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Simple wedge plate based lateral shearing interferometry technique for coherent alignment of tiled grating assembly
Optics Communications 459 (2020) 125067
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Simple wedge plate based lateral shearing interferometry technique for coherent alignment of tiled grating assembly D. Daiya a,b ,∗, R.K. Patidar a , A. Moorti a,b , N.S. Benerji a , A.S. Joshi a a b
Raja Ramanna Centre for Advanced Technology, Indore 452013, India HomiBhabha National Institute, Mumbai 400094, India
ARTICLE
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Keywords: Tiled gratings Chirped pulse amplification
ABSTRACT A simple, sensitive and robust technique is proposed for coherent alignment of tiled grating assembly using wedge plate lateral shearing interferometry. In this technique, two thick glass wedge plates are used to generate interferograms from reflected and diffracted laser beams, from the tiled grating assembly under alignment. This single optical element based interferometric technique is easy to set up, insensitive to the vibrations and provides simultaneous information of both reference as well as error interference patterns, which are used to estimate the alignment errors. Performance of this technique was tested through alignment of two reflective gratings (size 140 mm X 120 mm), with an angular and translation sensitivities of less than 1 μrad and 50 nm respectively.
1. Introduction Demonstration of Chirped Pulse Amplification (CPA) technique [1] by G. Mourou and D. Strickland in the year 1985 has set up the roadmap for the development of high-energy, high-power ultra-intense laser sources to explore various regimes of science and technology via laser–matter interaction [2–4]. The quest for achieving higher and higher laser peak power has reached to the level of ∼ 5 PW and to intensities of the order of 1022 W/cm2 [5–7]. Efforts are being made to further increase these values which would depend on the technological advancement of crucial optical components. One such crucial component is large size diffraction grating, required for temporal compression of 100’s of mm diameter amplified laser pulses. Development of such large size monolithic gratings with required optical figure, broadband diffraction efficiency, high laser induced damage threshold, groove density uniformity etc. is quite challenging and therefore limits its availability [8]. Coherent optical tiling of small aperture optics provides an alternative way to achieve very large size optical surface with dimensions larger than a meter or even more. One such example is large aperture astronomical telescopes, in which the several meter size objective is made by segmenting number of small aperture mirrors in coherent manner [9]. Similarly, in high-energy high-power PW class CPA lasers, small size gratings are put together to mimic as a single large size monolithic grating, in order to accommodate large size amplified laser beam [10–15]. However, the relative angular and translational motion between the gratings of a tiled grating assembly (TGA) introduces phase errors in the beam, which causes distortions in the focal plane
intensity distribution [16,17]. Therefore, grating tiling requires precise angular and translational motorized movements along with simple, sensitive and robust diagnostics to identify and measure the various possible alignment errors. For this purpose, diagnostics based on either near field measurements (interferometry based) [18,19] or far field measurements (by observing focal spot distributions) [20,21] or combining both the measurements [22] are being used. The far field based method is quite simpler and it requires only a convex lens for the tiling error measurements. However, for small tiling alignment errors, the interferometric methods prove to be more sensitive and accurate and can also distinguish between the various angular and translational errors present in the TGA [18]. The interferometry setups generally used for this purpose are based on either Michelson or Fizeau, or Mach–Zehnder interferometers etc., which require careful alignment of number of optical elements like beam splitter, reference flat etc. Along with this, these interferometers are prone to vibrations, due to which online alignment monitoring of gratings in the TGA based pulse compressors is a difficult task. In this paper, we propose a simple and sensitive technique for coherent alignment of gratings in the TGA using wedge plate lateral shearing interferometry. In the proposed technique, two thick glass wedge plates are used to generate two interferograms from the reflected and diffracted laser beams by the TGA, for the measurement of all the associated tiling errors. The various errors were estimated by calculating the spacing, slope and the relative shift of the fringes from the recorded interferograms. The use of single optical element (i.e. wedge plate) makes this technique easy to set up and insensitive
∗ Corresponding author at: Raja Ramanna Centre for Advanced Technology, Indore 452013, India. E-mail address: [email protected] (D. Daiya).
https://doi.org/10.1016/j.optcom.2019.125067 Received 5 October 2019; Received in revised form 25 November 2019; Accepted 2 December 2019 Available online 6 December 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
D. Daiya, R.K. Patidar, A. Moorti et al.
Optics Communications 459 (2020) 125067
Fig. 1. Various co-planar and in-plane alignment errors of a tiled grating assembly having two gratings. Tip angle error (𝜃x ), tilt angle error (𝜃y ), in-plane angle error (𝜃z ), lateral shift error (𝛥x), longitudinal piston error (𝛥z) and groove spacing mismatch error (𝛥d = d−d′ ).
to the vibrations. This also provides interferometric level of accuracies in the measurements of tiling alignment errors. For experimental demonstration, a wedge plate lateral shearing interferometer was set up using 20 mm thick glass wedge plates having ∼ 18 arcsec wedge angle and coherent alignment of a TGA having two gratings of size 140 mm × 120 mm was carried out. A He–Ne laser beam of ∼ 70 mm diameter was used and lateral shearing interferograms were recorded for both reflected and diffracted beams from the TGA.
part reflected (0th order beam) from the misaligned grating G2, with respect to the fixed reference grating G1, can be written as [23,24]: ) ( 𝑋 − 𝜃𝑥 𝑌 + 𝛥𝑧} (1) 𝜑𝑟𝑒𝑓 𝑙𝑒𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) = 2𝑘 cos 𝛼{𝜃𝑦 cos 𝛼 Where, (X, Y, Z) defines the coordinates of the observation plane, with 𝑌 -axis parallel to grating grooves and 𝑍-axis represents the laser propagation direction. Similarly, the additional error phase introduced in the beam part diffracted by the misaligned grating G2 at a diffraction angle 𝛽 can be written as [23]: ( ) 𝑋 𝜑𝑑𝑖𝑓 𝑓 𝑟𝑎𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) = 𝑘{cos 𝛼 + cos 𝛽}{𝜃𝑦 − 𝜃𝑥 𝑌 + 𝛥𝑧} cos 𝛽 ( ) 𝑋 𝛥𝑑 𝜆 + 𝑘{ + 𝜃𝑧 𝑌 + 𝛥𝑥} (2) cos 𝛽 𝑑 𝑑
2. Tiling alignment errors of a TGA The various possible tiling alignment errors, which may be present in the TGA having two gratings G1 and G2, are represented in the Fig. 1 [17]. G1 is assumed to be the fixed reference grating having surface normal and grooves orientation parallel to 𝑧-axis and 𝑦-axis respectively. Misalignment of the TGA will occur if the grating G2 will have different angular and positional orientation with respect to the reference grating G1. The three angular misalignment errors of the grating G2 with respect to G1 are: (a) Tip angle error (𝜃x ) due to rotation around 𝑥-axis, (b) Tilt angle error (𝜃y ) due to rotation around 𝑦-axis and (c) In-plane angle error (𝜃z ) due to rotation around 𝑧-axis. The two translational errors of the grating G2 along 𝑥-axis and 𝑧-axis are known as lateral shift error (𝛥x) and longitudinal piston error (𝛥z) respectively, whereas translation along 𝑦-axis does not introduce any additional phase [17]. Beside these alignment errors, the difference in groove spacing (denoted by d and d′ in Fig. 1) of the two gratings also lead to additional error phase in the laser beam diffracted by the TGA. These six tiling errors are categorized into two groups: (a) coplanar or mirror like errors and (b) in-plane or grating like errors [23]. The tip, tilt and longitudinal piston errors are grouped into co-planar errors, as in their presence the two gratings will not be having same plane. The laser beam reflected from the TGA (i.e. 0th order beam) will be affected by these errors only. The remaining three tiling errors, i.e. in-plane angle, lateral shift and groove spacing mismatch errors are known as in-plane (grating like) errors, as they affect the diffracted beam along with the co-planar errors. Hence, in order to coherently align the gratings of a TGA, first the 0th order reflected laser beam is diagnosed to remove co-planar errors and afterwards the in-plane errors are removed by diagnosing the diffracted laser beam [23]. Now, if we consider a laser beam of diameter D falling symmetrically (half of the beam part on each grating) on the TGA at an angle of incidence 𝛼. Then the additional error phase introduced in the beam
Where 𝑘 = 2𝜋∕𝜆 is the propagation vector, 𝜆 is the laser wavelength and 𝛥d = d−d′ is the groove spacing mismatch between the two gratings. These equations can be used to simulate the effect of tiling alignment errors on the focal spot intensity distribution and the alignment tolerances can be estimated by putting limit on the peak intensity reduction due to these errors [17]. 3. Description of wedge plate lateral shearing interferometry for TGA alignment Wedge plate lateral shearing interferometry provides a very simple way to diagnose tiling errors present in the TGA with required precision and accuracy. A scheme of the technique is shown in Fig. 2. A diagnostic laser beam is allowed to incident symmetrically on the TGA having two gratings (G1 and G2) at an angle of incidence equal to 𝛼. The reflected/diffracted laser beam from the TGA falls on a thick glass wedge plate of thickness t at an angle of incidence 𝜃i . The glass plate creates two space sheared replicas (along √ X-axis) in the plane of incidence, with shear distance 𝑆 = 𝑡 sin 2𝜃𝑖 ∕ (𝑛2 − sin2 𝜃𝑖 ), through partial reflections from its front and back surface. The wedge plate also introduces angle √ between the two replicas (along Y-axis) by an amount
2𝛾 = 2𝜃𝑤𝑒𝑑𝑔𝑒 (𝑛2 − sin2 𝜃𝑖 ), where n and 𝜃wedge are the refractive index and wedge angle of the glass plate respectively. The overlap area of the two sheared beams (where the interference occurs) can be divided into three regions 1, 2 and 3, which are separated by the dark strips created by the intra-grating separation (see screen image in Fig. 2). In region 1 and 3, the beam part reflected from G1 (represented by red color)
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D. Daiya, R.K. Patidar, A. Moorti et al.
Optics Communications 459 (2020) 125067
Fig. 2. Schematic of the wedge plate lateral shearing interferometry for coherent alignment of gratings of a TGA. Interference pattern with different regions is also shown on the screen. Details are in the text. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Simulated wedge plate lateral shearing interferograms for the 0th order reflected beam from a TGA having (a) perfect alignment with no errors (b) longitudinal piston error (𝛥z) of 250 nm (c) tip angle error (𝜃x ) of 50 μrad and (d) −50 μrad respectively, (e) tilt angle error (𝜃y ) of 50 μrad and (f) −50 μrad respectively.
and G2 (represented by green color) respectively interferes with itself and results into identical fringe patterns, having straight line fringes with orientation along 𝑋-axis and spacing given by 𝜆/(2𝛾). However, in region 2 the beam part reflected from G1 and G2 will be interfering and if there is alignment error in the TGA; the fringe pattern will be different from its neighboring regions due to additional tiling error phase (given by Eqs. (1) and (2)) between the interfering beam parts. So, the fringe patterns in region 1 or 3 serve as the reference for the alignment and the additional reference flat is not required, like other interferometric techniques. For the 0th order reflected laser beam, the resultant phase difference between the two interfering beams in the central region can be written using Eq. (1) as:
𝑓 𝑟𝑖𝑛𝑔𝑒 𝑠𝑙𝑜𝑝𝑒 =
𝑓 𝑟𝑖𝑛𝑔𝑒 𝑝ℎ𝑎𝑠𝑒 𝑠ℎ𝑖𝑓 𝑡 = 2𝑘𝛥𝑧 cos 𝛼
𝛥𝛷𝑑𝑖𝑓 𝑓 𝑟𝑎𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) = 𝜑𝑑𝑖𝑓 𝑓 𝑟𝑎𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) − 2𝑘𝛾𝑌 ( ) 𝑋 = 𝑘{cos 𝛼 + cos 𝛽}{𝜃𝑦 − 𝜃𝑥 𝑌 + 𝛥𝑧} cos 𝛽 ( ) 𝑋 𝛥𝑑 𝜆 +𝑘{ + 𝜃𝑧 𝑌 + 𝛥𝑥} − 2𝑘𝛾𝑌 cos 𝛽 𝑑 𝑑
(3)
From Eq. (3) the fringe spacing along the 𝑦-axis, slope of the fringes and fringe phase shift in the central region can be written as: 𝑓 𝑟𝑖𝑛𝑔𝑒 𝑠𝑝𝑎𝑐𝑖𝑛𝑔 =
𝜆 2{𝜃𝑥 cos 𝛼 + 𝛾}
(4b) (4c)
From Eq. (4a) it is clear that by measuring fringe spacing in the central region, one can estimate the tip angle error (𝜃x ). After calculating 𝜃x , the tilt angle error (𝜃y ) can be calculated by measuring fringe slope and using Eq. (4b). Finally, the longitudinal piston error (𝛥z) can be calculated by measuring fringe shift in the central region with respect to nearby reference regions. Similarly, the resultant phase difference between the two interfering beams in the central region of the diffracted beam’s interferogram can be written using Eq. (2) as:
𝛥𝛷𝑟𝑒𝑓 𝑙𝑒𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) = 𝜑𝑟𝑒𝑓 𝑙𝑒𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) − 2𝑘𝛾𝑌 = 2𝑘[𝜃𝑦 𝑋 − {𝜃𝑥 cos 𝛼 + 𝛾}𝑌 + 𝛥𝑧 cos 𝛼]
𝜃𝑦 𝜃 𝑥 cos𝛼 + 𝛾
(5)
From Eq. (5), various fringe parameters in the diffracted beam’s interferogram can be expressed in terms of the tiling error parameters
(4a) 3
D. Daiya, R.K. Patidar, A. Moorti et al.
Optics Communications 459 (2020) 125067
Fig. 4. Simulated wedge plate lateral shearing interferograms for the 1st order diffracted laser beam from a TGA having (a) longitudinal piston error (𝛥z) of 250 nm, (b) lateral shift error (𝛥x) of 250 nm, (c) tip angle error (𝜃x ) of −50 μrad, (d) in-plane angle error (𝜃z ) of 50 μrad, (e) tilt angle error (𝜃y ) of 50 μrad and (f) groove density mismatch (𝛥N) error of 0.1 gr/mm.
Fig. 5. Schematic of the experimental setup for precision alignment of a TGA with two reflective gratings using wedge plate lateral shearing interferometry. Typical interferograms recorded for the reflected (camera-1) and diffracted (camera-2) beams are also shown.
Fig. 6. (a) A typical shear interferogram recorded for beam reflected from the misaligned TGA showing region −1, 2 and 3. The ROI (1 and 2) are marked with two yellow rectangles in the regions 1 and 2, which served as the reference and the error interferograms respectively (b) Expanded view of the two ROIs used for calculating the fringe spacing and the orientation.
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Optics Communications 459 (2020) 125067
(similar to Eq. (4)) as: 𝜆 𝜃𝑥 (cos 𝛼 + cos 𝛽) + 2𝛾 − 𝜃𝑧 (𝜆∕𝑑) ( ) 𝜃𝑦 (1 + cos 𝛼∕ cos 𝛽) + cos𝜆 𝛽 𝛥𝑑 𝑑2
𝑓 𝑟𝑖𝑛𝑔𝑒 𝑠𝑝𝑎𝑐𝑖𝑛𝑔 =
(6a)
(6b) 𝜃 𝑥 ( cos 𝛼 + cos 𝛽) + 2𝛾 − 𝜃𝑧 (𝜆∕𝑑) 𝜆 𝑓 𝑟𝑖𝑛𝑔𝑒 𝑝ℎ𝑎𝑠𝑒 𝑠ℎ𝑖𝑓 𝑡 = 𝑘[𝛥𝑧(cos 𝛼 + cos 𝛽) + 𝛥𝑥] (6c) 𝑑 From Eq. (6), one may observe that the fringe parameters in the diffracted beam’s interferogram depend on all the six tiling errors. Therefore, after calculating the co-planar errors from the reflected beam’s interferogram using Eq. (4), remaining three tiling errors, i.e., the in-plane or grating like errors can be estimated from diffracted beam’s interferogram using Eq. (6). 𝑓 𝑟𝑖𝑛𝑔𝑒 𝑠𝑙𝑜𝑝𝑒 =
4. Simulation of the TGA errors on interferograms Wedge plate lateral shearing interferograms were simulated for the 0th order reflected beam from the TGA to investigate effect (given by Eq. (4)) of various tiling alignment errors on the fringe pattern. The TGA has two reflective gratings, each of size 140 mm × 120 mm and groove density (inverse of groove spacing) of N ∼ 1740 gr/mm. A He– Ne laser beam (𝜆 = 632.8 nm) of diameter 70 mm and 10th order super Gaussian profile falling at an angle of incidence of 𝛼 = 45◦ on the TGA is considered for the simulation. The wedge plate angle (𝜃wedge ) and lateral shear is kept as 18 arcsecs (along Y-axis) and 15 mm (along X-axis) respectively. The simulated interferograms for the longitudinal piston error of 250 nm, tip angle error of ±50 μrad and tilt angle error of ±50 μrad are shown in Fig. 3. The Fig. 3(a) represents the interferogram for a perfectly aligned TGA, leading to identical fringe pattern in all the three regions. The piston error leads to the vertical shift of the fringe pattern in the central region with respect to the side reference regions as shown in Fig. 3b. The tip angle error changes the fringe spacing in the central region of the interferogram, which decreases (Fig. 3(c)) or increases (Fig. 3(d)) depending upon the sign of the tip angle error. On the other hand, the tilt angle error rotates the fringes either in anti-clockwise direction (Fig. 3(e)) or in clockwise direction (Fig. 3(f)) depending upon the sign of the tilt angle error. However, the in-plane errors will not be having any effect on the fringe parameters of the 0th order interferogram. Next, shearing interferograms for the 1st order diffracted laser beam were also simulated using Eq. (6) and are shown in Fig. 4. In the diffracted beam’s interferograms, all six tiling errors will affect the fringe parameters in the central region of the interferogram. These six errors may be put into three groups of two each, according to their effect on the fringe pattern. The longitudinal piston error and the lateral shift error lead to shifting of the fringes, as can be seen in Figs. 4(a) and 4(b) respectively. The tip angle error and the in-plane angle error will change the fringe spacing (Fig. 4(c) & (d)); whereas the tilt angle error and the groove density mismatch will change the fringe slope (Fig. 4(e) & (f)). Hence, for measurement and distinguishing between the different tiling errors, first the 0th order shearing interferogram should be analyzed for the co-planar errors and then the in-plane errors should be estimated by the diffracted beam’s shearing interferogram.
Fig. 7. Experimentally observed (blue color asterisk symbol) and theoretically calculated (using Eq. (4) and shown as black line) variation of (a) fringe frequency with deliberately introduced tip angle error and (b) fringe slope with deliberately introduced tilt angle error in the TGA.
filtering to smoothen the laser beam profile. The collimated He–Ne laser beam was allowed to incident on the TGA in symmetric manner at an angle of incidence 𝛼 = 45◦ . The TGA reflects a part of the laser beam (0th order) perpendicular to the input beam (represented by the dotted line) and diffracts (-1st order) the remaining (represented by the dashed line) at diffraction angle of 𝛽 = 21.8◦ . In the experiment, the grating G1 was taken as fixed reference and the grating G2 was actuated using motorized six-axis alignment stage (model: 8095-M) from Newport having minimum translational and angular incremental step of less than 30 nm and 0.3 μrad respectively. Lateral shearing interferograms were generated for both 0th order reflected and 1st order diffracted laser beams using two identical wedge plates (each of diameter 150 mm, thickness 20 mm and wedge angle ∼ 18 arcsec), kept at an angle of incidence 𝜃i = 45◦ . The interferograms for the reflected and the diffracted laser beams were recorded using camera-1 and camera-2 respectively as shown in Fig. 5. These interferograms were further processed to estimate the translational and the angular misalignments between the two gratings of the TGA. Fig. 6 shows a typical recorded interferogram for the 0th order reflected laser beam. To find out the tiling alignment errors,
5. Testing of wedge plate shear interferometry: Results and analysis Experiments were carried out for the sub-wavelength precision tiling alignment of two nearly identical reflective gratings (size 140 mm × 120 mm and N = 1740 gr/mm) using the wedge plate lateral shearing interferometry. A schematic of the experimental setup is shown in Fig. 5 along with the typical interferogram recorded by two cameras. A He–Ne laser beam was expanded to ∼ 70 mm diameter, using a 20X microscope objective and an aberration corrected convex lens of focal length 550 mm. A 25 μm pinhole was used for spatial 5
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Optics Communications 459 (2020) 125067
Fig. 8. Step by step removal of co-planar errors (tilt, tip and piston) of TGA using reflected beam’s interferograms in the two selected ROIs (a) with tilt, tip and piston errors present, (b) with tip and piston errors present and tilt error removed, (c) with only piston error present and tip, tilt errors removed and (d) with nearly all the errors removed.
6. Conclusion
two regions of interest (ROI) were cropped in the left (region-1) and the central region (region-2) of the interferogram, as marked by two rectangles in Fig. 6(a). The two ROI interferograms were interpolated and smoothened by image processing, before further calculations. To estimate the spacing of the fringes in the two regions, the peaks of the fringes were located along vertical lines. The fringe peak positions along one such line is shown by the blue color asterisk symbol in Fig. 6(b). The tip angle error can be calculated from the fringe spacing in the two ROI interferograms using Eq. (4a). Similarly, to calculate tilt angle error using Eq. (4b), the fringes were fitted by straight lines (shown by red color line in Fig. 6(b)) to calculate the slope of the fringes in the two ROIs. To estimate the angular alignment sensitivities of the experimental setup, known amount of tip and tilt angular errors were introduced in the grating G2 using the motorized actuators. Fig. 7(a) shows a plot for the difference in fringe frequency (inverse of the fringe spacing) between the two selected ROIs with deliberately introduced tip angle error. For this, the grating G2 was actuated around the horizontal axis between ± 25 μrad with a step of ∼ 0.8μrad and fringe frequencies were calculated. The experimental values and theoretical values of the fringe frequency difference between the two ROIs nearly matches (Fig. 7a). Similarly, the tilt angle error was also introduced in the system with a step of ∼ 0.8 μrad and slope of the fringes was calculated. In this case also the experimental values found are nearly similar to the theoretically calculated values as shown in Fig. 7(b). Angular alignment sensitivity better than 1 μrad was achieved in the present experimental setup. These measurement accuracies can be improved further by using large size laser beam and using thicker shear plate with small wedge angle. Next, for coherent alignment of the TGA, the co-planar errors (tilt, tip and piston) were removed step by step using reflected beam’s interferogram, as shown in Fig. 8. First, the tilt angle error was removed by equalizing the fringe slopes in the reference and the central region (Fig. 8(b)). Then the fringe spacing was matched in the two regions for the removal of the tip angle error (Fig. 8(c)). After angular alignment for removal of tip and tilt, the grating G2 was translated along the normal of the grating plane to remove the remaining longitudinal piston error (Fig. 8(d)). The translation accuracy of better than 50 nm could be achieved by observing the fringe shift in the two ROIs. After removing co-planar errors, the diffracted beam’s interferogram (captured by the camera-2) was analyzed in similar manner to remove the in-plane errors. First of all, the fringe spacing in the central region of the diffracted beam’s interferogram (along Y-axis) was matched with the side reference region, to remove the in-plane angle error. Thereafter, the mismatch in the slopes of the fringes in the two regions, due to the groove density mismatch error, was compensated by tilting the grating G2 about 𝑌 -axis. Finally, the vertical shift of the horizontal fringe patterns in the two ROIs was corrected by translating G2 along 𝑋-axis to remove the lateral shift error. In this way, all the tiling alignment errors (both co-planar and in-plane errors) were identified and corrected to a precision better than 1μrad in the angle and 50 nm in the translation.
In conclusion, a simple, sensitive and robust technique, based on the wedge plate lateral shearing interferometry, has been proposed for the measurements of alignment errors present in a TGA. The technique has several advantages viz. easy to set up, insensitive to the vibrations and provides required measurement accuracies. Through simulations, effect of various tiling errors on the fringe parameters was investigated. Further, coherent tiling of two gratings was experimentally demonstrated and angular and translation sensitivities of less than 1 μrad and ∼ 50 nm respectively were achieved. These values can be further improved using larger size laser beam and thicker wedge plate with smaller wedge angle. The proposed technique could be beneficial for the online monitoring of the tiling alignment errors in a TGA based laser pulse compressor of high-energy high-power PW class CPA laser systems. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement D. Daiya: Conceptualization, Methodology, Software, Formal analysis, Investigation, Visualization, Writing - original draft, Writing - review & editing. R.K. Patidar: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. A. Moorti: Conceptualization, Supervision, Visualization, Writing - original draft, Writing review & editing. N.S. Benerji: Writing - original draft, Writing - review & editing. A.S. Joshi: Writing - original draft, Writing - review & editing. References [1] D. Strickl, G. Mourou, Opt. Commun. 56 (1985) 219. [2] C. Danson, D. Hillier, N. Hopps, D. Neely, High Power Laser Sci. Eng. 3 (2015) 1. [3] D.T. Reid, C.M. Heyl, R.R. Thomson, R. Trebino, G. Steinmeyer, H.H. Fielding, R. Holzwarth, Z. Zhang, P. DelHaye, T. Sudmeyer, G. Mourou, T. Tajima, D. Faccio, F.J.M. Harren, G. Cerullo, J. Opt. 18 (2016) 093006. [4] Opportunities in Intense Ultrafast: Reaching for the Brightest Light, The National Academic Press, ISBN 978-0-309-46769-8. http://dx.doi.org/10.17226/24939. [5] X. Zeng, K. Zhou, Y. Zuo, Q. Zhu, J. Su, X. Wang, X. Wang, X. Huang, X. Jiang, D. Jiang, Y. Guo, N. Xie, S. Zhou, Z. Wu, J. Mu, H. Peng, F. Jing, Opt. Lett. 42 (2017) 2014. [6] Y. Chu, Z. Gan, X. Liang, L. Yu, X. Lu, C. Wang, X. Wang, L. Xu, H. Lu, D. Yin, Y. Leng, R. Li, Z. Xu, Opt. Lett. 40 (2015) 5011. [7] J. Sung, H. Lee, J. Yoo, J. Yoon, C. Lee, J. Yang, Y. Son, Y. Jang, S. Lee, C. Nam, Opt. Lett. 42 (2017) 2058. [8] N. Bonod, J. Neauport, Adv. Opt. Photon. 8 (2016) 156. 6
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Simple wedge plate based lateral shearing interferometry technique for coherent alignment of tiled grating assembly D. Daiya a,b ,∗, R.K. Patidar a , A. Moorti a,b , N.S. Benerji a , A.S. Joshi a a b
Raja Ramanna Centre for Advanced Technology, Indore 452013, India HomiBhabha National Institute, Mumbai 400094, India
ARTICLE
INFO
Keywords: Tiled gratings Chirped pulse amplification
ABSTRACT A simple, sensitive and robust technique is proposed for coherent alignment of tiled grating assembly using wedge plate lateral shearing interferometry. In this technique, two thick glass wedge plates are used to generate interferograms from reflected and diffracted laser beams, from the tiled grating assembly under alignment. This single optical element based interferometric technique is easy to set up, insensitive to the vibrations and provides simultaneous information of both reference as well as error interference patterns, which are used to estimate the alignment errors. Performance of this technique was tested through alignment of two reflective gratings (size 140 mm X 120 mm), with an angular and translation sensitivities of less than 1 μrad and 50 nm respectively.
1. Introduction Demonstration of Chirped Pulse Amplification (CPA) technique [1] by G. Mourou and D. Strickland in the year 1985 has set up the roadmap for the development of high-energy, high-power ultra-intense laser sources to explore various regimes of science and technology via laser–matter interaction [2–4]. The quest for achieving higher and higher laser peak power has reached to the level of ∼ 5 PW and to intensities of the order of 1022 W/cm2 [5–7]. Efforts are being made to further increase these values which would depend on the technological advancement of crucial optical components. One such crucial component is large size diffraction grating, required for temporal compression of 100’s of mm diameter amplified laser pulses. Development of such large size monolithic gratings with required optical figure, broadband diffraction efficiency, high laser induced damage threshold, groove density uniformity etc. is quite challenging and therefore limits its availability [8]. Coherent optical tiling of small aperture optics provides an alternative way to achieve very large size optical surface with dimensions larger than a meter or even more. One such example is large aperture astronomical telescopes, in which the several meter size objective is made by segmenting number of small aperture mirrors in coherent manner [9]. Similarly, in high-energy high-power PW class CPA lasers, small size gratings are put together to mimic as a single large size monolithic grating, in order to accommodate large size amplified laser beam [10–15]. However, the relative angular and translational motion between the gratings of a tiled grating assembly (TGA) introduces phase errors in the beam, which causes distortions in the focal plane
intensity distribution [16,17]. Therefore, grating tiling requires precise angular and translational motorized movements along with simple, sensitive and robust diagnostics to identify and measure the various possible alignment errors. For this purpose, diagnostics based on either near field measurements (interferometry based) [18,19] or far field measurements (by observing focal spot distributions) [20,21] or combining both the measurements [22] are being used. The far field based method is quite simpler and it requires only a convex lens for the tiling error measurements. However, for small tiling alignment errors, the interferometric methods prove to be more sensitive and accurate and can also distinguish between the various angular and translational errors present in the TGA [18]. The interferometry setups generally used for this purpose are based on either Michelson or Fizeau, or Mach–Zehnder interferometers etc., which require careful alignment of number of optical elements like beam splitter, reference flat etc. Along with this, these interferometers are prone to vibrations, due to which online alignment monitoring of gratings in the TGA based pulse compressors is a difficult task. In this paper, we propose a simple and sensitive technique for coherent alignment of gratings in the TGA using wedge plate lateral shearing interferometry. In the proposed technique, two thick glass wedge plates are used to generate two interferograms from the reflected and diffracted laser beams by the TGA, for the measurement of all the associated tiling errors. The various errors were estimated by calculating the spacing, slope and the relative shift of the fringes from the recorded interferograms. The use of single optical element (i.e. wedge plate) makes this technique easy to set up and insensitive
∗ Corresponding author at: Raja Ramanna Centre for Advanced Technology, Indore 452013, India. E-mail address: [email protected] (D. Daiya).
https://doi.org/10.1016/j.optcom.2019.125067 Received 5 October 2019; Received in revised form 25 November 2019; Accepted 2 December 2019 Available online 6 December 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
D. Daiya, R.K. Patidar, A. Moorti et al.
Optics Communications 459 (2020) 125067
Fig. 1. Various co-planar and in-plane alignment errors of a tiled grating assembly having two gratings. Tip angle error (𝜃x ), tilt angle error (𝜃y ), in-plane angle error (𝜃z ), lateral shift error (𝛥x), longitudinal piston error (𝛥z) and groove spacing mismatch error (𝛥d = d−d′ ).
to the vibrations. This also provides interferometric level of accuracies in the measurements of tiling alignment errors. For experimental demonstration, a wedge plate lateral shearing interferometer was set up using 20 mm thick glass wedge plates having ∼ 18 arcsec wedge angle and coherent alignment of a TGA having two gratings of size 140 mm × 120 mm was carried out. A He–Ne laser beam of ∼ 70 mm diameter was used and lateral shearing interferograms were recorded for both reflected and diffracted beams from the TGA.
part reflected (0th order beam) from the misaligned grating G2, with respect to the fixed reference grating G1, can be written as [23,24]: ) ( 𝑋 − 𝜃𝑥 𝑌 + 𝛥𝑧} (1) 𝜑𝑟𝑒𝑓 𝑙𝑒𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) = 2𝑘 cos 𝛼{𝜃𝑦 cos 𝛼 Where, (X, Y, Z) defines the coordinates of the observation plane, with 𝑌 -axis parallel to grating grooves and 𝑍-axis represents the laser propagation direction. Similarly, the additional error phase introduced in the beam part diffracted by the misaligned grating G2 at a diffraction angle 𝛽 can be written as [23]: ( ) 𝑋 𝜑𝑑𝑖𝑓 𝑓 𝑟𝑎𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) = 𝑘{cos 𝛼 + cos 𝛽}{𝜃𝑦 − 𝜃𝑥 𝑌 + 𝛥𝑧} cos 𝛽 ( ) 𝑋 𝛥𝑑 𝜆 + 𝑘{ + 𝜃𝑧 𝑌 + 𝛥𝑥} (2) cos 𝛽 𝑑 𝑑
2. Tiling alignment errors of a TGA The various possible tiling alignment errors, which may be present in the TGA having two gratings G1 and G2, are represented in the Fig. 1 [17]. G1 is assumed to be the fixed reference grating having surface normal and grooves orientation parallel to 𝑧-axis and 𝑦-axis respectively. Misalignment of the TGA will occur if the grating G2 will have different angular and positional orientation with respect to the reference grating G1. The three angular misalignment errors of the grating G2 with respect to G1 are: (a) Tip angle error (𝜃x ) due to rotation around 𝑥-axis, (b) Tilt angle error (𝜃y ) due to rotation around 𝑦-axis and (c) In-plane angle error (𝜃z ) due to rotation around 𝑧-axis. The two translational errors of the grating G2 along 𝑥-axis and 𝑧-axis are known as lateral shift error (𝛥x) and longitudinal piston error (𝛥z) respectively, whereas translation along 𝑦-axis does not introduce any additional phase [17]. Beside these alignment errors, the difference in groove spacing (denoted by d and d′ in Fig. 1) of the two gratings also lead to additional error phase in the laser beam diffracted by the TGA. These six tiling errors are categorized into two groups: (a) coplanar or mirror like errors and (b) in-plane or grating like errors [23]. The tip, tilt and longitudinal piston errors are grouped into co-planar errors, as in their presence the two gratings will not be having same plane. The laser beam reflected from the TGA (i.e. 0th order beam) will be affected by these errors only. The remaining three tiling errors, i.e. in-plane angle, lateral shift and groove spacing mismatch errors are known as in-plane (grating like) errors, as they affect the diffracted beam along with the co-planar errors. Hence, in order to coherently align the gratings of a TGA, first the 0th order reflected laser beam is diagnosed to remove co-planar errors and afterwards the in-plane errors are removed by diagnosing the diffracted laser beam [23]. Now, if we consider a laser beam of diameter D falling symmetrically (half of the beam part on each grating) on the TGA at an angle of incidence 𝛼. Then the additional error phase introduced in the beam
Where 𝑘 = 2𝜋∕𝜆 is the propagation vector, 𝜆 is the laser wavelength and 𝛥d = d−d′ is the groove spacing mismatch between the two gratings. These equations can be used to simulate the effect of tiling alignment errors on the focal spot intensity distribution and the alignment tolerances can be estimated by putting limit on the peak intensity reduction due to these errors [17]. 3. Description of wedge plate lateral shearing interferometry for TGA alignment Wedge plate lateral shearing interferometry provides a very simple way to diagnose tiling errors present in the TGA with required precision and accuracy. A scheme of the technique is shown in Fig. 2. A diagnostic laser beam is allowed to incident symmetrically on the TGA having two gratings (G1 and G2) at an angle of incidence equal to 𝛼. The reflected/diffracted laser beam from the TGA falls on a thick glass wedge plate of thickness t at an angle of incidence 𝜃i . The glass plate creates two space sheared replicas (along √ X-axis) in the plane of incidence, with shear distance 𝑆 = 𝑡 sin 2𝜃𝑖 ∕ (𝑛2 − sin2 𝜃𝑖 ), through partial reflections from its front and back surface. The wedge plate also introduces angle √ between the two replicas (along Y-axis) by an amount
2𝛾 = 2𝜃𝑤𝑒𝑑𝑔𝑒 (𝑛2 − sin2 𝜃𝑖 ), where n and 𝜃wedge are the refractive index and wedge angle of the glass plate respectively. The overlap area of the two sheared beams (where the interference occurs) can be divided into three regions 1, 2 and 3, which are separated by the dark strips created by the intra-grating separation (see screen image in Fig. 2). In region 1 and 3, the beam part reflected from G1 (represented by red color)
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Optics Communications 459 (2020) 125067
Fig. 2. Schematic of the wedge plate lateral shearing interferometry for coherent alignment of gratings of a TGA. Interference pattern with different regions is also shown on the screen. Details are in the text. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Simulated wedge plate lateral shearing interferograms for the 0th order reflected beam from a TGA having (a) perfect alignment with no errors (b) longitudinal piston error (𝛥z) of 250 nm (c) tip angle error (𝜃x ) of 50 μrad and (d) −50 μrad respectively, (e) tilt angle error (𝜃y ) of 50 μrad and (f) −50 μrad respectively.
and G2 (represented by green color) respectively interferes with itself and results into identical fringe patterns, having straight line fringes with orientation along 𝑋-axis and spacing given by 𝜆/(2𝛾). However, in region 2 the beam part reflected from G1 and G2 will be interfering and if there is alignment error in the TGA; the fringe pattern will be different from its neighboring regions due to additional tiling error phase (given by Eqs. (1) and (2)) between the interfering beam parts. So, the fringe patterns in region 1 or 3 serve as the reference for the alignment and the additional reference flat is not required, like other interferometric techniques. For the 0th order reflected laser beam, the resultant phase difference between the two interfering beams in the central region can be written using Eq. (1) as:
𝑓 𝑟𝑖𝑛𝑔𝑒 𝑠𝑙𝑜𝑝𝑒 =
𝑓 𝑟𝑖𝑛𝑔𝑒 𝑝ℎ𝑎𝑠𝑒 𝑠ℎ𝑖𝑓 𝑡 = 2𝑘𝛥𝑧 cos 𝛼
𝛥𝛷𝑑𝑖𝑓 𝑓 𝑟𝑎𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) = 𝜑𝑑𝑖𝑓 𝑓 𝑟𝑎𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) − 2𝑘𝛾𝑌 ( ) 𝑋 = 𝑘{cos 𝛼 + cos 𝛽}{𝜃𝑦 − 𝜃𝑥 𝑌 + 𝛥𝑧} cos 𝛽 ( ) 𝑋 𝛥𝑑 𝜆 +𝑘{ + 𝜃𝑧 𝑌 + 𝛥𝑥} − 2𝑘𝛾𝑌 cos 𝛽 𝑑 𝑑
(3)
From Eq. (3) the fringe spacing along the 𝑦-axis, slope of the fringes and fringe phase shift in the central region can be written as: 𝑓 𝑟𝑖𝑛𝑔𝑒 𝑠𝑝𝑎𝑐𝑖𝑛𝑔 =
𝜆 2{𝜃𝑥 cos 𝛼 + 𝛾}
(4b) (4c)
From Eq. (4a) it is clear that by measuring fringe spacing in the central region, one can estimate the tip angle error (𝜃x ). After calculating 𝜃x , the tilt angle error (𝜃y ) can be calculated by measuring fringe slope and using Eq. (4b). Finally, the longitudinal piston error (𝛥z) can be calculated by measuring fringe shift in the central region with respect to nearby reference regions. Similarly, the resultant phase difference between the two interfering beams in the central region of the diffracted beam’s interferogram can be written using Eq. (2) as:
𝛥𝛷𝑟𝑒𝑓 𝑙𝑒𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) = 𝜑𝑟𝑒𝑓 𝑙𝑒𝑐𝑡𝑒𝑑 (𝑋, 𝑌 ) − 2𝑘𝛾𝑌 = 2𝑘[𝜃𝑦 𝑋 − {𝜃𝑥 cos 𝛼 + 𝛾}𝑌 + 𝛥𝑧 cos 𝛼]
𝜃𝑦 𝜃 𝑥 cos𝛼 + 𝛾
(5)
From Eq. (5), various fringe parameters in the diffracted beam’s interferogram can be expressed in terms of the tiling error parameters
(4a) 3
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Fig. 4. Simulated wedge plate lateral shearing interferograms for the 1st order diffracted laser beam from a TGA having (a) longitudinal piston error (𝛥z) of 250 nm, (b) lateral shift error (𝛥x) of 250 nm, (c) tip angle error (𝜃x ) of −50 μrad, (d) in-plane angle error (𝜃z ) of 50 μrad, (e) tilt angle error (𝜃y ) of 50 μrad and (f) groove density mismatch (𝛥N) error of 0.1 gr/mm.
Fig. 5. Schematic of the experimental setup for precision alignment of a TGA with two reflective gratings using wedge plate lateral shearing interferometry. Typical interferograms recorded for the reflected (camera-1) and diffracted (camera-2) beams are also shown.
Fig. 6. (a) A typical shear interferogram recorded for beam reflected from the misaligned TGA showing region −1, 2 and 3. The ROI (1 and 2) are marked with two yellow rectangles in the regions 1 and 2, which served as the reference and the error interferograms respectively (b) Expanded view of the two ROIs used for calculating the fringe spacing and the orientation.
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(similar to Eq. (4)) as: 𝜆 𝜃𝑥 (cos 𝛼 + cos 𝛽) + 2𝛾 − 𝜃𝑧 (𝜆∕𝑑) ( ) 𝜃𝑦 (1 + cos 𝛼∕ cos 𝛽) + cos𝜆 𝛽 𝛥𝑑 𝑑2
𝑓 𝑟𝑖𝑛𝑔𝑒 𝑠𝑝𝑎𝑐𝑖𝑛𝑔 =
(6a)
(6b) 𝜃 𝑥 ( cos 𝛼 + cos 𝛽) + 2𝛾 − 𝜃𝑧 (𝜆∕𝑑) 𝜆 𝑓 𝑟𝑖𝑛𝑔𝑒 𝑝ℎ𝑎𝑠𝑒 𝑠ℎ𝑖𝑓 𝑡 = 𝑘[𝛥𝑧(cos 𝛼 + cos 𝛽) + 𝛥𝑥] (6c) 𝑑 From Eq. (6), one may observe that the fringe parameters in the diffracted beam’s interferogram depend on all the six tiling errors. Therefore, after calculating the co-planar errors from the reflected beam’s interferogram using Eq. (4), remaining three tiling errors, i.e., the in-plane or grating like errors can be estimated from diffracted beam’s interferogram using Eq. (6). 𝑓 𝑟𝑖𝑛𝑔𝑒 𝑠𝑙𝑜𝑝𝑒 =
4. Simulation of the TGA errors on interferograms Wedge plate lateral shearing interferograms were simulated for the 0th order reflected beam from the TGA to investigate effect (given by Eq. (4)) of various tiling alignment errors on the fringe pattern. The TGA has two reflective gratings, each of size 140 mm × 120 mm and groove density (inverse of groove spacing) of N ∼ 1740 gr/mm. A He– Ne laser beam (𝜆 = 632.8 nm) of diameter 70 mm and 10th order super Gaussian profile falling at an angle of incidence of 𝛼 = 45◦ on the TGA is considered for the simulation. The wedge plate angle (𝜃wedge ) and lateral shear is kept as 18 arcsecs (along Y-axis) and 15 mm (along X-axis) respectively. The simulated interferograms for the longitudinal piston error of 250 nm, tip angle error of ±50 μrad and tilt angle error of ±50 μrad are shown in Fig. 3. The Fig. 3(a) represents the interferogram for a perfectly aligned TGA, leading to identical fringe pattern in all the three regions. The piston error leads to the vertical shift of the fringe pattern in the central region with respect to the side reference regions as shown in Fig. 3b. The tip angle error changes the fringe spacing in the central region of the interferogram, which decreases (Fig. 3(c)) or increases (Fig. 3(d)) depending upon the sign of the tip angle error. On the other hand, the tilt angle error rotates the fringes either in anti-clockwise direction (Fig. 3(e)) or in clockwise direction (Fig. 3(f)) depending upon the sign of the tilt angle error. However, the in-plane errors will not be having any effect on the fringe parameters of the 0th order interferogram. Next, shearing interferograms for the 1st order diffracted laser beam were also simulated using Eq. (6) and are shown in Fig. 4. In the diffracted beam’s interferograms, all six tiling errors will affect the fringe parameters in the central region of the interferogram. These six errors may be put into three groups of two each, according to their effect on the fringe pattern. The longitudinal piston error and the lateral shift error lead to shifting of the fringes, as can be seen in Figs. 4(a) and 4(b) respectively. The tip angle error and the in-plane angle error will change the fringe spacing (Fig. 4(c) & (d)); whereas the tilt angle error and the groove density mismatch will change the fringe slope (Fig. 4(e) & (f)). Hence, for measurement and distinguishing between the different tiling errors, first the 0th order shearing interferogram should be analyzed for the co-planar errors and then the in-plane errors should be estimated by the diffracted beam’s shearing interferogram.
Fig. 7. Experimentally observed (blue color asterisk symbol) and theoretically calculated (using Eq. (4) and shown as black line) variation of (a) fringe frequency with deliberately introduced tip angle error and (b) fringe slope with deliberately introduced tilt angle error in the TGA.
filtering to smoothen the laser beam profile. The collimated He–Ne laser beam was allowed to incident on the TGA in symmetric manner at an angle of incidence 𝛼 = 45◦ . The TGA reflects a part of the laser beam (0th order) perpendicular to the input beam (represented by the dotted line) and diffracts (-1st order) the remaining (represented by the dashed line) at diffraction angle of 𝛽 = 21.8◦ . In the experiment, the grating G1 was taken as fixed reference and the grating G2 was actuated using motorized six-axis alignment stage (model: 8095-M) from Newport having minimum translational and angular incremental step of less than 30 nm and 0.3 μrad respectively. Lateral shearing interferograms were generated for both 0th order reflected and 1st order diffracted laser beams using two identical wedge plates (each of diameter 150 mm, thickness 20 mm and wedge angle ∼ 18 arcsec), kept at an angle of incidence 𝜃i = 45◦ . The interferograms for the reflected and the diffracted laser beams were recorded using camera-1 and camera-2 respectively as shown in Fig. 5. These interferograms were further processed to estimate the translational and the angular misalignments between the two gratings of the TGA. Fig. 6 shows a typical recorded interferogram for the 0th order reflected laser beam. To find out the tiling alignment errors,
5. Testing of wedge plate shear interferometry: Results and analysis Experiments were carried out for the sub-wavelength precision tiling alignment of two nearly identical reflective gratings (size 140 mm × 120 mm and N = 1740 gr/mm) using the wedge plate lateral shearing interferometry. A schematic of the experimental setup is shown in Fig. 5 along with the typical interferogram recorded by two cameras. A He–Ne laser beam was expanded to ∼ 70 mm diameter, using a 20X microscope objective and an aberration corrected convex lens of focal length 550 mm. A 25 μm pinhole was used for spatial 5
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Optics Communications 459 (2020) 125067
Fig. 8. Step by step removal of co-planar errors (tilt, tip and piston) of TGA using reflected beam’s interferograms in the two selected ROIs (a) with tilt, tip and piston errors present, (b) with tip and piston errors present and tilt error removed, (c) with only piston error present and tip, tilt errors removed and (d) with nearly all the errors removed.
6. Conclusion
two regions of interest (ROI) were cropped in the left (region-1) and the central region (region-2) of the interferogram, as marked by two rectangles in Fig. 6(a). The two ROI interferograms were interpolated and smoothened by image processing, before further calculations. To estimate the spacing of the fringes in the two regions, the peaks of the fringes were located along vertical lines. The fringe peak positions along one such line is shown by the blue color asterisk symbol in Fig. 6(b). The tip angle error can be calculated from the fringe spacing in the two ROI interferograms using Eq. (4a). Similarly, to calculate tilt angle error using Eq. (4b), the fringes were fitted by straight lines (shown by red color line in Fig. 6(b)) to calculate the slope of the fringes in the two ROIs. To estimate the angular alignment sensitivities of the experimental setup, known amount of tip and tilt angular errors were introduced in the grating G2 using the motorized actuators. Fig. 7(a) shows a plot for the difference in fringe frequency (inverse of the fringe spacing) between the two selected ROIs with deliberately introduced tip angle error. For this, the grating G2 was actuated around the horizontal axis between ± 25 μrad with a step of ∼ 0.8μrad and fringe frequencies were calculated. The experimental values and theoretical values of the fringe frequency difference between the two ROIs nearly matches (Fig. 7a). Similarly, the tilt angle error was also introduced in the system with a step of ∼ 0.8 μrad and slope of the fringes was calculated. In this case also the experimental values found are nearly similar to the theoretically calculated values as shown in Fig. 7(b). Angular alignment sensitivity better than 1 μrad was achieved in the present experimental setup. These measurement accuracies can be improved further by using large size laser beam and using thicker shear plate with small wedge angle. Next, for coherent alignment of the TGA, the co-planar errors (tilt, tip and piston) were removed step by step using reflected beam’s interferogram, as shown in Fig. 8. First, the tilt angle error was removed by equalizing the fringe slopes in the reference and the central region (Fig. 8(b)). Then the fringe spacing was matched in the two regions for the removal of the tip angle error (Fig. 8(c)). After angular alignment for removal of tip and tilt, the grating G2 was translated along the normal of the grating plane to remove the remaining longitudinal piston error (Fig. 8(d)). The translation accuracy of better than 50 nm could be achieved by observing the fringe shift in the two ROIs. After removing co-planar errors, the diffracted beam’s interferogram (captured by the camera-2) was analyzed in similar manner to remove the in-plane errors. First of all, the fringe spacing in the central region of the diffracted beam’s interferogram (along Y-axis) was matched with the side reference region, to remove the in-plane angle error. Thereafter, the mismatch in the slopes of the fringes in the two regions, due to the groove density mismatch error, was compensated by tilting the grating G2 about 𝑌 -axis. Finally, the vertical shift of the horizontal fringe patterns in the two ROIs was corrected by translating G2 along 𝑋-axis to remove the lateral shift error. In this way, all the tiling alignment errors (both co-planar and in-plane errors) were identified and corrected to a precision better than 1μrad in the angle and 50 nm in the translation.
In conclusion, a simple, sensitive and robust technique, based on the wedge plate lateral shearing interferometry, has been proposed for the measurements of alignment errors present in a TGA. The technique has several advantages viz. easy to set up, insensitive to the vibrations and provides required measurement accuracies. Through simulations, effect of various tiling errors on the fringe parameters was investigated. Further, coherent tiling of two gratings was experimentally demonstrated and angular and translation sensitivities of less than 1 μrad and ∼ 50 nm respectively were achieved. These values can be further improved using larger size laser beam and thicker wedge plate with smaller wedge angle. The proposed technique could be beneficial for the online monitoring of the tiling alignment errors in a TGA based laser pulse compressor of high-energy high-power PW class CPA laser systems. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement D. Daiya: Conceptualization, Methodology, Software, Formal analysis, Investigation, Visualization, Writing - original draft, Writing - review & editing. R.K. Patidar: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. A. Moorti: Conceptualization, Supervision, Visualization, Writing - original draft, Writing review & editing. N.S. Benerji: Writing - original draft, Writing - review & editing. A.S. Joshi: Writing - original draft, Writing - review & editing. References [1] D. Strickl, G. Mourou, Opt. Commun. 56 (1985) 219. [2] C. Danson, D. Hillier, N. Hopps, D. Neely, High Power Laser Sci. Eng. 3 (2015) 1. [3] D.T. Reid, C.M. Heyl, R.R. Thomson, R. Trebino, G. Steinmeyer, H.H. Fielding, R. Holzwarth, Z. Zhang, P. DelHaye, T. Sudmeyer, G. Mourou, T. Tajima, D. Faccio, F.J.M. Harren, G. Cerullo, J. Opt. 18 (2016) 093006. [4] Opportunities in Intense Ultrafast: Reaching for the Brightest Light, The National Academic Press, ISBN 978-0-309-46769-8. http://dx.doi.org/10.17226/24939. [5] X. Zeng, K. Zhou, Y. Zuo, Q. Zhu, J. Su, X. Wang, X. Wang, X. Huang, X. Jiang, D. Jiang, Y. Guo, N. Xie, S. Zhou, Z. Wu, J. Mu, H. Peng, F. Jing, Opt. Lett. 42 (2017) 2014. [6] Y. Chu, Z. Gan, X. Liang, L. Yu, X. Lu, C. Wang, X. Wang, L. Xu, H. Lu, D. Yin, Y. Leng, R. Li, Z. Xu, Opt. Lett. 40 (2015) 5011. [7] J. Sung, H. Lee, J. Yoo, J. Yoon, C. Lee, J. Yang, Y. Son, Y. Jang, S. Lee, C. Nam, Opt. Lett. 42 (2017) 2058. [8] N. Bonod, J. Neauport, Adv. Opt. Photon. 8 (2016) 156. 6
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