Investigation of combining-efficiency loss induced by a diffractive optical element in a single-aperture coherent beam combining system

Meizhong Liu; Meizhong Liu; Hui Shen; Yifeng Yang; Yifeng Yang; Yuqiao Xian; Yuqiao Xian; Jingpu Zhang; Jingpu Zhang; Hanbin Wang; Hanbin Wang; Binglin Li; Binglin Li; Xiaxia Niu; Xiaxia Niu; Bing He; Bing He

doi:10.1364/OE.413459

1. Introduction

Coherent beam combining (CBC) is a promising approach to achieve high brightness because this approach can inherently maintain the spectral bandwidth and coherence of the source [1]. Furthermore, as a diffractive element with high diffraction efficiency and excellent intensity uniformity with respect to the generated beamlets, a diffractive optical element (DOE) was extensively used as the beam combiner in CBC systems [1–8]. One-dimensional or two-dimensional DOEs have been used to obtain multi-kilowatt, high-combining-efficiency, and near-diffraction-limited laser outputs [2–4]. In these DOE-based CBC systems, a fiber array was directed to overlap with the DOE at angles similar to the diffractive order to perform the filled-aperture geometry. The filled-aperture geometry is crucial to eliminate the far-field sidelobes and reduce the combining losses when compared with those associated with the tiled-aperture combining scheme [2,9]. However, thermal expansion of the fiber array, imperfected DOE fabrication, and optical-element mount errors lead to beam deviation of beamlets (contained pointing error and position error), which eliminate the common-aperture condition and decrease the combining efficiency consequently [10].

In decades, beam combining efficiency of CBC system influenced by some parameters of fiber array such as pointing error, piston phase error, wavefront error, polarization error, path mismatch error was studied in detail [11,12]. Based on these discussions, the combining-efficiency losses induced by fiber array can be calculated in the actual CBC systems [1,10,13–16]. In addition, DOE is another important factor that causes losses of beam combining efficiency in DOE-based CBC system. For systems where the fiber array has been matched, the mismatch of the DOE induces the pointing error and the position error of the diffracted beams, which consequently reduces the power in the common aperture. More importantly, the period of the DOE may amplify these mismatches by affecting the sensitivity of DOE to the pointing error. Therefore, it is necessary to develop a theoretical model to analyze the combining-efficiency losses caused by the DOE-mismatch-induced beam deviation. However, the research on the combining-efficiency loss with DOE mismatch is sparse.

In this study, we proposed a theoretical beam propagation model to investigate the combining efficiency with beam deviation in the DOE-based CBC architecture. Using the developed theoretical model, the combining-efficiency losses associated with the DOE-mount-tilt error, emitter-incident angular error, and DOE-groove-tilt error are discussed in this study. The effect of the emitter-incident angular error on the combining-efficiency loss is analyzed by applying a random angular perturbation. Particularly, the combining-efficiency loss when considering the DOE-groove-tilt error is analyzed by expanding the theoretical model to a two-dimensional situation. Eleven array emitters are used to represent the general beam-combining case. The DOE-mount-tilt error, emitter-incident angular error, and DOE-groove-tilt error are set in the intervals of [0, 1 mrad], [0, 1 mrad], and [0, 0.2 mrad] during the simulations, respectively. The combining-efficiency losses gradually increase to 14.34%, 8.58%, and 36.41%, respectively, with the increasing error angle. To verify the proposed theory, we conducted a proof-of-principle experiment with respect to the beam deviation, which conformed to the theoretical prediction. The quantitative discussion and analysis of the combining-efficiency loss are potentially valuable for designing practical CBC systems to maximize the combining efficiency.

2. Theoretical model

In this study, a theoretical model is established to analyze the combining-efficiency loss caused by DOE. The theoretical model is divided into two parts, i.e., analysis of the beam deviation induced by DOE and the combining-efficiency loss.

2.1 Beam deviation

The simplified model diagram of the DOE-based diffraction is shown in Fig. 1. We consider the DOE to include an x₀-dependent DOE surface-relief layer and a silicon-based substrate with thickness T. The pointing error and position error associated with the diffracted beams can be induced by the angular error of the incident beam and the tilt angle of DOE. Therefore, an incident wave with an initial incident angle and DOE with a tilt angle are considered. The optical axis is perpendicular to the incident plane. In addition, the observation plane is perpendicular to the optical axis during the evolution of the aforementioned angles.

Fig. 1. Schematic of the simplified model and beam propagation process for the DOE-based diffraction; the positive angular direction is assumed to be clockwise.

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We consider the incident laser field to be a Gaussian beam having a waist width of ω₀ at a distance of L₀ and an angle of α₀. A tilt angle of β can be observed between the DOE and the optical axis. The field distribution of an initial Gaussian beam can be written as follows [17]:

(1)$${E_m}(x^{\prime\prime},z^{\prime\prime}) = {E_0}\frac{{{\omega _0}}}{{\omega (z^{\prime\prime})}}\exp \left[ { - \frac{{{{x^{\prime\prime}}^2}}}{{{\omega^2}(z^{\prime\prime})}}} \right]\exp ({ikz^{\prime\prime}} ),$$

where ω(z') represents the beam size and k = 2π/λ. Based on the Cartesian coordinate translation and rotation [18], the field distribution on the front surface of the DOE can be written as

(2)$${U_0}({x_0},z) = g({x_0},z)\exp ({ik{x_0}\sin {\alpha_m}} )\quad \textrm{and}$$

(3)$$\begin{array}{c} g({x_0},z) = {E_0}\frac{{{\omega _0}}}{{\omega ({x_0}\sin {\alpha _m} + z\cos {\alpha _m}\textrm{ + }{L_m})}}\exp \left[ { - \frac{{{{({{x_0}\cos {\alpha_m} - z\sin {\alpha_m}} )}^2}}}{{{\omega^2}({x_0}\sin {\alpha_m} + z\cos {\alpha_m}\textrm{ + }{L_m})}}} \right]\\ \cdot \exp [{ik({z\cos {\alpha_m}\textrm{ + }{L_m}} )} ], \end{array}$$

where α_m = α₀ − β and L_m = L₀/cos |α_m|. To simplify the discussion, the DOE is assumed to be a lossless and pure-phase element with period Λ. The transmission function of DOE is considered to be the phase function exhibiting a periodic distribution along the x₀-direction. The transmission function of the DOE with a beam-splitting ratio of 2N + 1 can be expressed as a complex Fourier series [19].

(4)$$t({x_\textrm{0}}) = \sum\limits_{j ={-} N}^N {{D_j}\exp (ik{x_\textrm{0}}\frac{{j\lambda }}{\Lambda })} + \phi ({x_\textrm{0}}),$$

where D_j is a coefficient in the Fourier series and ϕ(x₀) is the higher-order Fourier series. Thus, the field distribution before diffraction can be expressed as

(5)$${U^{\prime}_0}({x_0},z) = g({x_0},z)\exp ({ik{x_0}\sin {\alpha_m}} )t({x_0}).$$

According to the scalar diffraction theory [17], the diffraction field can be written as

(6)$$U(x,z) = \frac{1}{{i\lambda z}}\exp (ikz)\exp \left( {ik\frac{{{x^\textrm{2}}}}{{2z}}} \right)G({{f_x}} )\cdot \sum\limits_{j ={-} N}^N {{D_j}\delta \left[ {{f_x} - \left( {\frac{j}{\Lambda } + \frac{{\sin {\alpha_m}}}{\lambda }} \right)} \right],}$$

where G(f_x) = F{g(x₀, z)}. In addition, the influence of the higher-order Fourier series is neglected because of the presence of the small Fourier coefficient D_j, [20]. When studying the intensity distribution, the angular frequency in the x direction can be expressed as x/λz. Meanwhile, the coordinates in the x direction of the observation plane are translated due to the tilt of the DOE, and the offset is −zsinβ. Thus, Eq. (6) can be corrected as

(7)$$U(x,z) = \frac{1}{{i\lambda z}}\exp (ikz)\exp \left( {ik\frac{{{x^\textrm{2}}}}{{2z}}} \right)\cdot \sum\limits_{j ={-} N}^N {{D_j}G\left[ {\frac{x}{{\lambda z}} - \left( {\frac{j}{\Lambda } + \frac{{\sin {\alpha_m} + \sin\beta }}{\lambda }} \right)} \right]} .$$

The crosstalk between each field distribution of the diffractive beam can be omitted. When compared with the substrate, the contribution of the surface-relief layer to the total optical path difference can be ignored because the thickness of the substrate is considerably greater than that of the surface-relief layer. Therefore, each diffracted beam is independently incident on the parallel plate-like substrate at an angle α_m,j. Further, an x-direction offset is added due to refraction. The relation between the offset Δx and angle α_m,j can be expressed as

(8)$$\begin{array}{l} \Delta x = T\tan [{\arcsin ({{{\sin {\alpha_{m,j}}} / {{n_1}}}} )} ],\\ {\alpha _{m,j}} = \arctan \left( {\sin {\alpha_m} + \frac{{j\lambda }}{\Lambda }} \right) = \arctan \left[ {\sin ({{\alpha_0} - \beta } )+ \frac{{j\lambda }}{\Lambda }} \right], \end{array}$$

where n₁ is the refractive index of substrate. The field distribution of the diffracted beams associated with DOE can be written as

(9)$$\begin{array}{c} {U_j}(x,z) = \frac{{{D_j}}}{{i\lambda (z + T)}}\exp [{ik(z + T{n_1})} ]\exp \left( {ik\frac{{{{(x + \Delta x)}^\textrm{2}}}}{{2(z + T)}}} \right)\\ \cdot G\left[ {\frac{{x + \Delta x}}{{\lambda (z + T)}} - \left( {\frac{j}{\Lambda } + \frac{{\sin {\alpha_m} + sin\beta }}{\lambda }} \right)} \right]. \end{array}$$

The light field of the diffracted beams propagates to a distance L′ and converges at the observation plane. Compared with the x-axis, the observation plane is rotated by −β. The line equation of the observation plane in the coordinate system of xoz can be expressed as z = xtanβ + L′ and x = x′cos β. By substituting these two values into Eq. (9), we can obtain the intensity distribution of the diffracted beams on the observation plane as follows:

(10)$${I_j}(x^{\prime}) = {|{{U_j}} |^2}\textrm{ = }{\left[ {\frac{{{D_j}}}{{\lambda (x^{\prime}\tan \beta + L^{\prime} + T)}}} \right]^2}{G^2}\left[ \begin{array}{l} \frac{{x^{\prime}\cos \beta + \Delta x}}{{\lambda (x^{\prime}\cos \beta \tan \beta + L^{\prime} + T)}}\\ - \left( {\frac{j}{\Lambda } + \frac{{\sin {\alpha_m} + \sin\beta }}{\lambda }} \right) \end{array} \right].$$

The function g(x₀, z) is a Gaussian function and its Fourier transform remains Gaussian [12]. Based on Eq. (10), the center position of the far-field Gaussian beams deviated during the evolution of the tilt angle β and incident angle α_m. By setting the independent variable of function G(ξ) to 0, the center position of the diffracted beams can be solved

(11)$${x_{j,\textrm{cent}}} = \frac{{(L^{\prime} + T)[{{{j\lambda } / {\Lambda + \sin({\alpha_\textrm{0}} - \beta ) + \sin \beta }}} ]- \Delta x}}{{\cos \beta \{{1 - \tan \beta [{{{j\lambda } / {\Lambda + \sin({\alpha_\textrm{0}} - \beta ) + \sin \beta }}} ]} \}}}.$$

Based on Eq. (11), the beam deviation of the diffracted beams is the difference between the center position after the introduction of the error sources and the initial center position. For a CBC architecture, these error sources include the DOE-mount-tilt error (β), emitter-incident angular error (α₀), and DOE-groove-tilt error (corresponding to the error in the DOE period of Λ). Further, the offset Δx in the x direction becomes more obvious with the increasing DOE thickness T and diffraction angle α_m,j.

2.2 Combining-efficiency loss

Because of the introduction of error sources, the original refraction and diffraction processes would be modulated during the beam propagation process. An additional position error will be introduced by the modulated refraction process. The modulated diffraction process will change the pointing of the diffracted beams. Hence, the beam deviation of the diffraction beams is obtained based on the pointing error and position error (Fig. 2). From Eq. (11), the beam deviation can be decomposed into two parts.

(12)$$\begin{array}{l} \delta {x_{j,\textrm{cent}}} = L^{\prime}\delta {\theta _m} + \delta {x_m},\\ \delta {\theta _m} = \frac{{{{j\lambda } / {\Lambda + \sin({\alpha _\textrm{0}} - \beta ) + \sin \beta }}}}{{\cos \beta \{{1 - \tan \beta [{{{j\lambda } / {\Lambda + \sin({\alpha_\textrm{0}} - \beta ) + \sin \beta }}} ]} \}}} - {{(j\lambda } / {\Lambda + \sin{\alpha _\textrm{0}}}}),\\ \delta {x_m} = \frac{{T[{{{j\lambda } / {\Lambda + \sin({\alpha_\textrm{0}} - \beta ) + \sin \beta }}} ]- \Delta x}}{{\cos \beta \{{1 - \tan \beta [{{{j\lambda } / {\Lambda + \sin({\alpha_\textrm{0}} - \beta ) + \sin \beta }}} ]} \}}} - T({{j\lambda } / {\Lambda + \sin{\alpha _\textrm{0}}}}), \end{array}$$

where δθ and δx_m represent the pointing error and position error, respectively. They reduce the overlapping area of the combined beams in the near and far fields, eliminating the common-aperture condition and reducing the beam combining efficiency. For a CBC architecture having 2N + 1 channels, the combining-efficiency loss induced by the pointing error and position error can be expressed as follows [12]:

(13)$$\Delta \eta = \Delta {\eta _x} + \Delta {\eta _\theta } = \frac{{\sigma _x^2}}{{\omega _0^2}} + \frac{{\sigma _\theta ^2}}{{\theta _\omega ^2}},$$

where θ_ω = λ/πω₀ is the far-field divergence of the Gaussian beam and σ_x and σ_θ are the root mean squares (RMSs) of the diffracted beams when considering position error and pointing error, respectively. σ_x and σ_θ can be given as follows:

(14)$$\sigma _i^2 = \frac{1}{{2N + 1}}\sum {{{\left( {\delta {i_m} - \frac{1}{{2N + 1}}\sum {\delta {i_m}} } \right)}^2}} - {\left[ {\frac{1}{{2N + 1}}\sum {\left( {\delta {i_m} - \frac{1}{{2N + 1}}\sum {\delta {i_m}} } \right)} } \right]^2},$$

where i represents x or θ.

Fig. 2. Beam deviation of the diffracted beam based on pointing error and position error.

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According to Eqs. (12)–(14), the period of DOE and the beam size of the array emitters can affect the combining-efficiency loss. In Eq. (13), the beam size is the denominator of Δη_x and the numerator of Δη_θ. In case of a large beam size, the combining-efficiency loss is dominated by the pointing error. Otherwise, the combining-efficiency loss is dominated by the position error. Theoretically, there is a beam size that minimizes the combining-efficiency loss when the remaining influencing factors are maintained constant.

3. Simulation and discussion

3.1 Combining-efficiency loss versus the DOE-mount-tilt error

The DOE beam splitter is used as a combiner in a CBC architecture, which requires the incident array emitters to match the DOE’s angle of incidence. The ideal incident angle of α₀ can be obtained using

(15)$$\frac{{\sin {\alpha _\textrm{0}}}}{\lambda } + \frac{j}{\Lambda } = 0.$$

The perturbation of the perpendicularity between the optical axis and DOE is always accompanied by the formation of the actual CBC system. In the beam-combining case, the pointing error and position error can be calculated using Eqs. (12) and (15), respectively, when the diffracted beams are considered for combining based on the DOE-mount-tilt error β.

For simplicity and avoiding losses, a laser array comprising eleven individual emitters with an operating wavelength of 1071 nm is used in the simulations. The thickness and refractive index of the DOE are assumed to be 2 mm and 1.45, respectively. Considering that the tilt angle in the actual CBC system is controlled from a few µrad to hundreds of µrad, the perturbation of the tilt angle is in the interval [0, 1 mrad]. According to the above formula, the combining-efficiency loss is represented as a function of the tilt angle. As shown in Fig. 3(a), the beam size and period of DOE are 10 mm and 50 µm, respectively. The curve spreads as a second-order polynomial, and the combining-efficiency loss increases to 0.35% when the tilt angle is 1 mrad.

Fig. 3. (a) The combining-efficiency loss as a function of the tilt angle, (b) the critical beam size versus the tilt angle, (c) the combining-efficiency loss versus the tilt angle and period of DOE, and (d) the combining-efficiency loss versus the tilt angle and beam size; the red curves in (c) and (d) represent a combining-efficiency loss of 1%.

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In Eq. (13), the combining-efficiency losses associated with the pointing error and position error are influenced by the beam size, and the total combining-efficiency loss can be theoretically minimized. According to the inequality of arithmetic and geometric means (AM–GM inequality), the total combining-efficiency loss is minimal when Δη_x and Δη_θ are equal. Thus, we can solve the corresponding size (critical size) of the Gaussian beam. The critical size is shown in Fig. 3(b) as a function of the tilt angle. For the DOE with a period of 50 µm, the critical size is approximately 14.5 µm at a tilt angle of 0–1 mrad. However, considering the power density, the CBC system can barely operate in a state with minimal combining-efficiency loss because the critical beam size is considerably small. Therefore, the combining-efficiency loss is dominated by the pointing error.

A reduced combining-efficiency loss can be obtained by selecting a large period of DOE and small beam size [Figs. 3(c) and 3(d)]. At a tilt angle of 1 mrad, the combining-efficiency loss decreases from 14.34% to 0.022% when the period of DOE is increased from 20 to 100 µm. Similarly, the combining-efficiency loss decreases from 3.2% to 0.004% as the beam size decreases from 30 to 1 mm. Therefore, for an actual CBC system, a considerable tilt angle tolerance can be obtained by increasing the period of the DOE and decreasing the size of the incident beam. However, the concomitant ideal incident angle is reduced, which may result in insufficient space for the array emitters. The power density increases as the beam size decreases, indicating that the optical elements in the CBC system require a high damage threshold. Figures 3(c) and 3(d) show the areas in which the combining-efficiency losses are less than 1%. The appropriate period of DOE and beam size of array emitters can be determined based on these areas to balance the combining-efficiency loss and engineering feasibility.

3.2 Combining-efficiency loss versus random perturbation of the emitter-incident angular error

In the CBC architecture, the ideal combining efficiency can be obtained by matching the incident angle of the array emitters with the DOE’s incident angle. However, it is difficult to precisely place the beamlets according to the designed incident angles. Let us assume that the actual incident angle is similar to the angle α₀ with an incident angular error of δα. Then, the corresponding pointing error and position error can be obtained using Eqs. (12) and (15), respectively.

We consider random emitter-incident angular errors. Thus, the corresponding combining-efficiency loss should be statistically analyzed. According to the central limit theorem, a large number of independent random events tend toward a normal distribution in probability and statistical theory [17]. Consequently, the incident angular error spreads as a normal distribution with the probability density function

(16)$$f(\delta \alpha ) = \frac{1}{{\sqrt {2\pi } {\chi _{\delta \alpha }}}}\exp \left( { - \frac{{{{(i - {\mu_{\delta \alpha }})}^2}}}{{2\chi_{\delta \alpha }^2}}} \right)\sim N({\mu _{\delta \alpha }},\chi _{\delta \alpha }^2),$$

where µ_δα denotes the mean and χ_δα denotes the standard deviation.

The combining-efficiency loss is calculated for the standard deviation of the incident angular error at 1 mrad. The period of DOE and beam size are 50 µm and 10 mm, respectively. The remaining parameters used in the calculation are the same as those explained in Section 3.1. This calculation is repeated 1000 times to show rational statistical regularity. The combining-efficiency loss is obtained by dividing the samples into 20 samples of sizes uniformly for considering the relative frequency [Fig. 4(a)]. The statistical combining-efficiency loss agrees with the normal distribution with a mean of 0.35% and standard deviation of 0.29%.

Fig. 4. (a) The combining-efficiency loss with random perturbation of the emitter-incident angular error, (b) the statistical results of combining-efficiency loss based on the incident angular error, (c) the combining-efficiency loss versus the emitter-incident angular error and period of DOE, and (d) the combining-efficiency loss versus the emitter-incident angular error and beam size; the red curves in (c) and (d) represent a combining-efficiency loss of 1%.

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The trend of the combining-efficiency loss versus the standard deviation of the incident angular error is shown in Fig. 4(b). The mean and standard deviation associated with the combining-efficiency loss increase gradually with the incident angular error. A second-order polynomial is used to fit the mean of the combining-efficiency loss, and the result of the fitting is Δη = 0.247δα² + 0.116δα.

The statistical mean of the combining-efficiency loss is used to represent the corresponding combining-efficiency loss to study the effect of the DOE period and beam size on the combining-efficiency loss. The trends associated with the combining-efficiency loss versus the incident angular error, the period of DOE, and beam size are similar to the case observed in Section 3.1 [Figs. 4(c) and (d)]. However, compared with Section 3.1, the level of combining-efficiency loss in Figs. 4(c) and 4(d) is lower at the same angle. Furthermore, for an emitter-incident angular error of 1 mrad, the combining-efficiency loss increases to 8.58% as the period of DOE decreases to 20 µm and increases to 1.87% as the beam size increases to 30 mm. Moreover, the areas in which the combining-efficiency loss is less than 1% are large [Figs. 4(c) and 4(d)]. Hence, the tolerance of the emitter-incident angular error is greater compared to that of the perpendicularity error of DOE in an actual CBC system.

3.3 Combining-efficiency loss versus the DOE-groove-tilt error

The influence of the aforementioned errors on the combining-efficiency loss can be effectively reduced using a reflection-based DOE and sophisticated architecture [6]. However, an additional beam deviation of the diffraction beams is involved because of the perturbation of the perpendicularity between the DOE groove and the plane formed by the array emitters (Fig. 5). This case can be usually attributed to the tilt attitude of DOE or manufacturing errors.

Fig. 5. Schematic of the simplified model and beam propagation process with groove-tilt DOE.

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In case of the groove-tilt DOE, the field distribution must be extended to two dimensions. In the Appendix, we derive the position error and pointing error in the x′ and y′ directions of the observation plane. The position error and pointing error can be attributed to the conversion of the angular frequency in the y′-axis direction to the x′-axis direction during the evolution of the γ angle. Based on the discussion presented in the Appendix, the combining-efficiency losses are simulated as a function of the groove tilt angle, DOE period, and beam size [Figs. 6(a) and (b)]. The perturbation of the groove tilt angle is in the range of [0, 0.2 mrad]. At a groove tilt angle of 0.2 mrad, the combining-efficiency loss is observed to increase from 1.01% to 25.29% as the period of DOE increases from 100 to 20 µm. Simultaneously, a more significant increase in combining-efficiency loss is observed as the beam size is increased to 30 mm. The corresponding combining-efficiency loss reaches 36.41%, which is considerably greater than that caused by the DOE-mount-tilt error and emitter-incident angular error. The combining-efficiency loss becomes approximately 10% even if the groove tilt angle is only a few tens of µrad. Compared with the remaining two error sources, the area in which the combining-efficiency loss is less than 1% is the smallest when considering the DOE-groove-tilt error. Therefore, the combining-efficiency loss is most sensitive to the DOE-groove-tilt error. The DOE-groove verticality should be appropriately controlled in an actual CBC system to minimize the groove tilt angle.

Fig. 6. (a) The combining-efficiency loss versus the groove tilt angle and period of DOE and (b) the combining-efficiency loss versus the groove tilt angle and beam size; the red curves in (a) and (b) represent a combining-efficiency loss of 1%.

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4. Experiment

To verify the theory of beam deviation presented in Section 2.1, a Fourier lens with a focal length f is adopted after considering a DOE to focus the diffracted beams onto the observation plane, as shown in Fig. 7. All the parameters used in the numerical simulation are presented in Table 1. The magnification of the zooming system with the lens is l′ − D/l′, where l′ is the image distance and can be calculated based on geometric optics. The beam deviation of the diffracted beams as a function of the tilt angle of DOE is shown in Fig. 8(a). Further, the beam deviation of five diffracted beams with the tilt angle in the interval [−20°, 20°] is simulated.

Fig. 7. Experimental optical arrangement of the beam deviation of the diffracted beams, and the profile of the −2^nd, −1^st, 0^th, 1^st, and 2^nd order of diffracted beams monitored via CCD.

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Fig. 8. Beam deviation of the −2^nd, −1^st, 0^th, 1^st, and 2^nd order of diffracted beam as a function of the tilt angle. (a) is the simulated result, and (b) is the measured value.

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Table 1. Simulation parameters

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The beam deviation of the −2^nd, –1^st, 0^th, 1^st, and 2^nd order of diffracted beams as a function of the tilt angle is shown in Fig. 8. The curve of the zero-order diffracted beam is approximately linear, whereas those of the non-zero-order diffracted beams spread as a second-order polynomial. The difference between the two curves can be attributed to the zero-order-diffracted beam that can be attributed to the deviation of the diffraction angle (i.e., pointing error). The maximum beam deviation is 204 µm for a propagation distance of 2.3 × 10⁵ µm and a magnification of 0.161. The beam deviation of the high-order diffracted beams is more significant than that of the low-order diffracted beams. In the ideal case, the diffracted beams are equally spaced on an arbitrary observation plane perpendicular to the optical axis, and the beam–beam spacing is proportional to λ/Λ [21]. According to Eq. (11), the beam–beam spacing is modulated by the term cos β during the tilting of DOE, causing the beam–beam spacing to not remain constant. The variation of beam–beam spacing as a function of the tilt angle of DOE is presented in Fig. 9. In case of high-channel DOE, the beam deviation of the high-order diffraction beams is more sensitive to the tilt angle of DOE.

Fig. 9. Comparison between the simulated result and measured value of the variation of the beam–beam spacing.

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The corresponding experiment was performed to verify the aforementioned conclusions. In our experiment, a 20 mm × 20 mm DOE (fabricated by LIMO company) with a period of 50 µm was used to implement the beam-splitting function, which has a continuous surface-relief phase profile on the SiO₂ substrate with a thickness of approximately 0.85 mm. The beam-splitting ratio is 11:1, and the beam–beam angle is 21.3 mrad (approximately 1.22°). The experimental setup was established according to the parameters in Table 1 (Fig. 7). The DOE was mounted onto a rotary stage with a resolution of 0.1′. A collimated single-frequency (SF) laser with an operating wavelength of 1071 nm was directly incident on the DOE surface. A lens with a focal length of approximately 6.8 cm was placed after the DOE, and a charged coupled device (CCD) camera was used to monitor the diffracted beams. A beam block was inserted after the DOE to capture the useless diffracted beams.

In our experiment, the tilt angle was considered to have a step size of 4°, and the intensity distribution of the five diffracted beams was recorded with each angle. The center position of the diffracted beams was characterized based on the centroid of its intensity distribution. The beam deviation of the diffracted beams in the x direction was calculated by comparing with the initial situation and is plotted in Fig. 8(b). The beam–beam spacing was obtained by linearly fitting the position distribution of the five diffracted beams along the x-axis. The variation of the beam–beam spacing was calculated by comparing with the initial situation (Fig. 9). The measurements of the beam deviations and the variations of beam–beam spacing agree well with the theoretical predictions. A small difference can be observed between the simulated values and the measurements of the beam deviations at a large tilt angle, which may be caused by the inapplicability of the paraxial approximation.

5. Conclusion

Thus, we developed a theoretical DOE-based beam propagation model in this study. The theoretical model was applied to investigate the effect of the beam deviation induced by the mount mismatch of the DOE on the beam combining efficiency. Further, the combining-efficiency losses obtained by considering the DOE-mount-tilt error, emitter-incident angular error, and DOE-groove-tilt error were extensively studied. The simulation indicated that the curves of the combining-efficiency losses with these error sources spread as a second-order polynomial. The combining-efficiency loss increased significantly with the decreasing period of DOE and the increasing beam size. For a DOE period of 20 µm and a beam size of 10 mm, the combining-efficiency loss was 14.34%, 8.58%, and 25.29% when the DOE-mount-tilt error was 1 mrad, the emitter-incident angular error was 1 mrad, and the DOE-groove-tilt error was 0.2 mrad, respectively. For a DOE period of 50 µm and a beam size of 30 mm, the combining-efficiency loss was 3.2%, 1.87%, and 36.41% at the above error angles, respectively. The ranges of the three error sources when the combining-efficiency loss was less than 1% followed the relation: DOE-groove-tilt error < DOE-mount-tilt error < emitter-incident angular error. The theoretical model of beam deviation was tested in a corresponding experiment. The combining-efficiency loss is most sensitive to the DOE-groove-tilt error and shows the largest tolerance to the emitter-incident angular error. These analyses will be beneficial to quantify the combining-efficiency loss of the CBC systems, providing a valid basis for optimizing the design of the CBC systems and the manufacturing of DOE.

Appendix

Herein, we derive the combining-efficiency loss associated with a two-dimensional beam propagation model. The periodic structure of initial DOE is distributed along the y′-axis, as shown in Fig. 5. Let us assume that the Gaussian beam is incident upon the DOE (the origin of the ox₀y₀z coordinate system) at an angle α_m. After two Cartesian coordinate translations and rotations, the field distribution on the front surface of the DOE can be written as

(17)$${U_0}({x_0},{y_0},z) = g({x_0},{y_0},z)\exp [{ik({{y_0}\cos \gamma \sin {\alpha_m} - {x_0}\sin \gamma \sin {\alpha_m}} )} ]\quad \textrm{and}$$

(18)$$\begin{array}{c} g({x_0},{y_0},z) = {E_0}\frac{{{\omega _0}}}{{\omega [{({y_0}\cos \gamma - {x_0}\sin \gamma )\sin {\alpha_m} + z\cos {\alpha_m}\textrm{ + }{L_m}} ]}}\cdot \exp [{ik({z\cos {\alpha_m}\textrm{ + }{L_m}} )} ]\\ \cdot \exp \left[ { - \frac{{{{({{y_0}\sin \gamma + {x_0}\cos \gamma } )}^2} + {{[{({y_0}\cos \gamma - {x_0}\sin \gamma )\cos {\alpha_m} - z\sin {\alpha_m}} ]}^2}}}{{{\omega^2}[{({y_0}\cos \gamma - {x_0}\sin \gamma )\sin {\alpha_m} + z\cos {\alpha_m}\textrm{ + }{L_m}} ]}}} \right], \end{array}$$

where γ is the tilt angle of the groove and the positive angular direction is assumed to be counterclockwise. When compared with the theoretical model in Fig. 1, only one additional axis is established, and the remaining processes are similar. Consequently, the field distribution of the diffracted beams associated with the DOE is obtained based on Eqs. (3–9).

(19)$$\begin{array}{c} {U_j}(x,y,z) = \frac{{{D_j}}}{{i\lambda (z + T)}}\exp [{ik(z + T\cdot {n_1})} ]\exp \left( {ik\frac{{{{(x + \Delta x)}^2} + {{(y + \Delta y)}^2}}}{{2(z + T)}}} \right)\\ \cdot G\left[ {\frac{{x + \Delta x}}{{\lambda (z + T)}} + \frac{{\sin \gamma \sin {\alpha_m}}}{\lambda },\frac{{y + \Delta y}}{{\lambda (z + T)}} - \left( {\frac{j}{\Lambda } + \frac{{\cos \gamma \sin {\alpha_m}}}{\lambda }} \right)} \right], \end{array}$$

where Δ_x and Δ_y are the x-direction and y-direction offsets, respectively, and are given by

(20)$$\begin{array}{l} {\Delta _x} = {{T\tan {\alpha _{z,j}}\cdot \sin \gamma \sin {\alpha _m}} / {\sqrt {{{({\sin \gamma \sin {\alpha_m}} )}^2} + {{\left( {\cos \gamma \sin {\alpha_m} + \frac{{j\lambda }}{\Lambda }} \right)}^2}} }},\\ {\Delta _y} = {{T\tan {\alpha _{z,j}}\cdot \left( {\cos \gamma \sin {\alpha_m} + \frac{{j\lambda }}{\Lambda }} \right)} / {\sqrt {{{({\sin \gamma \sin {\alpha_m}} )}^2} + {{\left( {\cos \gamma \sin {\alpha_m} + \frac{{j\lambda }}{\Lambda }} \right)}^2}} }},\\ {\alpha _{z,j}} = \arcsin \left( {{{\sin \left\{ {\arctan \left[ {\sqrt {{{({\sin \gamma \sin {\alpha_m}} )}^2} + {{\left( {\cos \gamma \sin {\alpha_m} + \frac{{j\lambda }}{\Lambda }} \right)}^2}} } \right]} \right\}} / {{n_1}}}} \right). \end{array}$$

Based on the coordinate rotation between the planes of xoy and x′oy′, a system of transformation equations should be substituted into Eq. (19), resulting in the intensity distribution of the diffracted beams on the observation plane.

(21)$${I_j}(x^{\prime},y^{\prime}) = {|{{U_j}} |^2} = {\left[ {\frac{{{D_j}}}{{\lambda (L^{\prime} + T)}}} \right]^2}\cdot {G^2}\left[ \begin{array}{l} \frac{{\Delta x - y^{\prime}\sin \gamma + x^{\prime}\cos \gamma }}{{\lambda (L^{\prime} + T)}} + \frac{{\sin \gamma \sin {\alpha_m}}}{\lambda },\\ \frac{{y^{\prime}\cos \gamma + x^{\prime}\sin \gamma + \Delta y}}{{\lambda (L^{\prime} + T)}} - \left( {\frac{j}{\Lambda } + \frac{{\cos \gamma \sin {\alpha_m}}}{\lambda }} \right) \end{array} \right].$$

Similarly, the beam deviation of the diffracted beams on the observation plane can be obtained by setting the independent variable of the function G(ξ) to 0.

(22)$$\left\{ {\begin{array}{l} {{x_{j,\textrm{cent}}} = (L^{\prime} + T)\sin \gamma {{\cdot j\lambda } / \Lambda } - \Delta x\cos \gamma - \Delta y\sin \gamma }\\ {{y_{j,\textrm{cent}}} = (L^{\prime} + T)({\sin{\alpha_m} + \cos \gamma \cdot {{j\lambda } / \Lambda }} )+ \Delta x\sin \gamma - \Delta y\cos \gamma } \end{array}} \right..$$

For the beam-combining case, the pointing error and position error of the diffracted beams for combining with the tilt groove of DOE can be rewritten as

(23)$$\left\{ {\begin{array}{l} {\delta {\theta_{x,m}} = \sin \gamma {{\cdot j\lambda } / \Lambda }}\\ {\delta {x_m} = T\sin \gamma {{\cdot j\lambda } / \Lambda } - \Delta x\cos \gamma - \Delta y\sin \gamma } \end{array}} \right.\quad \textrm{and}$$

(24)$$\left\{ {\begin{array}{l} {\delta {\theta_{y,m}} = ({\cos \gamma - 1} )\cdot {{j\lambda } / \Lambda }}\\ {\delta {y_m} = T\cdot ({\cos \gamma - 1} )\cdot {{j\lambda } / \Lambda } + \Delta x\sin \gamma - \Delta y\cos \gamma } \end{array}} \right.,$$

where δθ_x,m, δx_m, δθ_y,m, and δy_m are the pointing error in the x′ direction, position error in the x′ direction, pointing error in the y′ direction, and position error in the y′ direction, respectively. Furthermore, the RMS of the diffracted beams associated with the combination of the position error and pointing error should be modified to that observed in a two-dimensional situation.

(25)$$\begin{array}{c} {\sigma ^2} = \frac{1}{{2N + 1}}\sum {\left[ {{{\left( {\delta {i_m} - \frac{1}{{2N + 1}}\sum {\delta {i_m}} } \right)}^\textrm{2}}\textrm{ + }{{\left( {\delta {j_m} - \frac{1}{{2N + 1}}\sum {\delta {j_m}} } \right)}^\textrm{2}}} \right]} \\ - {\left[ {\frac{1}{{2N + 1}}\sum {\sqrt {{{\left( {\delta {i_m} - \frac{1}{{2N + 1}}\sum {\delta {i_m}} } \right)}^2} + {{\left( {\delta {j_m} - \frac{1}{{2N + 1}}\sum {\delta {j_m}} } \right)}^2}} } } \right]^2}, \end{array}$$

where i represents x or θ_x and j represents y or θ_y.

Funding

Key-Area Research and Development Program of Guangdong Province (2018B090904001); National Key R&D Program of China (2018YFB0504500); National Natural Science Foundation of China (61735007, 61805261, 61705243); Youth Innovation Promotion Association CAS (2020252); .

Disclosures

The authors declare no conflicts of interest.

References

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Parameters	Value	Unit	Parameters	Value	Unit
Laser wavelength λ	1071	nm	Period of DOE Λ	50.0	µm
Object distance l or L'	23.0	cm	Thickness of DOE T	0.85	mm
Lens focal length f	6.8	cm	Refractive index of DOE n₁	1.45	none
Detection distance D	8.1	cm	Initialing incident angle α₀	3	°

Investigation of combining-efficiency loss induced by a diffractive optical element in a single-aperture coherent beam combining system

Abstract

1. Introduction

2. Theoretical model

2.1 Beam deviation

2.2 Combining-efficiency loss

3. Simulation and discussion

3.1 Combining-efficiency loss versus the DOE-mount-tilt error

3.2 Combining-efficiency loss versus random perturbation of the emitter-incident angular error

3.3 Combining-efficiency loss versus the DOE-groove-tilt error

4. Experiment

5. Conclusion

Appendix

Funding

Disclosures

References

Cited By

Figures (9)

Tables (1)

Equations (25)

Optics Express