2 Neural network method for laser-pointing error measurement

Our prototype laser-pointing system contains a laser source, a thin lens and a CCD detector, as shown in Figure 1. In the prototype system, for a beam tilt T the corresponding spot image M on the CCD is simulated through a virtual optical system method for a tilted beam^[Reference Xia, Gao and Han^24]. The distance u between the source plane and the lens is set to be equal to the focal length f, so that the image beam waist can be approximately located on the focal plane for its largest waist radius ${w}_{02}\approx \lambda f/\left(\pi {w}_{01}\right)$ (half width at 1/e ² center intensity), and the spot radius w ₂ on the CCD can then be expressed as ${w}_2\approx {w}_{02}\sqrt{1+{\left({\delta}_z/{Z}_{R2}\right)}^2}$ , where Z _R2 is the Rayleigh length in the image space, w ₀₁ is the waist radius at the source plane and λ is the wavelength of the laser source.

We chose a laser of the wavelength λ = 632.8 nm, and a beam waist radius w ₀₁ = 2.0 mm with Gaussian distribution at the beam waist, corresponding to a typical transverse electromagnetic mode (TEM₀₀) He-Ne laser source. The CCD had a pixel size of 0.0057 mm, with an output gray level of 12 bits, providing an intensity range 0–4095. The pixel intensities of an image are all integers to simulate analog-to-digital (A/D) conversion, which is equivalent to introducing a detection noise of less than 0.5. We limit the position offsets a ₀ and b ₀ within the range [−0.5, 0.5] mm, and the inclination angles θ_x and θ_y within [−25, 25] Ҽrad. To compare the prediction performance for image sets with different system parameters, the intensity of the collimated beam at the center of the CCD is fixed to a particular value by adjusting the intensity of the laser source. A dataset composed of 12,000 spot images of 36 × 36 pixel regions with the corresponding randomly generated beam tilts can then be obtained.

Figure 1 Prototype laser-pointing system. S is the laser source; L is the thin lens; M is the spot image on the CCD; T is the beam tilt of the waist center on the source plane; a ₀, θ_x are the position offset and inclination angle of the beam relative to the optical axis in the x direction, respectively, and b ₀, θ_y are those in the y direction; u is the distance between the source plane and the lens; f is the focal length; δ_z is the defocus distance of the CCD. The optical axis of the system is along the z direction.

The neural network used in this study is implemented by Python^[Reference Nielsen^25] without any specific package. It is a feed-forward network^[Reference Wizinowich, Lloyd-Hart and McLeod^19] with three layers: an input layer of 36 × 36 = 1296 nodes for normalized pixel intensities of an image, a hidden layer of 100 nodes and an output layer for the prediction of the normalized beam tilt. Samples from a dataset (10,000) are used to train the neural network for 2000 epochs with the back-propagation technique^[Reference Rumelhart, Hinton and Williams^26], and the remaining 2000 samples are used to test the performance of the neural network after each epoch of training. For the jth training epoch, the prediction error E_j of the neural network is evaluated as

(1) \begin{align}{E}_j=\frac{1}{N}\sum \limits_{n=0}^{N-1}\frac{1}{\sqrt{m}}\left\Vert {\mathbf{y}}_n-{\mathbf{a}}_n\right\Vert, \end{align}

where a_n is the nth output of the network, y_n is the nth actual beam tilt (normalized), m is the dimension of vector y_n and N is the number of test samples. Finally, the mean value of E_j in the last 500 epochs of a total of 2000 epochs is employed to represent the prediction performance E _mean of the neural network.

We first start with a simpler case with the beams tilted only along one direction (e.g., here the x direction) to illustrate the effect of our method. We consider a vertical CCD and choose a focal length f = 100 mm and a defocus distance δ_z = Z _R2 (instead of δ_z = 0 as in the traditional method), so that the position and angular errors can contribute nearly the same to the bound of the beam displacement on the CCD. The performance of the neural network is shown for this case in Figure 2(a). Clearly, the prediction error ultimately remains at a high level, about 0.33, implying that the network cannot well separate the position and angular errors from images on the vertical CCD. To determine the cause of the failure, we analyzed the difference between two spot images with the same centroid on the CCD. To achieve the maximum image difference we choose two beams with the tilts T₁ = (−0.49646, −25)^T and T₂ = (0.49646, 25)^T, and both with the spot centroid at the center of the CCD. The corresponding image difference (M₂ − M₁) is shown in Figure 2(b), where the maximum pixel intensity is only 0.00096, resulting in the failure of the error decoupling.

Figure 2 Prediction errors E_j for all epochs, spot image M₂ and image difference for beam tilts in the x direction. (a) and (b) show the prediction errors, the spot image and image difference on the vertical CCD, respectively. (c) and (d) show those on the tilted CCD with a rotation of 60° around the y-axis. (e) and (f) show those on the tilted CCD with a rotation of 60° around the x-axis.

In order to enlarge the image difference, we designed the system with a tilted CCD rotated by 60° around either the y or x axis. Figures 2(c) and 2(e) show the elliptical spots of image M₂ with tilt T₂ on the CCD rotated around the y and x axes, respectively. For the same beam tilts T₁ and T₂, the corresponding image differences are largely improved to −65.3 and 22.8 as shown in Figures 2(d) and 2(f), respectively. The significant enhancement can be attributed mainly to the magnified difference in incident angles and the no-longer centro-symmetric intensity distribution for the tilted CCD. As shown in the lower-left corner of Figure 2(d), since the incident angle of beam 2 is obviously larger than that of beam 1, the pattern for beam 2 exhibits a broader distribution with a lower peak value on the tilted CCD. For the pixels in Figure 2(f), the patterns of both beams deviate from the centro-symmetric distribution, giving the observed quadrupole image difference. The prediction results of the neural network are consistent with the changes in the image differences. As shown in Figures 2(c) and 2(e), the prediction performances E _mean with the image rotation around the y and x axes are 0.010 and 0.014, respectively. It can therefore be inferred that the tilted CCDs can help to decouple the position and angular errors, and the rotation perpendicular to the beam tilt direction is better.

For non-Gaussian beams, the maximum difference is expected to appear at different positions but with a similar magnitude. For flat-topped beams common in high-energy systems, the maximum difference would occur around the edges of the pattern with a similar magnitude; it is expected that this can be recognized accurately by a neural network, as discussed below.

3 Results and discussion of prediction in two directions

For practical prediction of beam tilts in two directions, we consider the prototype system under different combinations of parameters θ, f and δ_z. The tilting angle θ is chosen from 0°, 15°, 30°, 45° and 60°. The y = −x axis is chosen as the rotation axis of the CCD, allowing the spot to be stretched diagonally and contained in a smaller square pixel region. The focal length f is taken as 40, 60, 80, 100 or 120 mm. The spot images of the positive and negative defocus distances are symmetrical about the focal plane, so only the positive defocus distances δ_z/Z _R2 = 0, 0.5, 1, 1.5 or 2 are considered. The positive defocus distance δ_z/Z _R2 is set to a constant here, so that the spot radius can be changed with different focal length f. Finally, 125 generated datasets are substituted into the neural network to obtain the prediction performances of the beam tilts.

We first analyze the prediction performances at different tilting angles θ. As can be seen from each curve in Figures 3(a) and 3(b), the prediction is generally better as the tilting angle increases. It can be understood that larger spot size and intensity contrast due to the tilted CCD give the neural network better performance. In addition, for the vertical CCD with θ =  0°, the prediction performances E _mean are always very large around 0.38, regardless of the focal lengths and the defocus distances, indicating that the introduction of a tilted CCD is essential to decouple the position and angular errors.

Figure 3 Prediction performances E _mean with different focal lengths f, tilting angles θ and defocus distances δ_z. (a) and (b) show spot samples and the prediction performance with typical focal lengths f = 40 and 100 mm, respectively. The partially enlarged plot in the dotted rectangle represents the prediction results for θ = 60°. (c) and (d) show the prediction performance when the tilting angles are 45° and 60°, respectively.

To clearly elucidate the effect of the defocus distance, we focus on the prediction performances with the tilting angles θ =  45° and 60° as shown in Figures 3(c) and 3(d). In these cases, the factors that improve the neural network performance, such as larger spot size and image contrast, are competing with each other. A larger defocus distance can increase the spot size, but reduce other positive effects. For f ≥ 60 mm, the spot size may be sufficient for the network, and other factors play the leading role. Hence, the prediction performances deteriorate with the defocus distance. When f = 40 mm, the spot size is so small (5 × 5 for δ_z = 0, θ = 60° as shown in Figure 3(a)) that it plays a greater role in prediction. The two curves therefore show a downward trend, or zigzag downward.

Similarly, we can derive the influence of the focal length from Figures 3(c) and 3(d). With a large defocus distance (δ_z/Z _R2 = 1, 1.5 or 2), the prediction result obviously gets worse as the focal length increases. When the defocus distance is further reduced (δ_z/Z _R2 = 0 or 0.5), a larger focal length (60 or 80 mm) achieves a better prediction result, while focal lengths of 40 and 120 mm at both ends give worse prediction results. As the focal length decreases, the negative effect of spot size is magnified, as well as the positive effect from other factors. When the spot size is insufficient due to the small defocus distance, the two effects are comparable and then balanced at the larger focal length.

According to the analysis above, some useful rules can be drawn. The position and angular errors cannot be decoupled by spot images on the vertical CCD. A tilted CCD can help to solve this problem, and better prediction performance can be acquired at a larger tilting angle of the CCD. A smaller focal length and defocus distance have greater potential in prediction, but they may also cause a too small spot size, which leads to performance degradation. Factors that can increase the spot size may improve the performance. For the pixel size and error ranges in the prototype system, the optimal parameter combination is the focal length f of 60 mm, defocus distance δ_z of 0 and tilting angle θ of 60°.

We briefly discuss the potential errors in this method below. Due to its data-based character^[Reference Hornic^15], the neural network technique may have an anti-interference ability in some systems^[Reference Wizinowich, Lloyd-Hart and McLeod^19, Reference Guo, Korablinova, Ren and Bille^21]. With the distinct difference in the images provided by the tilted CCD, the current method is expected to find a one-to-one correspondence between images and pointing errors from complex systems with some disturbances. For actual detection noise (far less than 0.5 introduced during A/D conversion), the influence on the prediction is considered to be rather small. For more complex wavefront errors induced by turbulence and thermal effects in the propagation process, however, the impact on our technique requires further study.