Refractive index measurement deflectometry for measuring gradient refractive index lens

Zekun Zhang; Ruiyang Wang; Xinwei Zhang; Renhao Ge; Wanxing Zheng; Manwei Chen; Dahai Li; Dahai Li

doi:10.1364/OE.518670

1. Introduction

In the realm of optical design, designers frequently manipulate parameters such as the curvature radius of lens surfaces to improve optical performance. However, as material technology continues to advance, there has been a growing interest in exploring the internal refractive index distribution of lens as a means of achieving superior optical design solutions. One example of such lens is gradient refractive index (GRIN) lens, which are imaging elements designed and fabricated from media exhibiting a gradient refractive index. These lens offer a number of advantages, including compact size, ease of installation, flat surfaces that can be readily bonded to other optical elements to form integrated devices [1,2]. This implies that in many applications, it may be preferable than traditional devices. If the manufacturing method used allows for precise control of radial refractive index variations, the performance of GRIN lens could be quite high, exhibiting only weak spherical aberrations like aspherical lens. Furthermore, with the advancement of fabrication techniques, cost-effective mass production of GRIN lens has also become a possibility. As a result, GRIN lens have sparked considerable research interest [3–5] and find applications in a broad range of fields, including optical communication, microscopy systems, medical devices, and others [6–8]. As these fields continue to evolve, the demand for GRIN lens with greater accuracy and smaller dimensions has increased, leading to a concomitant need for greater accuracy in measuring the refractive index of GRIN lens. Currently, most optical test methods focus on surface performance measurement, but for GRIN lens, their optical properties are no longer determined solely by surface morphology, thus these methods have reached their limitations.

Currently, common methods for measuring internal gradient refractive index include atomic force microscopy (AFM) [9], refractive near-field method [10,11], focusing method [12], and digital holography [13]. These methods mostly use rotation of lens under test (LUT) or multi-angle imaging to measure refractive index distribution. Among them, AFM etches the surface of the LUT and observes its 3D morphology to reflect the refractive index curve, taking advantage of the relationship between the refractive index distribution and the etching rate. However, the AFM has the drawbacks of being expensive, having a small field of view, and slow measurement speed. Near-field refractive index measurement measures the refractive index distribution based on the linear proportional relationship between the optical intensity of the refractive mode and the refractive index, and it is widely used as a high-precision method, but it is limited by the surface flatness of LUT and high requirements for the environment. The focusing method obtains the relative refractive index distribution by observing the focused optical intensity distribution after LUT, and this method is simple and easy to perform but has high requirements for the uniformity and quality of the illumination light source. The digital holography method combines digital holographic microscopy systems and computer tomography scanning methods, with high measurement accuracy, but the equipment cost, and complexity is also high. In addition, Wei et al. [14] proposed a reverse scheme based on ray deflection measurement to reconstruct the refractive index field of gradient refractive index media, but it requires more light beam scanning of LUT, which is complex and time-consuming. For the direct measurement of GRIN lens, the application of interferometric techniques [15,16] is also widespread, as they achieve full-field measurements by utilizing the relationship between reflectivity and refractive index. However, experimental setups using interferometry are susceptible to environmental factors and other disturbances. Binkele et al. [17] used phase-shifting diffraction interferometry (PSDI) and experimental ray tracing method (ERT) to characterize the gradient refractive index parameters. PSDI can obtain refractive index information of the entire field of view at one time, but the measurement device is more complex. ERT obtains results through point-by-point scanning, with higher local accuracy but lower efficiency. Gómez-Correa et al. [5] present a precise numerical method that based on invariants. In the given light ray paths, they reconstruct different known symmetrical GRIN media using rays propagated in two and three dimensions.

In this paper, we propose the refractive index measurement deflectometry (RIMD) based on phase measuring deflectometry. Deflectometry is a high-accuracy optical element measurement technology that offers several advantages, including simple equipment, large dynamic range, and fast measurement speed. The key to deflectometry lies in inversely calculating the information of the optical element from the measured deflection of light, which is comparable to solving the inverse problem [4,5]. Since its proposal, it has been widely applied, and currently exhibits good performance in the measurement of surface shape of specular elements [18–20] and wavefront information of transmissive elements [21,22]. In the experiment, only a liquid crystal display (LCD) is required to display sinusoidal fringe pattern, and a camera is used to record the fringe images passing through LUT. By applying the phase shifting algorithm, the phase distribution of the LUT is obtained, and the deflection angle of the light passing through the LUT is calculated. Then the equation of light propagation in non-uniform media is used to obtain the gradient of local refractive index change, and finally, the radial refractive index distribution of the entire element is obtained through the integration algorithm. The principle of RIMD is introduced, and the correctness and accuracy of the method are verified through numerical simulation. The influence of lens surface shape and camera calibration error on the measurement results is analyzed. In the experiment, a gradient refractive index lens is successfully measured.

2. Principle

The refractive Index measurement deflectometry (RIMD) testing system consists of an LCD, a camera, and the GRIN element under test, as illustrated in Fig. 1. The optical axes of the lens of camera and the LUT as well as the LCD are aligned. From the perspective of the reverse Hartmann test system [23], the pinhole camera can be regarded as a “point source” that emits ray from its projection center. Each ray corresponds to a pixel on CCD and represents a light source point on the LCD. Consequently, any change in the trajectory of the ray can be ‘seen’ by the camera. In the absence of a lens, the light ray propagates in a straight line. However, when a GRIN lens is placed in the path of the light ray, the ray experiences deflection due to refractive index variation. Since there is no curvature variation on the surface of the GRIN lens, the deflection of light ray is solely caused by the refractive index change. Therefore, the refractive index variation can be determined by calculating the deflection angle of the light ray. The relationship between them can be deduced through the following derivation.

Fig. 1. The diagram of the measurement setup for RIMD.

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According to Fermat's principle, the optical path of light propagation takes a stationary value, which can be expressed mathematically as:

(1)$$\delta \int_A^B {n({x,y,z} )ds = 0}, $$

where $\delta $ is the variation symbol and s represent the trajectory of the light ray. Using the Euler-Lagrange equation, the vector form of the equation for light propagation in non-uniform media can be obtained:

(2)$$\frac{d}{{ds}}\left( {n\frac{{d{\bf r}}}{{ds}}} \right) = \nabla n, $$

where r represents the vector coordinate of any point on the optical path, and $\nabla $ is the gradient operator. The equation can be rewritten as two first-order systems of equations:

(3)$$n\frac{{d{\bf r}}}{{ds}} = {\bf d},\textrm{ }\frac{{d{\bf d}}}{{ds}} = \nabla n, $$

where d represents the local direction vector of the light ray. The directions of the light ray upon entering and leaving the LUT are denoted as ${{\textbf d}_{in}}$ and ${{\textbf d}_{out}}$ respectively. The difference between them is the deflection angle $\beta $ of the light ray, which can be expressed as the integral of refractive index gradient along the path of the light ray:

(4)$$\beta = {{\bf d}_{out}} - {{\bf d}_{in}} = \int\limits_l {\nabla n} ds, $$

From this, the relationship between the angle of ray deflection and the variation in refractive index is obtained. During the actual measurement process, the angle of light deflection is calculated according to the model shown in Fig. 2. The light ray originates from the camera's projection center C and intersect with the LCD at points S and R when the lens is absent and present, respectively. In order to facilitate the acquisition of the deflection angle, which is due to the gradient of the refractive index integrated along the optical path is considered as a refraction of the light ray at the center M of the LUT. To determine the deflection angle of the light ray, it is necessary to calculate the direction of the ray as it enters and exits the lens. The direction of the incident ray can be determined using points C and R. However, due to the unknown and varying refractive index of the LUT, the location of the exit point of the ray cannot be determined. Therefore, to determine the direction of the outgoing ray, all refractions of the ray within the lens are approximated to occur at the middle point M of the LUT [24], connecting M and the point S on the LCD to determine the direction of the outgoing ray. Due to the thickness of the lens being much smaller than the distance from the lens to the LCD generally, such an approximation also does not introduce significant errors.

Fig. 2. The computational model for RIMD.

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Therefore, the deflection angles ${\beta _x}$ and ${\beta _y}$ of the ray in the x and y directions can be expressed by the Eq. (5) respectively.

(5)$${\beta _x} = arctan\frac{{\frac{{{x_m} - {x_c}}}{{{z_m} - {z_c}}} - \frac{{{x_s} - {x_m}}}{{{z_s} - {z_m}}}}}{{1 + \frac{{({x_m} - {x_c})({x_s} - {x_m})}}{{({z_m} - {z_c})({z_s} - {z_m})}}}},\;{\beta _y} = arctan\frac{{\frac{{{y_m} - {y_c}}}{{{z_m} - {z_c}}} - \frac{{{y_s} - {y_m}}}{{{z_s} - {z_m}}}}}{{1 + \frac{{({y_m} - {y_c})({y_s} - {y_m})}}{{({z_m} - {z_c})({z_s} - {z_m})}}}}, $$

where point $C({{x_c},{y_c},{z_c}} )$ is the world coordinate of the camera projection center, which can be obtained through camera calibration; points $S({{x_s},{y_s},{z_s}} )$ and $R({{x_r},{y_r},{z_r}} )$ are the world coordinates of the intersection points of the deflected and undeflected light ray with the LCD, which can be obtained through N-step phase-shifting method [25]. Taking the solution process for point $S({{x_s},{y_s},{z_s}} )$ as an example. In the absence of the lens, capturing sine phase-shift fringes in the x and y directions displayed on the screen using the camera, the phase $\phi ({x,y} )$ can be obtained using Eq. (6):

(6)$$\phi (x,y) = \arctan \frac{{\sum\limits_{i = 1}^N {{I_i}(x,y,{\delta _i})\sin {\delta _i}} }}{{\sum\limits_{i = 1}^N {{I_i}(x,y,{\delta _i})\cos {\delta _i}} }}, $$

where I represents the fringe pattern, i is the fringe index, N is the number of phase-shift steps, and ${\delta _i}$ is the phase shift. As Eq. (1) involves the arctangent operation, the phase is truncated to [-π,π]. Therefore, phase unwrapping is required. The wrapped phase $\phi ({x,y} )$ can be converted to continuous phase $\varphi ({x,y} )$ through the phase unwrapping algorithm. Then, using Eq. (7), the coordinates ${x_s}$ and ${y_s}$ on the screen can be obtained.

(7)$${x_s} = \frac{{{\varphi _\textrm{x}}}}{{2\pi }} \cdot T \cdot pixel,{y_s} = \frac{{{\varphi _\textrm{y}}}}{{2\pi }} \cdot T \cdot pixel, $$

where T is the period of the sine fringe pattern, measured in screen pixels, and $pixel$ represents the physical size of a screen pixel. The coordinate in the z-direction can be obtained through system calibration. $R({{x_r},{y_r},{z_r}} )$ can be similarly determined when the lens is in place using the same approach. Point $M({{x_m},{y_m},{z_m}} )$ is the world coordinate of the refraction point, where ${z_{m2s}}$ is defined as the distance from point M to the LCD, and ${z_m} = {z_r} - {z_{m2s}}$, and ${x_m}$ and ${y_m}$ can be obtained through Eq. (8).

(8)$$\left[ {\begin{array}{{c}} {{x_m}}\\ {{y_m}}\\ 1 \end{array}} \right] = \alpha \left[ {\begin{array}{{ccc}} 1&0&{ - {x_c}}\\ 0&1&{ - {y_c}}\\ 0&0&0 \end{array}} \right]\left[ {\begin{array}{{c}} {{x_r}}\\ {{y_r}}\\ 1 \end{array}} \right] + \left[ {\begin{array}{{c}} {{x_c}}\\ {{y_c}}\\ 1 \end{array}} \right], $$

where the coefficient α is defined as $\alpha = ({{z_m} - {z_c}} )/({{z_r} - {z_c}} )$.

Since the thickness W of the LUT is much smaller than the distance from the camera projection center to the LUT, the path of the light propagation inside the lens can be approximated as the thickness of the lens. Therefore, Eq. (4) can be simplified as:

(9)$$\beta = W\nabla n, $$

After obtaining the deflection angle of the ray and the thickness of the lens, the refractive index distribution n of the LUT can be obtained through the modal method or the zonal method. The above process also illustrates that during the measurement, calculating $\beta $ only requires knowing the distances between the camera, lens, and LCD, without any specific component placement requirements. The camera position can be freely aligned to achieve the best imaging quality based on the element configuration and LCD, without necessarily being placed at the lens's focal point. This is the distinguishing feature between RIMD and wavefront aberration measurement deflectometry.

3. Numerical simulation

In order to verify the correctness and accuracy of our proposed method, numerical simulations are conducted based on the model in Fig. 2, testing two lenses with different refractive index distributions. The diameter of the lens is 5 mm, the thickness is 1.7 mm, and the refractive index distribution is $n = {n_0} + {n_{r1}}{r^2} + {n_{r2}}{r^4}$, with specific parameters set as shown in Table 1. The distance between the pinhole camera and the LUT ${z_{m2c}}$ is set to 400 mm, and the distance between the LUT and the LCD ${z_{m2s}}$ is set to 200 mm.

Table 1. The ideal parameter settings for the two lens

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When light ray enters or exits the lens, its change in direction follows Snell's Law. However, due to the continuously changing refractive index within the GRIN lens, conventional ray tracing algorithms like Korsch raytracing [26] are not applicable. Instead, algorithms suitable for non-uniform media need to be chosen, such as Non-normalized Physical Ray Tracing [3] for non-normalized media or the Runge-Kutta method. Here, the fourth order Runge-Kutta method [27] is chosen for ray tracing. The position and direction of the light ray passing through the lens can be updated based on the local refractive index gradient as follows:

(10)$$\left\{ \begin{array}{l} {R_{n + 1}} = {R_n} + \Delta t[{T_n} + 1/6(A + 2B)]\\ {T_{n + 1}} = {T_n} + 1/6(A + 4B + C)] \end{array} \right., $$

where $\Delta t = \Delta s/n$, $\Delta s$ is iteration step size. R and T represent the position and direction respectively, and are given by:

(11)$$R = \left( {\begin{array}{{c}} x\\ y\\ z \end{array}} \right),T = n\left( {\begin{array}{{c}} {dx/ds}\\ {dy/ds}\\ {dz/ds} \end{array}} \right),D = \frac{1}{2}\left( {\begin{array}{{c}} {\partial {n^2}/\partial x}\\ {\partial {n^2}/\partial y}\\ {\partial {n^2}/\partial z} \end{array}} \right), $$

$D$ represents the product of refractive index and refractive index gradient. The constants A, B and C are functions of the refractive index gradients and are given by:

(12)$$\left\{ {\begin{array}{{l}} {A = \Delta tD({R_n})}\\ {B = \Delta tD({R_n} + \frac{{\Delta t}}{2}{T_n} + \frac{{\Delta t}}{8}A)}\\ {C = \Delta tD({R_n} + \Delta t{T_n} + \frac{{\Delta t}}{2}B)} \end{array}} \right., $$

The center of the lens is set as the origin of the world coordinate system, and the pinhole camera is treated as a point light source with coordinates $({{x_c},{y_c},{z_c}} )$. After emitting from the camera, the light ray are uniformly sampled at 0.05 mm intervals on the front surface of the lens. Connecting the sampled points to the camera provides the direction of incident ray. Upon entering the lens, they undergo refraction, and the changed direction serves as the initial values for the Runge-Kutta tracing method, with an iteration step size set to 0.01 mm. Through iteration and one refraction, the direction of emergent ray is obtained. Calculating their intersection with the screen allows us to determine the coordinates $S({{x_s},{y_s},{z_s}} )$ and $R({{x_r},{y_r},{z_r}} )$. Finally, the refractive index distribution can be reconstructed using Eq. (5), (8), and (9). The results are shown in Fig. 3.

Fig. 3. The reconstructed refractive index distributions of GRIN lens: (a) and (d) show the ideal refractive index distributions of sample 1 and sample 2, respectively; (b) and (e) show the refractive index distributions reconstructed using RIMD; (c) and (f) show the reconstruction errors obtained by subtracting (b) from (a) and (e) from (d), respectively.

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Figure 3(a) and (d) show the ideal refractive index distributions for sample 1 and sample 2, respectively. Figure 3(b) and 3(e) show the refractive index distributions obtained by fitting using the Southwell method [28]. Figure 3(c) and (f) reflect the reconstruction errors of the proposed method for different refractive index distributions, with reconstruction errors Root Mean Square (RMS) value of approximately $1.17 \times {10^{ - 6}}$ and $2.12 \times {10^{ - 4}}$ for sample 1 and sample 2, respectively. In addition, the refractive index distribution data along the diameter of the lens is also extracted, and a comparison is made with the ideal refractive index distribution, as depicted in Fig. 4. Figure 4(a) shows the comparison between the ideal gradient of sample1 and the fitted refractive index gradient of the reconstructed data. The red line reflects the error of the refractive index gradient, which is at the order of ${10^{ - 7}}$. Meanwhile, Fig. 4(b) shows the comparison result of the refractive index distribution fitted by the refractive index gradient. The error curve indicates that the RIMD has gradually increased reconstruction error from the center to the edge when reconstructing the refractive index. However, the error magnitude at the edge area is also controlled within ${10^{ - 6}}$. Figure 4(c) and (d) show the comparison results of sample2, whose refractive index gradient error is about ${10^{ - 5}}$, and the absolute error magnitude of the fitted refractive index is at the order of ${10^{ - 4}}$. It can be seen that when the refractive index changes more complexly and the gradient is larger, the accuracy of RIMD will decrease to some extent, but it still maintains a relatively high level. Therefore, numerical simulations can prove the correctness and accuracy of this method.

Fig. 4. The data comparison along the lens diameter: (a) and (c) present the comparison results of the refractive index gradients for sample1 and sample2, respectively, where the red lines indicate the refractive index gradient errors. (b) and (d) present the comparison results of the ideal distribution curve and the fitting curve of reconstructed data, where the red lines indicate the absolute errors of RIMD reconstruction.

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Moreover, by solely varying ${z_{m2c}}$, the distance between the camera and the LUT, deviations can be observed in the reconstructed results. The calculated RMS and Peak-to-Valley (PV) values of these errors are presented in Table 2. The reconstruction error consistently decreases as the distance increases, supporting our hypothesis. In the assumption of Eq. (9), due to the small thickness of the LUT and the relatively small angle of light incidence, the propagation path of light can be approximated as the thickness of the element. As the distance between the camera and LUT increases, the light ray emitted from the camera approaches parallel incidence upon reaching the front surface of the LUT. Consequently, the angle of entry into the LUT decreases, leading to smaller radial components. This reduction in the introduced error from the approximation in Eq. (9) is consistent with the results in Table 2. In the experiment, it is advisable to appropriately increase the distance between the camera and LUT while ensuring the quality of lens imaging to reduce the measurement errors of the refractive index.

Table 2. The influence of different distances on the measurement results

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4. Experiments

4.1 Procedures of RIMD

In the preceding section, the effectiveness and theoretical accuracy of RIMD are validated through numerical simulation. In this section, the same experimental setup, as illustrated in Fig. 2, is established to validate the feasibility and measurement accuracy of RIMD in practical experiments. According to the principles of RIMD, it is necessary to obtain the coordinates in the world coordinate system for the camera projection center $C({{x_c},{y_c},{z_c}} )$, as well as the points $R({{x_r},{y_r},{z_r}} )$ and $S({{x_s},{y_s},{z_s}} )$ on the LCD. Additionally, the distance from the middle of LUT to the LCD needs to be determined. In the experiment, these parameters are established through the following steps, as illustrated in Fig. 5(a). (1) Calibration of the camera and measurement system to obtain the world coordinate of the camera projection center C; (2) Capturing the fringe patterns when the lens is placed to obtain the coordinate of S ; (3) Capturing the fringe patterns without lens to obtain the coordinate of R; (4) Calculating the coordinates of the refracted point M; (5) Calculating the deflection angle $\beta $ and using the integral reconstruction algorithm to obtain the refractive index distribution.

Fig. 5. The measurement process of RIMD and system calibration schematic diagram: (a) measurement process of RIMD; (b) Adjustment of the camera, LUT, and LCD screen using a point light source microscope with ruler during the calibration process; (c) shows the external aperture added to the camera; (d) shows the target used for LUT adjustment and distance measurement.

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Step 1 of the measurement of GRIN lens using the deflectometry technique involves aligning and calibrating the measurement system, mainly using a three-coordinate measuring auxiliary device. In this experiment, a point source microscope (PSM) with an electric displacement stage is used as the auxiliary device [29]. Its structure is shown in Fig. 5(b). First, the LCD screen is adjusted to coincide with the $xoy$ plane of the established virtual Cartesian world coordinate system. Then, an external aperture with a diameter of approximately 1 mm is added in front of the camera lens which is shown in Fig. 5(c). The PSM is used to scan the position of the aperture, ensuring the accurate determination of the camera coordinates $C({{x_c},{y_c},{z_c}} )$ in the world coordinate system. Further, the PSM and a high-precision displacement stage are employed to adjust the pose of the LUT, ensuring its alignment along the optical axis of the camera and parallel to the $xoy$ plane of the world coordinate system.

Additionally, individual target is placed adjacent to the camera, the LUT, and the screen, as depicted in Fig. 5(d), to precisely measure the distances from the camera to the test element ${z_{m2c}}\; $ and from the test element to the screen ${z_{m2s}}$.

4.2 Measurement of GRIN lens

The experimental setup is established as shown in Fig. 6 to measure a GRIN lens and the results are compared with the nominal design parameters. The parameters of the GRIN lens are consistent with those of sample 2 and diameter of the lens is 5 mm and the thickness is 1.7 mm. The lens alignment is performed using a point source microscope with ruler, and a CCD camera (TXG50, Baumer, Switzerland) equipped with an external pinhole (serving as the camera's projection center) is used for on-axis measurements. An LCD screen (E-2M21GM, MTIPH, China) s installed at the vertical position of the system's optical axis. The CCD camera has a resolution of 2448× 2050 pixels, with a pixel size of 3.75 um, while the screen has a resolution of 1600 × 1200 pixels, with a pixel size of 0.2705 mm. In our experiment, the distance between the camera and the lens under test (LUT) is 249.5 mm, and the distance between the LUT and the LCD is 417.4 mm.

Fig. 6. Photo of the setup for the experimental measurement using RIMD.

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Due to limitations imposed by the lens holding device, only a partial area of the lens is measured. The experimental results obtained are illustrated in Fig. 7(a), while Fig. 7(b) shows the ideal refractive index distribution fitted based on the nominal parameters of the lens and the red circle represents the area corresponding to the current measurement. A comparison between the two reveals that the reconstructed refractive index distribution accurately reflects the overall structure where the refractive index is higher in the center compared to the edges. Refractive index data is extracted along the two red dashed lines represented in Fig. 7(a) and compared with the ideal refractive index. The results are shown in Fig. 7(c) to Fig. 7(d). Figure 7(c) and (e) presents the contrast in refractive index gradients in x and y direction respectively, while Fig. 7(d) and (f) shows the refractive index distribution in x and y direction respectively. From a one-dimensional data perspective, the reconstructed results exhibit a strong agreement with the fitting curve, displaying a highly symmetrical convex shape. The center region of the GRIN lens exhibits the smallest deviation between the fitting curve and experimental data, while the errors at the edge region are slightly higher than those at the center. In the numerical simulation, the maximum error associated with this part is approximately ${10^{ - 4}}$. The causes of any additional errors will be analyzed in the discussion.

Fig. 7. The reconstructed refractive index distributions of GRIN lens: (a) Refractive index distribution reconstructed using RIMD; (b) Ideal refractive index distribution obtained by fitting based on nominal parameters; (c) Comparison between the extracted reconstructed data along the x direction and the ideal refractive index gradient; (d) Comparison between the extracted reconstructed data along the x direction and the ideal refractive index; (e) Comparison between the extracted reconstructed data along the y direction and the ideal refractive index gradient; (f) Comparison between the extracted reconstructed data along the y direction and the ideal refractive index.

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However, there are indeed some regions of data missing and offset in the figure. These differences can be attributed to severe defects in the lens in those areas, which prevent obtaining the corresponding phase distribution through the fringe analyze, or significant deviation of light ray in those regions. However, according to the principles of RIMD, the probing light ray emitted by the camera are independent of each other. Therefore, errors caused by the defective regions do not affect other parts.

5. Discussion

5.1 Analysis of calibration error

In the experimental section, the discrepancy between the RIMD measurement results and the ground truth can be partially attributed to the accuracy of the calibration tools. This hypothesis can be validated through numerical simulations. Constrained by the precision of the electric displacement stage and point source microscope, deviations may occur when determining the position and adjusting the orientation of the LUT. Building upon previous work [22,30,31], calibration errors are introduced into the simulated experiments of Sample 2 in section 3, with the LUT experiencing a 1 mm off-axis displacement in the x-direction and a 1° rotation around the x-axis. Additionally, to simulate realistic camera imaging noise, 1% Gaussian white noise is added to the images [32]. The resulting outcomes are depicted in Fig. 8. Figure 8(a) represents the refractive index distribution obtained from RIMD, Fig. 8(b) shows the reconstruction error under this condition. The curves obtained by comparing the refractive index distribution extracted along the diameter of the lens with the ideal refractive index distribution are shown in Fig. 8(c) and (d). Figure 8(c) presents the comparison results of the refractive index gradient, while Fig. 8(d) shows the refractive index distribution. From the simulation results in Fig. 8, it can be observed that when the lens is not precisely aligned with the measurement system's optical axis, additional errors are introduced into the measurement results, which agree with the conclusions drawn from the experiments. Therefore, it can be anticipated that using more accurate alignment tools can further enhance the measurement accuracy.

Fig. 8. The reconstructed results with added calibration errors and noise: (a) Refractive index distribution reconstructed using RIMD; (b) Reconstruction error under this condition; (c) Comparison between the fitted curve of the extracted reconstructed data along the diameter of the lens and the ideal refractive index gradient; (d) Comparison between the fitted curve of the extracted reconstructed data along the diameter of the lens and the ideal refractive index.

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5.2 Analysis of lens surface shape

Ideal GRIN lens relies on changes in refractive index to control the light field, and their front and rear surfaces are usually considered as ideal planes as shown in Fig. 9(a). However, in the actual processing of GRIN lens, surface shape deviations exist, as shown in Fig. 9(b). These surface shape deviations also cause the light ray to undergo deflection during propagation, leading to the deviation of the GRIN lens's ability to control the light field from the model design. To investigate the effect of surface shape on refractive index measurement results, the front and rear surfaces of the LUT are measured using the LuphoScan profilometer, and the obtained results are shown in Fig. 9(c) and (d). Due to constraints imposed by the holding device, the diameter of the measurement region is also 3.6 mm, consistent with the size of the experimentally measured region. The RMS value of the front surface shape of the lens is 1.940μm with a PV value of 9.743μm. The RMS value of the rear surface shape is 0.638μm with a PV value of 2.848μm.

Fig. 9. Analysis of lens surface shape: (a) Light ray undergo a certain degree of deflection when passing through an ideal GRIN lens; (b) Light ray undergo additional deflection when passing through a GRIN lens with an added surface shape; (c) Surface shape corresponding to the front surface of LUT; (d) Surface shape corresponding to the rear surface of LUT.

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When the lens has surface shape, it causes additional displacement in the normal direction when light ray undergoes refraction. Figure 10(a) and (b) show the x-directional and y-directional displacement of the normal direction respectively introduced by the surface shape on the front surfaces. Figure 10(c) and (d) are the displacement of the rear surface. Introducing these displacements into a lens model with a constant refractive index and calculating the deflection angle of the light, the results are compared with the deflection angle calculated for the GRIN lens model without surface shape introduced in section 3. The results are illustrated in Fig. 10(e) and (f). They show the deflection angles in the x and y directions obtained by adding the surface shape to the lens. In this case, the deflection of light is caused purely by the variation of the surface shape. Figure 10(g) and (h) show the deflection angles of the GRIN lens model, where the deflection of light is caused purely by the variation of the refractive index. From the comparison of the RMS and PV values between the two models, the light deflection caused by the surface shape is a small fraction of the light deflection caused by the refractive index variation.

Fig. 10. Angle comparison of deflection : (a) and (b) are the displacement of the normal direction caused by the front surface shape in x and y direction respectively; (c) and (d) are the displacement of the normal direction caused by the rear surface shape in x and y direction respectively; (e) The deflection angles in the x direction caused by changes in lens surface shape; (f) The deflection angles in the y direction caused by changes in lens surface shape; (g) The deflection angles in the x direction caused by refractive index changes; (h) The deflection angles in the y direction caused by refractive index changes.

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Lenses with additional surface shapes are reconstructed for refractive index distribution using the procedure described in section 2. The refractive index distribution data along the diameter of the lens are extracted, and the difference between the reconstructed data and the ideal distribution is obtained, as shown in Fig. 11. Figure 11(a) represent the difference between the reconstructed results after introducing the surface shape and the design parameters. Figure 11(b) and (c) are the reconstruction errors of refractive index gradient and refractive index, respectively. It can be clearly seen that the reconstruction errors increase noticeably after introducing surface shapes but remains at a controllable level. From this perspective, it can be concluded that the influence of surface shape deviation on the measurement results is small.

Fig. 11. Analysis of the reconstructed results after introducing the surface shape: (a) represents the difference between the reconstructed results after introducing the surface shape and the design parameters; (b) Comparison between the fitted curve of the extracted reconstructed data along the diameter of the lens and the ideal refractive index gradient; (c) Comparison between the fitted curve of the extracted reconstructed data along the diameter of the lens and the ideal refractive index.

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5.3 Analysis of results from the perspective of focal length

Due to limitations in the laboratory setup, currently, there are no means to utilize other devices such as an atomic force microscope to obtain data for the LUT and compare it with the results from RIMD. To validate the reasonableness of the results, the measurement results are also analyzed from the perspective of focal lengths. From the properties of the GRIN lens [33], a plano element can be determined, the index of which varies as a function of the radial distance r according to:

(13)$$n(r) = {n_0}(1 - K{r^2})$$

and has a length L will have a focal length given by:

(14)$$f = \frac{1}{{{n_0}\sqrt {2K} \sin (L\sqrt {2K} )}}$$

In the experiment, the nominal refractive index distribution of the LUT is $n(r )= 1.5376 - 0.019{r^2} + 2.553 \times {10^{ - 5}}{r^4}$ and the nominal focal length is 15.665 mm. The focal length of the LUT is measured using a focal collimator, as depicted in Fig. 12. A reticle is placed at the focal point of the collimating lens. The focal collimator is illuminated by an extended source, and the sample to be tested is placed in the emergent beam. A filar eyepiece inspects the image formed at the focal plane of the test lens. The focal length of the sample is given by:

(15)$$f = \frac{{{y^{\prime}}}}{y}{f_0}$$

where $y^{\prime}$ is the measured size of the image, y is the size of the reticle, and ${f_o}$ is the focal length of the collimator objective. The measurement results are presented in Table 3.

Fig. 12. Lens Focal Length Measurement Setup

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Table 3. The measurement results of focal length

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The average value of the LUT focal length obtained from multiple experiments is $f = 15.37mm$. Considering the errors in the focal length measurement setup, it can be assumed that the actual refractive index distribution of the LUT is close to the nominal values. Extracting the refractive index calculated through RIMD from Fig. 7(f), using the same expression as the nominal distribution, and performing a linear least-squares solution, the expression is obtained corresponding to the reconstructed refractive index as: $n(r )= 1.538 - 0.01951{r^2} + 2.824 \times {10^{ - 5}}{r^4}$. Since the coefficients corresponding to r to the fourth power have a very small magnitudes, it does not significantly impact the focal length. Using Eq. (13) and (14), the focal length corresponding to the RIMD reconstructed refractive index distribution is calculated to be $f = 15.268mm$. From the calculated focal lengths, it can be observed that the results obtained by RIMD are close to both the nominal values and the experimentally measured focal length. This, to some extent, verifies the accuracy of the RIMD measurement results.

6. Conclusions

In conclusion, a radial gradient refractive index measurement method based on deflectometry is proposed, providing a detailed description of the principle of measuring GRIN lens using RIMD. Compared to other methods for measuring gradient refractive index, our approach eliminates the need for sample rotation or multi-angle imaging, allowing for full-field refractive index information to be obtained with a single measurement. The proposed method is characterized by its simplicity, speed, ease of instrument use, low cost, and insensitivity to environmental factors. In the experiments, a radial GRIN lens is measured, and the results are compared with the design parameters, achieving errors within the order of ${10^{ - 3}}$. The correctness and accuracy of the method are validated through numerical simulations, examining the impact of calibration error and lens surface shape. Additionally, result reliability is assessed by considering focal lengths. Future work can focus on improving measurement accuracy by refining the calibration method. Additionally, this method can be applied to the inverse measurement of other refractive index field distributions, such as flow fields and flame structures.

Funding

National Natural Science Foundation of China (U20A20215, 62375190); Sichuan University (2020SCUNG205).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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	$n_{0}$	$n_{r 1}$	$n_{r 2}$	$f / m m$
Sample1	1	$8 \times 10^{- 4}$	0	−367.362
Sample2	1.5376	−0.019	$2.553 \times 10^{- 5}$	15.665

	$z_{m 2 c}$	RMS	PV
Sample2	100mm	$5.476 \times 10^{- 4}$	$1.956 \times 10^{- 3}$
	200mm	$5.130 \times 10^{- 4}$	$1.851 \times 10^{- 3}$
	300mm	$4.023 \times 10^{- 4}$	$1.418 \times 10^{- 3}$
	400mm	$2.126 \times 10^{- 4}$	8.146 $\times 10^{- 4}$

Experiment	$y^{'}$ /mm	$y$ /mm	$f_{0}$ /mm	$f$ /mm
1	0.702	18	394.2	15.37
2	0.714	18	394.2	15.64
3	0.693	18	394.2	15.11
Average	/	/	/	15.37

	$n_{0}$	$n_{r 1}$	$n_{r 2}$	$f / m m$
Sample1	1	$8 \times 10^{- 4}$	0	−367.362
Sample2	1.5376	−0.019	$2.553 \times 10^{- 5}$	15.665

	$z_{m 2 c}$	RMS	PV
Sample2	100mm	$5.476 \times 10^{- 4}$	$1.956 \times 10^{- 3}$
	200mm	$5.130 \times 10^{- 4}$	$1.851 \times 10^{- 3}$
	300mm	$4.023 \times 10^{- 4}$	$1.418 \times 10^{- 3}$
	400mm	$2.126 \times 10^{- 4}$	8.146 $\times 10^{- 4}$

Refractive index measurement deflectometry for measuring gradient refractive index lens

Abstract

1. Introduction

2. Principle

3. Numerical simulation

4. Experiments

4.1 Procedures of RIMD

4.2 Measurement of GRIN lens

5. Discussion

5.1 Analysis of calibration error

5.2 Analysis of lens surface shape

5.3 Analysis of results from the perspective of focal length

6. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Tables (3)

Equations (15)

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