Investigation of spatio-temporal stretching in a duplex grating compressor

Yifei Chen; Yifei Chen; Yifei Chen; Cheng Wang; Cheng Wang; Zongxin Zhang; Xiaojun Yang; Yi Xu; Yuxin Leng; Yuxin Leng; Zhizhan Xu; Zhizhan Xu

doi:10.1364/OE.27.031667

1. Introduction

The invention of chirped pulse amplification (CPA) technique [1] and optical parametric chirped-pulse amplification (OPCPA) [2] enable laser pulses to be amplified safely by stretching pulses before the main amplifier. In the past decade, the final compressed peak power has been increased from hundreds of terawatt to multi-petawatt in ultra-high intensity laser facilities utilizing CPA and OPCPA [3–6]. More lasers are in the design phase or under construction for 10 PW worldwide, such as ELI [7], SULF [8], Vulcan-10PW [9], and Apollon-10PW [10]. The grating compressor is one of the key components in a CPA system which is inevitably exposed to the highest optical load for final high peak power pulsed laser output [11]. Gratings with larger aperture are commonly required in a CPA system in order to reduce the power density on the grating surface. Since the manufacture of large diffraction gratings is difficult and costly, alternative solutions have been proposed to improve the capacity of a compressor by relatively small-scale gratings. Tiled grating technique [12] allows sub-gratings to be tiled to form a large aperture grating; Bidirectional grating compressor (BGC) [13] presented a duplexing of the whole compressor in different directions; Object-image-grating self-tiling (OIGST) [14] uses a virtual image in the mirror to enlarge the effective grating aperture. Both OIGST and BGC can be regarded as duplex grating compressor (DGC).

In practice, spatio-temporal couplings (STCs) play a vital role in the transport and compression of an ultrashort pulsed beam. Recently, STCs act as a new degree of freedom for the control over the size, position, and movement of beam focus [15–18]. Although STCs can be useful in the optimization of nonlinear processes [19] and the generation of an attosecond pulse [20, 21], those couplings can lead to a strong reduction of intensity on the target in high power laser system [22]. Researchers are quite familiar with the pulse front tilt and angular dispersion caused by misaligned stretchers or compressors. Efforts have been made to analyze their influence on temporal change of pulses both theoretically and experimentally [23–27].

However, the spatio-temporal evolution of a pulse in DGC is more complicated than the pulse in a classical grating compressor because DGC commonly requires an inclination angle of incidence (also called out-of-plane tilt angle [28] upward/downward angle [29] or off-horizontal angle [30]) to avoid the back-reflection and to output the compressed pulse. In such condition, the evolution of an input pulse cannot be simplified as dispersing within the diffraction plane. Thus, application of DGC deserves careful investigation toward STCs, especially those stretching the compressed pulse duration.

The general theory of first-order STCs of Gaussian pulses and beams [31] provides complete definitions for first-order STCs, as well as a simple method to quantify them. The mathematical approach has a close relationship with ray-pulse matrices [32] because matrix formalism is convenient in dealing with the propagation of electromagnetic radiation in linear optical elements. In the past decades, several kinds of matrix approaches have been introduced to describe the beam propagation in pulse compressors and dispersive systems [33–36]. Kostenbauder introduced 4×4 ray-pulse matrices to model dispersive elements [32], which were further extended to 6×6 transfer matrix by Lin to analyze STCs in a grating-pair compressor [37]. Matrix treatments were not limited in describing Gaussian pulse. Yao generalized the formalism to describe random electromagnetic pulsed beam-like radiation interacting with linear optical elements [38]. Although 6×6 transfer matrix is capable in dealing with the evolution in both transverse coordinates, such transfer matrix has never been applied to model the 3D spatio-temporal evolution of a pulse in the grating-based system.

In this work, we modified the 6×6 matrix formalism to describe diffraction of a reflecting grating with arbitrary orientation and deal with general incident conditions. Matrix approach is applied to quantify STCs of compressed pulses in DGC. We find that the inclination angle in DGC leads to pulse front tilt and spatial chirp, which can stretch the output pulse. These calculations reveal the relationship of STCs with the attitude and position of gratings in DGC. Our approach can be used to guide the application and optimization of DGC in a high power laser system.

2. Matrix formalism

2.1. Definition of transfer matrix and generalized ABCD law

Before deriving the transfer matrix of grating with general incident conditions, we may review the definition of 6×6 transfer matrix and generalized ABCD law [37]. We define the beam path of the central frequency as the optical axis and construct the transverse coordinate at the input and output of a dispersive element. The output pulse parameters (subscript 2) of the dispersive element have the following relationship with the input pulse parameters (subscript 1):

(1)$$\left( \begin{array}{c} {x_2}\\ {y_2}\\ c({t_2} - {t_0})\\ {n_2}x_2^{\prime}\\ {n_2}y_2^{\prime}\\ ({\nu_0} - {\nu_2})/{\nu_0} \end{array} \right) = \left( {\begin{array}{cccccc} {}&{}&{}&{}&{}&{}\\ {}&{\textbf A}&{}&{}&{\textbf B}&{}\\ {}&{}&{}&{}&{}&{}\\ {}&{}&{}&{}&{}&{}\\ {}&{\textbf C}&{}&{}&{\textbf D}&{}\\ {}&{}&{}&{}&{}&{} \end{array}} \right)\left( \begin{array}{c} {x_1}\\ {y_1}\\ c({t_1} - {t_0})\\ {n_1}x_1^{\prime}\\ {n_1}y_1^{\prime}\\ ({\nu_0} - {\nu_1})/{\nu_0} \end{array} \right),$$

where c is the velocity of light in vacuum. x, $y$ and $c(t - {t_0})$ represent transverse and longitude positions. $x^{\prime}$ and $y^{\prime}$ are deviation angle with respect to the reference optical axis, n is the refractive index. ${\nu _0}$ is the reference frequency. ${t_0}$ is the arrival time of a certain transverse plane. t and $\nu $ are the arrival time and frequency of the ray. The matrix in Eq. (1) is defined as the transfer matrix of such a dispersive element. For example, the matrix describing free space propagation with path length x has the form:

(2)$${\textbf F}(x) = \left( {\begin{array}{cc} {{{\textbf I}_{3 \times 3}}}&{{\textbf B}(x)}\\ {{{\textbf 0}_{3 \times 3}}}&{{{\textbf I}_{3 \times 3}}} \end{array}} \right),\textrm{ where }{\textbf B}(x) = \left( {\begin{array}{ccc} x&0&0\\ 0&x&0\\ 0&0&0 \end{array}} \right).$$

Once the transfer matrix is given, the spatial and temporal amplitude distribution of Gaussian beam propagating through this system can be calculated by matrix formalism. The spatio-temporal electric field of a Gaussian pulse can be written as following matrix form:

(3)$$\textrm{E}(x,y,t) \propto \exp \left\{ { - i\frac{\pi }{{{\lambda_0}}}{{\left( {\begin{array}{c} x\\ y\\ {c(t - {t_0})} \end{array}} \right)}^T}{\textbf Q}_1^{{\textbf - }1}\left( {\begin{array}{c} x\\ y\\ {c(t - {t_0})} \end{array}} \right)} \right\},$$

where ${\textbf Q}_1^{{\textbf - }1}$ is a 3×3 matrix called complex curvature tensor. The pulse parameters at the output plane of the dispersive system satisfy the generalized ABCD propagation law [37]:

(4)$${\textbf Q}_2^{ - 1}{ = ({\mathbf{C}} + {\mathbf{DQ}}}_1^{ - 1}{)({\mathbf{A}} + {\mathbf{BQ}}}_1^{ - 1}{{)}^{ - 1}}.$$

2.2. Transfer matrix of arbitrarily oriented grating

To determine the transfer matrix of a grating, we construct a reference optical axis corresponding to the path of the central frequency. Then we define a grating coordinate system ${\hat{x}_g},{\hat{y}_g},{\hat{z}_g}$ and place the origin at the intersection of the optical axis and grating plane. Diffraction plane is defined as ${\hat{x}_g} - {\hat{z}_g}$ plane. We define local beam coordinate (${\hat{x}_1},{\hat{y}_1},{\hat{z}_1}$ for incident beam and ${\hat{x}_2},{\hat{y}_2},{\hat{z}_2}$ for diffracted beam) by constructing the transverse coordinate perpendicular to the reference axis and keeping ${\hat{x}_1},{\hat{x}_2}$ parallel to the diffraction plane. As is shown in Fig. 1, the orientation of a grating with respect to the incident optical axis is determined by $\alpha $ and $\theta $ [26], where $\alpha $ is the inclination angle, $\theta $ is the in-plane incident angle.

Fig. 1. Generic case of diffraction on a grating. The plane shown in light green is the diffraction plane. Reference coordinate systems are constructed to derive the transfer matrix.

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The in-plane diffraction angle $\phi $ for the central frequency ${\nu _0}$ satisfies the grating equation:

(5)$$\sin \phi + \sin \theta = \frac{{m{\lambda _0}}}{{d\cos \alpha }},$$

where m is the diffraction order, d is the groove spacing, ${\lambda _0}\textrm{ = }{c \mathord{\left/ {\vphantom {c {{\nu_0}}}} \right.} {{\nu _0}}}$ is the wavelength of the central frequency. The transfer matrix of the arbitrarily orientated grating is then determined by three variables:

(6)$${\textbf G}(\theta ,\phi ,\alpha ) = \left( {\begin{array}{cc} {{{\textbf A}_G}}&{{{\textbf {B}}_G}}\\ {{{\textbf C}_G}}&{{{\textbf D}_G}} \end{array}} \right).$$

To derive the submatrix ${{\textbf A}_G}$ and ${{\textbf C}_G}$, one may consider a ray parallel to the chief ray but with spatial offset $({{x_1},{y_1}} )$ in the incident beam coordinate, the intersection of such ray and the grating will be $({{x_1}/\cos \theta ,{y_1}/\cos \alpha - {x_1}\tan \alpha \tan \theta } )$ in the grating coordinate. The central frequency is then diffracted to $({ - {x_1}\cos \phi /\cos \theta ,{y_1} - {x_1}\sin \alpha (\sin \phi + \sin \theta )/\cos \theta } )$ in the output beam coordinate. In addition, the path length is ${x_1}\cos \alpha (\sin \theta + \sin \phi )/\cos \theta$ longer than the chief ray and provides extra time delay. Likewise, an angularly deviated ray can help derive the submatrix ${{\textbf {B}}_G}$ and ${{\textbf D}_G}$. In general, submatrices will have the following form:

(7)$$\begin{aligned}{{\textbf A}_G} &= \left({\begin{array}{ccc} { - \cos \phi /\cos \theta }&0&0\\ { - \sin \alpha (\sin \phi + \sin \theta )/\cos \theta }&1&0\\ {\cos \alpha (\sin \theta + \sin \phi )/\cos \theta }&0&1 \end{array}} \right),\quad {{\textbf B}_G} = {{\textbf C}_G} = 0,\\ {{\textbf D}_G} &= \left( {\begin{array}{ccc} { - \frac{{\cos \theta }}{{\cos \phi }}}&{ - \frac{{\sin \alpha (\sin \phi + \sin \theta )}}{{\cos \phi }}}&{\frac{{\cos \alpha (\sin \phi + \sin \theta )}}{{\cos \phi }}}\\ 0&1&0\\ 0&0&1 \end{array}} \right). \end{aligned}$$

The existence of inclination angle $\alpha $ makes the spatio-temporal evolution more complex than ideal incident condition. With the help of generalized transfer matrix, we can analyze the STCs in grating-based systems without confining the incident wave vector in the diffraction plane.

2.3. Transfer matrix of a duplex grating compressor

Let us consider a simple DGC setup in which the pulsed beam with an inclination angle $\alpha $ is sent into a parallel grating compressor. Grating grooves are set along the vertical direction. The transfer matrix of such compressor shown in Fig. 2 can be obtained by the ordered product of free propagation matrices and gratings’ transfer matrices:

(8)$${\textbf M} ={{\textbf G}_4}{\textbf F}({b_2}){{\textbf G}_3}{\textbf F}(L) {{\textbf G}_2}{\textbf F}({b_1}){{\textbf G}_1},$$

where ${b_1} = {b^{\prime}_1}/(\cos \phi \cos \alpha )$ are the propagation path length between G₁ and G₂,$L = L^{\prime}/\cos \alpha$ are the path length between G₂ and G₃. ${\textbf F}(L)$ means the propagation matrix with path length L. The transfer matrix of the compressor shown in Fig. 2 will be quite lengthy, our MATLAB function [39] can help derive an analytical expression for such a generic condition.

Fig. 2. Symbol stipulations and definitions of our matrix formalism in a parallel grating system. The inclination angle$\alpha $ is omitted since the system is shown in top view. ${b^{\prime}_1},{b^{\prime}_2}$ and $L^{\prime}$ are distances measured in the horizontal plane.

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In order to simplify our further analysis, consider the compressor with mirror symmetry where ${\theta _\textrm{1}}\textrm{ = }{\theta _\textrm{2}}\textrm{ = }\theta {\phi _\textrm{1}}\textrm{ = }{\phi _\textrm{2}}\textrm{ = }\phi$ and ${b_1} = {b_2} = b$. Each submatrix in ${\textbf M}$ is found by substituting Eq. (2) and Eq. (6) into Eq. (8)

(9)$${{\textbf A}_{\textbf M}}{\textbf = }{{\textbf D}_{\textbf M}}{\textbf = }\left( {\begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right), {{\textbf B}_{\textbf M}} = \left( {\begin{array}{ccc} {{B_{11}}}&0&0\\ 0&{{B_{22}}}&{{B_{23}}}\\ 0&{{B_{32}}}&{{B_{33}}} \end{array}} \right), {{\textbf C}_{\textbf M}}{\textbf = }\left( {\begin{array}{ccc} 0&0&0\\ 0&0&0\\ 0&0&0 \end{array}} \right),$$

where

(10)$$\begin{aligned}{B_{11}} &= (L{{\cos }^2}\phi + 2b{{\cos }^2}\theta )/{{\cos }^2}\phi\\ {B_{22}} &= L + 2b + [2b{{(\sin \phi + \sin \theta )}^2}{{\sin }^2}\alpha /{{\cos }^2}\phi ]\\ {B_{33}} &= 2b{{(\sin \phi + \sin \theta )}^2}{{\cos }^2}\alpha /{{\cos }^2}\phi\\ {B_{23}}&= {B_{32}} ={-} [b{{(\sin \phi + \sin \theta )}^2}\sin 2\alpha /{{\cos }^2}\phi].\end{aligned}$$

Assuming the input pulse is free of spatio-temporal coupling, only diagonal terms exist in the complex curvature tensor. By expressing the incident field as ${\textbf Q}_1^{ - 1} = diag(Q_{xx}^{ - 1},Q_{yy}^{ - 1},Q_{tt}^{ - 1})$, the field at the output of the compressor is found by substituting Eq. (10) into Eq. (4):

(11)$${\textbf Q}_{\textbf 2}^{{\textbf - 1}}{\textbf = (}{{\textbf Q}_{\textbf 1}}{\textbf + }{{\textbf B}_{\textbf M}}{{\textbf )}^{{\textbf - 1}}} = {\left( {\begin{array}{ccc} {{Q_{xx}} + {B_{11}}}&0&0\\ 0&{{Q_{yy}} + {B_{22}}}&{{B_{23}}}\\ 0&{{B_{32}}}&{{Q_{tt}} + {B_{33}}} \end{array}} \right)^{ - 1}},$$

where ${B_{23}}$ and ${B_{32}}$ yields a coupling term in the $y - t$ domain. The expression suggests that, when the pulse comes in with inclination, STCs will arise in the vertical direction.

3. Result and discussion

In this section, we calculated a chirped Gaussian pulse propagating in an ideal DGC as an example to help reveal STCs and their contribution to the compressed pulse duration. A simple method is also provided to compress the pulse to its transform-limited pulse duration. Our simulation based on a 15-mm radius, 25-fs transform-limited pulsed Gaussian beam centered at 800 nm which is stretched in an all-reflective Öffner type stretcher. The incident angle, the equivalent perpendicular distance of grating pair and the grating density are 52°, 985 mm and 1480 lines mm^-1 respectively. The electric field of the stretched pulse is given by:

(12)$${\textbf Q}_i^{{\textbf - }1} = \left( {\begin{array}{ccc} {{ - 1}{.13} \times {1}{{0}^{ - 3}}i}&0&0\\ 0&{{ - 1}{.13} \times {1}{{0}^{ - 3}}i}&0\\ 0&0&{{ - 0}{.28 - 1}{.253} \times {1}{{0}^{ - 5}}i} \end{array}} \right)$$

The stretched pulse can be compressed to its original duration by a stretcher-matched grating compressor. In our calculation, the pulse is sent to such compressor with an inclination of 5°, so that we can analyze the influence of inclination angle on the compressed pulse. The compressed electric field has the following form:

(13)$${\textbf Q}_o^{{\textbf - }1} = \left( {\begin{array}{ccc} {{2}{.53} \times {1}{{0}^{ - 6}}{ - 1}{.13} \times {1}{{0}^{ - 3}}i}&0&0\\ 0&{{6}{.11} \times {1}{{0}^{ - 7}}{ - 1}{.13} \times {1}{{0}^{ - 3}}i}&{{ - 3}{.7} \times {1}{{0}^{ - 5}}{ - 9}{.72} \times {1}{{0}^{ - 3}}i}\\ 0&{{ - 3}{.7} \times {1}{{0}^{ - 5}}{ - 9}{.72} \times {1}{{0}^{ - 3}}i}&{{ 27}{.252 - 0}{.202}i} \end{array}} \right)$$

As is mentioned in Eq.11, the inclination angle changes not only the dispersion in the compressor but also brings coupling terms into the pulse. The temporal duration of such pulse with STCs can be decomposed into four parts [40]: the stretched duration with respect to its Fourier limit is dominated by spectral phase over the whole spatial extent, amplitude coupling in the space-spectral domain and phase coupling in the space-spectral domain. Therefore, DGC setups must provide approaches to minimize these stretching effects:

(i) Global spectral phase: Group Delay Dispersion (GDD) and Third Order Dispersion (TOD). The global spectral phase is the global synchronization of the spectral components over the whole beam section. It is not a kind of STCs, but its compensation is quite crucial in designing a pulse compressor. For the standard grating pair compressor, the angle of incidence and the distance between gratings act as two degrees of freedoms and enables control over GDD and TOD [11]. In DGC setups, the same procedure can be applied [28]. Figure 3(a) shows the residual GDD when continuously adjusting the position and incident angle of the gratings in DGC. Dash line in Fig. 3(a) gives the incident angle and corresponding distance which compensate the TOD. The intersection point provides compensation of GDD and TOD simultaneously. From now on, our analysis will be confined nearby this point. The third order and higher order dispersion will be neglected.
(ii) Amplitude coupling in the space-spectral domain: Spatial Chirp (SPC). From a geometrical perspective, the inclination angle, as well as the wavelength-dependent path length in DGC, can be regarded as the source of SPC. Since the wave vector has a vertical component, it is obvious that wavelength-dependent path length will cause vertical displacement [30]. Typically, the SPC should be normalized by the space and spectral width because the stretched duration depends on the spectral narrowing at the upper and lower edge of the beam’s cross section. Here in Fig. 3(b), SPC is normalized by the square root of both space and spectral width to provide a dimensionless coefficient. As shown in Fig. 3(b), the increased beam radius will minimize such spectral narrowing effect. The compressed pulse duration advances gradually to its transform-limited duration. For petawatt laser systems with meter-scale aperture, the stretching effect caused by SPC is negligible.
(iii) Phase coupling in the space-spectral domain: Wave-front-tilt dispersion (WFD). In our calculation, the beam is well collimated and is free of angular dispersion. Therefore, we do not find significant WFD. Since the pulse width converges towards its transform-limited duration with enlarged beam size, the same conclusion can be directly derived by duration decomposing that the pulse is barely affected by phase coupling in the space-spectral domain.
(iv) Another distinct coupling: Pulse-front tilt (PFT). Here in our calculation, when the grating distance slightly deviates from the optimum value (marked ‘A’ in Fig. 3 and Fig. 4), we found PFT to be raised. PFT is a well-known amplitude coupling in the spatial-temporal domain. It is usually observed in the system with angular dispersion [41] and is regarded as a fundamental coupling which stretches a pulse duration. However, PFT in DGC is generated by simultaneous temporal and spatial chirp [42]. Figure 4 shows the PFT normalized by the square root of space and temporal width and corresponding GDD when a 100 mm diameter pulse is sent in DGC. Following the decomposing procedure, we found that the stretching effect is already included in the global spectral phase mentioned above.

Fig. 3. (a) Residual GDD calculated in various incident angle and grating distance. Dash line shows parameter combinations for TOD compensation. The point marked ‘A’ denotes the optimized situation when both GDD and TOD are compensated. (b) Calculated radius-dependence of compressed pulse duration and normalized spatial chirp.

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Fig. 4. Normalized pulse-front tilt in output pulse. The blue line shows corresponding GDD. The point marked ‘A’ is the optimal point, as is shown in Fig. 3(a).

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4. Conclusion

A generalized transfer matrix is derived for reflective grating. The matrix formalism is applied to investigate the first-order STCs implicated in DGC setups. We have demonstrated that SPC caused by inclination angle $\alpha $ will not seriously degrade the pulse duration in high power lasers which have huge beam apertures. Fine tuning of the grating distance is necessary to eliminate PFT and residual GDD in output pulse. When the gratings are set properly, DGC is expected to improve the output power by small gratings without inducing fatal STCs.

Funding

Strategic Priority Research Program of the Chinese Academy of Sciences (CAS) (XDB16030100); International S&T Cooperation Program of China (2016YFE0119300); Program of Shanghai Academic Research Leader (18XD1404200); Shanghai Municipal Science and Technology Major Project (2017SHZDZX02).

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Investigation of spatio-temporal stretching in a duplex grating compressor

Abstract

1. Introduction

2. Matrix formalism

2.1. Definition of transfer matrix and generalized ABCD law

2.2. Transfer matrix of arbitrarily oriented grating

2.3. Transfer matrix of a duplex grating compressor

3. Result and discussion

4. Conclusion

Funding

References

Cited By

Figures (4)

Equations (13)

Optics Express