The influence of position errors on stray light in compensation interferometry

1.   INTRODUCTION

With the advancement of Earth observation and deep space exploration, the required aspheric mirrors need increasingly large diameters, with higher precision requirements^[1]. Null compensation interferometry is an essential method for testing aspheric mirrors^[2], with the accuracy being the main factor determining manufacturing precision of the aspheric mirrors. Consequently, it is extremely important to improve the accuracy of compensation interferometry^[3]. Stray light during compensation interferometry will affect testing accuracy, and in severe cases, can even make testing impossible.

Null compensation interferometry is divided into two categories: diffraction compensation interferometry and refraction and reflection compensation interferometry^[4]. In diffraction compensation interferometry, stray light refers to the light of diffraction orders that are not necessary for testing, which enter the interferometer. The result is ghost images being produced and superimposed on the testing fringes, which adversely affect the accuracy. In refraction and reflection compensation interferometry, stray light is generated by multiple reflections and transmissions among the optical elements of the testing optical path. The crosstalk fringes generated by multi-beam interference between the stray light, reference light, and testing light are superimposed on the testing fringes, adversely affecting accuracy.

Position errors include decenter error, tilt error, and distance error^[5]. During refraction and reflection compensation interferometry, the position errors of optical components significantly impact the testing accuracy of the test aspheric mirrors, as well as the crosstalk fringes generated by stray light^[6]. Using simulations to analyze the influence of position errors on the resulting crosstalk fringes can guide the design of aspheric mirrors testing schemes and improve their feasibility, thereby reducing engineering risks and improving efficiency.

Here, we mainly focus on the crosstalk fringes generated by stray light in null compensation interferometry of a convex aspheric mirror. The crosstalk fringes are simulated based on a specific compensation interferometry optical path, using a simulation analysis of the position errors. In Section 2, we introduce the transmissive Hindle compensation interferometry, and simulate the crosstalk fringes generated by stray light in the transmissive Hindle compensation testing optical path of a given convex aspheric mirror. We also propose evaluation indices of the crosstalk fringes generated by stray light in refraction and reflection compensation interferometry and evaluate the simulation results. In Section 3, we simulate the influence of position errors on the crosstalk fringes generated by stray light in compensation interferometry, and analyze the simulation results based on the evaluation indices. In Section 4, we summarize the simulation results.

2.   SIMULATION AND EVALUATION OF CROSSTALK FRINGES

2.1   Transmissive Hindle Compensation Interferometry Method

Quadratic conical mirrors can be examined using stigmatic null-testing^[7]. When the test aspheric mirror is convex, the reference sphere used in the testing is called a Hindle sphere^[8]. Traditional reflective Hindle sphere testing presents challenges due to central obstruction^[9], necessitating that the size of the Hindle sphere is significantly larger than the size of the test aspheric mirror. This restriction significantly complicates the testing of large-diameter quadratic conical mirrors^[10].

These limitations can be addressed by transitioning from a reflective Hindle sphere to a transmissive Hindle shell. The optical path for transmissive Hindle shell testing (shown in Fig. 1) has no central obscuration, and the required size of the transmissive Hindle sphere is decreased, facilitating the testing of large-diameter secondary conical mirrors.

Figure  1.  Schematic diagram of the light path used in transmissive Hindle compensation interferometry.

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In this study, simulations are carried out on the transmissive Hindle compensation interferometry for a test convex quadratic conical mirror. The parameters of the test convex quadratic conical mirror are shown in Table 1. The testing optical path of transmissive Hindle compensation interferometry is shown in Fig. 2, and the testing wavefront of transmissive Hindle compensation interferometry is shown in Fig. 3. The compensation interferometry optical path includes a compensator and the test aspheric mirror. The light emitted from the interferometer passes through the compensat or, reflects off the aspheric mirror to be tested (toward the rear surface of the compensator), and then reflects to the interferometer, where it interferes with the reference light.

Table  1.  Test aspheric mirror parameters

Diameter/mm	R/mm	K	Order term
300	2 858.183	−1.461 7	0

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Figure  2.  Testing optical path for transmissive Hindle compensation interferometry.

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Figure  3.  The wavefront of transmissive Hindle compensation interferometry.

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2.2   Simulation of Crosstalk Fringes

We simulate the crosstalk fringes resulting from stray light in the compensation interferometry optical path. When stray light passes through the optical elements, any stray light returning to the interferometer after a single reflection is called single reflected stray light. Light reflected back after three reflections is called triple reflected stray light, and so on. The quantity model of stray light in the compensation interferometry is expressed as^[10]

where m is the number of compensators, and A, z, n, and y are temporary variables. The quantities of single, triple, and quintuple reflected stray light optical paths in the compensation testing are confirmed, and calculated to be 3, 5, and 13, respectively. We simulate these stray light optical paths and the testing optical path, separately tracing the coordinates and phase of the light rays on the reference surfaces in each optical path. The simulation results are then integrated. Amplitude information of the testing light and the stray light rays is simulated according to the relationship between the ray coordinates and amplitude. Subsequently, we simulate the complex amplitude distribution of the testing light and the stray light rays according to the relationship between the size of the reference surface and the number of pixels of the interferometer.

In compensation interferometry, the stray light, testing light, and reference light are all emitted by the laser in the interferometer. Therefore, these light rays are mutually coherent. The derived multi-beam interference intensity, I, is expressed as

where E is complex amplitude, A is amplitude, Φ is the phase, w is angular frequency, t is time, and n is the number of beams. Using the simulated complex amplitude distribution of stray light, testing light, and reference light, we simulate the intensity distribution of the testing fringes and crosstalk fringes resulting from the multi-beam interference of multiple light sources.

Simulation results are as shown in Fig. 4. The interferometer pixel resolution in the simulation is 2048 × 2048, and the center pixel coordinates of the testing fringe and the crosstalk fringe are both (0, 0). The edge pixel coordinates of the testing fringe are 484 and 1564, and the edge pixel coordinates of the crosstalk fringe are 1007 and 1041. The crosstalk fringe is superimposed on the center of the testing fringe.

Figure  4.  Simulation results of testing and crosstalk fringes.

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2.3   Evaluation Indices of Crosstalk Fringes

To evaluate the influence of crosstalk fringes on testing fringes, two evaluation indices were introduced: position deviation and diameter ratio.

2.3.1   Position deviation

We first determine the position deviation of the crosstalk fringes. The x- and y-position deviations are expressed as

where $\left(\mathit{x}_{\text{crosstalk center}}\text{,}\ y_{\text{crosstalk center}}\right)$ is the center coordinate of the crosstalk fringes, and $\left(\mathit{x}_{\text{test center}}\text{,}\ y_{\text{test center}}\right)$ is the center coordinate of the testing fringes.

This position deviation is the difference between the center coordinate values of the crosstalk fringes and the testing fringes. We find the effective aperture of the test aspheric mirror according to the testing requirements. If the crosstalk fringes are outside the effective aperture of the test aspheric mirror, the crosstalk fringes do not affect the testing results. If the crosstalk fringes are within the effective aperture of the test aspheric mirror, the influence of the crosstalk fringes on the testing results needs to be judged based on another index.

When the position deviation is negative, the crosstalk fringe center is located on the left side of the test fringe center; when the position deviation is positive, the crosstalk fringe center is located on the right side of the test fringe center, and; when the position deviation is zero, the crosstalk fringes are exactly in the center of the test fringes.

2.3.2   Diameter ratio

After determining the position deviation between the crosstalk fringes and the testing fringes, it is necessary to judge the influence area of the crosstalk fringes on the testing fringes; the diameter ratio is the ratio of the diameter of the crosstalk fringes to the diameter of the testing fringes, expressed as

where ${D_{{\text{crosstalk}}}}$ is the diameter of the crosstalk fringes and ${D_{{\text{test}}}}$ is the diameter of the test fringes.

The influence of crosstalk fringes diminishes as the diameter ratio decreases. In the design of the compensation testing scheme, crosstalk fringes do not need to be infinitely small, provided that they are as far as possible outside the effective aperture.

2.3.3   Analysis of simulated results of crosstalk fringes

We analyze the results of the simulation (shown in Table 2) according to the two evaluation indices. The x- and y-position deviations of the crosstalk fringes generated by stray light in the compensation testing optical path are both 0, the diameter ratio with the testing fringes is 1/32, the crosstalk fringes are superimposed on the center of the testing fringes, and the diameter of the area occupied by the crosstalk fringes is 1/32 of the diameter of the testing fringes. According to the testing requirements, the effective aperture of the test aspheric mirror is the same as the center of the test aspheric mirror, and the minimum diameter of the effective aperture is 1/12 of the aperture of test aspheric mirror. A schematic diagram of the effective aperture and position of the crosstalk fringes is shown in Fig. 5. The crosstalk fringes do not affect the testing fringes.

Table  2.  Simulation results of the testing fringes and crosstalk fringes, and the evaluation index of the crosstalk fringes

Parameter	Testing fringe	Crosstalk fringe
Center pixel coordinates	(1024, 1024)	(1024, 1024)
Diameter/pix	1080	34
Position deviation	0
rdiameter	1/32

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Figure  5.  Diagram showing the positions of the effective aperture, ineffective aperture, and crosstalk fringes.

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3.   SIMULATION OF THE EFFECT OF POSITION ERROR ON STRAY LIGHT

3.1   Simulation of Stray Light Optical Paths

The main position errors of the compensator are decenter error, tilt error, and distance error^[11]. Based on the two evaluation indices proposed in Section 2.3, we investigate the influence of position errors on the testing light and stray light in compensation interferometry, and analyze the effects of the position errors of the compensator on the stray light in compensation interferometry.

A schematic diagram of the interferometer is shown in Fig. 6, mainly consisting of a laser, pinhole, collimating lens, beam splitter, transmission sphere, imaging lens, and charge-coupled device (CCD). The transmission sphere consists of several lenses, and the pinhole serves as a spatial filter. In compensation interferometry, most stray light is filtered by the pinhole, so that only a small amount of stray light can pass through the pinhole and enter the interferometer. This light interferes with the testing light and reference light to generate crosstalk fringes that are superimposed on the testing fringes. The structure of the interferometer is difficult to simulate. To achieve filtering in the simulation, a pinhole is added at the focus.

Figure  6.  Schematic diagram of the interferometer internal structure. The blue box is the internal structure of the interferometer and the focus of the interferometer is marked with an F.

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The compensation interferometry optical path is shown in Fig. 7. The optical components are made from silica, with a transmittance of 96% and a reflectivity of 4%. The intensity of light passing reflected off and transmitted through optical elements multiple times is considered to be negligible. Consequently, only stray light with an intensity higher than (4%)³ will be considered. Four stray light optical paths with intensities meeting the requirements and high light densities are chosen. The stray light rays are named according to the serial number of the reflective surface, and a simulation analysis is conducted for "②", "③", "②①③", and "③①②". Simulation results of the stray light optical paths are shown in Fig. 8. The reference surface is set as the surface of the interferometer transmission sphere closest to the Hindle shell in the physical experiment.

Figure  7.  Schematic diagram of the transmissive Hindle compensation interferometry. L₁ is the front distance of the Hindle shell, and L₂ is the rear distance of the Hindle shell. Point 1 marks the front surface of the Hindle shell, point 2 marks the rear surface of the Hindle shell, point 3 marks the test aspheric mirror.

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The simulation of the influence of position error on the position and size of the testing light spot and stray light spots is sufficient to investigate the influence of position errors on the position and area of crosstalk fringes. Since this testing optical path is rotationally symmetric, the influence of x- and y-decenter or tilt on the crosstalk fringes and testing fringes is the same. We can determine the impact of position error on the testing fringes and crosstalk fringes by modifying the x-decenter, y-tilt, and the front and rear distances of the compensator, and quantitatively simulating the effect of position error on the testing light and stray light.

Figure  8.  Simulation results of the optical paths of stray light. The stray light rays are named according to the serial number of the reflective surface:

(A) Stray light optical path ②. (B) Stray light optical path ③. (C) Stray light optical path ②①③. (D) Stray light optical path ③①②. The light source is located at the center of the pinhole.

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3.2   Simulation of The Influence of Decenter Error

We simulate the testing light and stray light with x-decenter values of 0 mm, 0.05 mm, 0.1 mm, and 0.15 mm, and determine the center and edge coordinates of the light beam in the reference plane of the four stray light optical paths and the test optical path. This allows us to calculate the two evaluation indices of x-position deviation and diameter ratio. The simulation results are shown in Table 3, and the testing light and stray light spots are shown in Fig. 9. As the decenter increases, the average variation of x-position deviation is 0.32 mm, 0.59 mm and 0.45 mm. All have an average variation in diameter ratio of 0.

Table  3.  The x-position deviation and r_diameter, and variation in their corresponding eccentricities

Decenter	Optical path	x-position deviation/mm	rdiameter
x-decenter: 0 mm y-decenter: 0 mm	②	0	1/43
	③	0	1/43
	②①③	0	1/39
	③①②	0	1/33
x-decenter: 0.05 mm y-decenter: 0 mm	②	−0.36	1/43
	③	−0.20	1/43
	②①③	−0.35	1/39
	③①②	−0.38	1/33
x-decenter: 0.1 mm y-decenter: 0 mm	②	−0.92	1/43
	③	−0.89	1/43
	②①③	−0.90	1/39
	③①②	−0.95	1/33
x-decenter: 0.15 mm y-decenter: 0 mm	②	−1.37	1/43
	③	−1.32	1/43
	②①③	−1.30	1/39
	③①②	−1.43	1/33

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Figure  9.  The testing light spot and stray light spots.

(A) No decenter. The stray light spot is in the center of the testing light spot. (B) 0.15 mm x-decenter. The testing and stray light spots are offset, and the stray light spots are not in the center of the testing spot.

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As the decenter increases, the center of the stray light and the testing light changes, with the position deviation of the stray light increasing and the stray light optical paths moving further away from the center of the testing fringes. At the same time, the change in the diameter of the testing light and the stray light is less than 0.01 mm, which can be considered effectively unchanged. The diameter ratio of the testing light and the stray light rays under different x-decenter conditions is the same, i.e. the decenter has an influence on the position of both the testing light and the stray light, and with a greater influence on the position of the testing light. The area of both the testing light and stray light rays is unaffected.

3.3   Simulation of The Influence of Tilt Error

We simulate the light spots of testing light and stray light at a y-tilt of 0′, 0.2′, 0.4′, and 0.6′ (with results shown in Table 4). The testing light spot and stray light spots in the y-direction with no tilt and 0.6′ y-tilt are shown in Fig. 10. As the tilt increases, the average variation in x-position deviation is 0.24 mm, and the average variation of diameter ratio is 0.

Table  4.  The x-position deviation and r_diameter correspond to different tilts

Tilt	Optical path	x-position deviation/mm	rdiameter
x-tilt: 0′ y-tilt: 0′	②	0	1/43
	③	0	1/43
	②①③	0	1/39
	③①②	0	1/33
x-tilt: 0′ y-tilt: 0.2′	②	0.24	1/43
	③	0.24	1/43
	②①③	0.24	1/39
	③①②	0.24	1/33
x-tilt: 0′ y-tilt: 0.4′	②	0.48	1/43
	③	0.48	1/43
	②①③	0.48	1/39
	③①②	0.48	1/33
x-tilt: 0′ y-tilt: 0.6′	②	0.72	1/43
	③	0.72	1/43
	②①③	0.72	1/39
	③①②	0.72	1/33

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Figure  10.  The testing light spot and stray light spots.

(A) No tilt. The stray light spots are in the center of the testing light spot. (B) 0.6′ y-tilt. The testing light spot is offset, and the stray light spots have almost no offset. The stray light spots are not in the center of the testing spot.

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As the tilt increases, the center of the stray light spots remains unchanged, while the center of the testing light changes. The position deviation change only depends on the change of the testing light. The position deviation of stray light remains the same in the y-tilt situation. Meanwhile, the change in the diameter of the testing light and the stray light is less than 0.01 mm, which can be considered effectively unchanged. The diameter ratio of the testing light and the stray light is the same at different y-tilts, i.e. the tilt error has little effect on the stray light and only affects the position of the testing light.

3.4   Simulation of The Influence of Distance Error

We simulate the positions of testing light and stray light spots at different front and rear distances of +1 mm, +3 mm, −1 mm, and −3 mm, where the positive direction is toward the test mirror (results are shown in Table 5 and Table 6). The position deviation value is always 0, so it is not shown, and the testing light spot and stray light spots in the different front distance and rear distance are shown in Fig.11. As the front and rear distance increases, the average variation in x-position deviation is 0, and the average variation of diameter ratio is less than 0.01 mm, which is considered to be effectively unchanged.

Table  5.  The r_diameter and corresponding variation in front distances. The position deviation value is always 0, so it is not shown

Front distance	Optical path	rdiameter	Front distance	rdiameter
+1 mm	②	1/43	−1 mm	1/43
	③	1/43		1/40
	②①③	1/43		1/38
	③①②	1/35		1/33
+3 mm	②	1/43	−3 mm	1/42
	③	1/41		1/40
	②①③	1/39		1/38
	③①②	1/33		1/32

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Table  6.  The r_diameter and corresponding variation in rear distances. The position deviation value is always 0, so it is not shown

Rear distance	Optical path	rdiameter	Rear distance	rdiameter
+1 mm	②	1/43	−1 mm	1/43
	③	1/43		1/40
	②①③	1/43		1/38
	③①②	1/35		1/33
+3 mm	②	1/40	−3 mm	1/38
	③	1/38		1/36
	②①③	1/37		1/37
	③①②	1/31		1/29

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As the front distance or the rear distance increases, the center of the testing light and stray light remains unchanged, therefore the position deviation remains unchanged. The diameter change of the stray light spots is less than 0.01 mm at all distances, which is considered to be effectively unchanged. The diameter ratio is only affected by the testing light, i.e. the distance error has almost no effect on the stray light and only affects the area of the testing light.

Figure  11.  The light spots of testing light and stray light under different frontal and rear distance.

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4.   CONCLUSIONS

We simulate crosstalk fringes and testing fringes for a transmissive Hindle sphere testing scheme, and perform simulations and quantitative analyses of the influence of position errors during testing on crosstalk fringes. Position errors include decenter error, tilt error, and distance error. As the decenter increases, the variation in position deviation is less than 0.6 mm, and the variation of the diameter ratio is 0. As the tilt increases, the variation in the position deviation is less than 0.3 mm, and the diameter variation is 0. As the distance increases, the rate of change of the position deviation is 0, and the rate of change of the diameter ratio is less than 0.01 mm, which considered to be effectively unchanged.

Decenter errors have a minor influence on stray light, and the sensitivity of stray light to decenter errors is much lower than that of the testing light. Tilt errors and distance errors have nearly no effect on stray light. In the design stage of the testing scheme, when the accuracy of the testing light is not significantly affected, stray light has no additional impact on accuracy if the position of optical components is changed. When only minor modifications are needed for the testing optical path, the influence of crosstalk fringes generated by stray light does not need to be considered. This work can be useful in the design of compensation testing schemes, reducing engineering risks and improving efficiency.