermal loading) and to stabilize the AD to a set target.
Fig. 3. Algorithm used to automatically tune the measured angular dispersion of a broadband pulse by the rotation of two identical glass wedges at angles about the optical axis.
3. RESULTS AND DISCUSSION
For the purpose of testing the capabilities of the correction system, the initial laser source was purposefully dispersed by a misaligned fused silica prism pair, giving an approximate 4 µrad/nm AD that it was desired to compensate. The correction factor was set to be $\varepsilon = 0.1$ for mathematical stability. Before-and-after camera snapshots are presented in Fig. 4 to highlight the benefit of the algorithm as implemented by the procedure and diagnostic described above. Figure 4(a) shows this source on the diagnostic, with compensator wedge angles at the undeviated positions, ${\theta _1} = 0$, ${\theta _2} = \pi$ (the 0th order is masked). Figure 4(b) shows the final, optimized pulse on the diagnostic. Visually, the tilt in the diffracted wings have been flattened, indicating suppression of the AD. Measurably, the AD was eliminated after 50 iterations with an error of 5 nrad/nm.
Fig. 4. (a) Angular dispersion diagnostic camera images of the laser pulse dispersed by a misaligned prism compressor and (b) corrected by the automated loop after approximately 50 iterations. Contours show 10% intensity increase, the 0th order far field has been masked.
Figure 5(a) presents the evolution of recorded angular dispersion over the prism movement iterations. Following activation of the loop, a dramatic reduction in angular dispersion is evident (linearized by the limited step size for stability) and is kept low thereafter.
Fig. 5. (a) Absolute angular dispersion evolution per correction iteration of the automated loop and (b) for targeted values in $x$ and $y$.
The limit to the achievable target AD value is dependent upon the beam spatial quality and camera resolution. The speed of the optimization in this test was limited by the wedge movement step size; the gain coefficient (in this specific case, $\gamma = 0.1$), on the discrete step sizes ensured the pointing was not lost to the diagnostic. It is important to note that the starting AD was intentionally and unreasonably high; in real applications on laser systems with lower repetition rates, the step size $\gamma$ could be increased to achieve the target much faster. Alternatively, a pilot beam could be introduced to act as a model for the main pulse.
The solution can address AD wherever measured; however, for practical implementation on a working laser system, a full-aperture leakage beam after the final compressor is the ideal location for the diagnostic. If this is not available, however, a leakage at the input to the compressor would suffice, assuming the compressor is optimally aligned. Compensating wedges should be placed ideally when the beam is somewhat enlarged to lessen the strength of the unavoidable spatial chirp. It is also necessary to consider the added temporal chirp, which could be accounted for by an acousto-optic programmable dispersive filter, or by adjusting the stretcher or compressor. The upper limit of the correction rate is set by the speed of the motors of the wedges and alignment loop. The lower limit is set by the laser repetition rate at the diagnostic; fluctuations in the AD faster than this will not allow stable corrections.
As a more versatile tool, one can substitute $\sigma \to \sigma - \phi _x^t$ and $\rho \to \rho - \phi _y^t$ to aim for a target ${\boldsymbol \Phi} = (\phi _x^t,\phi _y^t)$. Figure 5(b) shows the implementation of the loop to achieve a target AD of ${-}0.5\;\unicode{x00B5} {\rm rad/nm}$ and 1 µrad/nm in the $x$ and $y$ axes, respectively, with an error of 10 nrad/nm. This provides a great benefit to high-power laser systems for controlling PFT and therefore steering secondary sources in TNSA and LWFA experiments, for example.
4. CONCLUSION
An automatic feedback loop has been successfully tested to minimize the absolute angular dispersion of a dispersed ultrabroadband NIR laser pulse source (800–960 nm) to a precision of 5 nrad/nm. Motorized BK-7 prisms were rotated about the optical axis in discrete steps as determined by a low-cost diagnostic in a feedback loop. This method can be tuned to reach and then stabilize the angular dispersion to a desired set point, within the limits based on the chosen wedge angle and material. The system offers a simple addition to broad bandwidth laser systems in order to minimize or control angular dispersion, necessary to optimize the compressed pulse on target, as well as providing more flexibility of the pulse front tilt during experiments. The experimental benefit of this system has earned its inclusion in the new Extreme Photonics Applications Centre [22] for providing optimization of the petawatt pulse on shot, as well as allowing control of secondary sources such as electron beams, X-rays, and ion acceleration. The system will be extended to include higher-order correction for more complex laser sources.
APPENDIX A
1. Derivation of Angular Dispersion
The rotation matrix in two dimensions is
For two wedges, treating the ray deviation using a small angle approximation, the total deviation can be written as a sum of the wedge contributions,
Here ${\boldsymbol \beta}(\lambda)$ is now in two dimensions; that is, the angle from the optical ($z$) axis in $x$ and $y$. Using the trigonometric identities,
If we approximate the refractive index as linear with respect to $\lambda$, then the angular dispersion can be written as
2. Derivation of Wedge Movements
We seek small wedge movements to converge the system until ${\boldsymbol \delta} = 0$, i.e., ${\boldsymbol \Phi}({S^{i + 1}},{T^{i + 1}}) - {\boldsymbol \Phi}({S^i},{T^i}) = {\boldsymbol \delta}$. That is,
The first term from Eq. (A8) cancels with ${\boldsymbol \Phi}({S^i},{T^i})$ from Eq. (A6), and the fourth term can be ignored, as it is small. This leaves ${\boldsymbol \delta}$ equal to the sum of terms 2 and 3 from Eq. (A8). Solving these simultaneous equations for $dS$ and $dT$ finds the solution to the wedge movements in terms of the wedge rotation and glass properties,
Funding
Engineering and Physical Sciences Research Council (EPSRC) Centre for Doctoral Training in Industry-Inspired Photonic Imaging, Sensing and Analysis (EP/S022821/1).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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