1. Introduction

Recent development in the information technology (IT) industry have been mostly attributed to the miniaturized and integrated manufacturing processes of the electro-optic elements constituting the devices, modules and equipment. These processes, however, make it very difficult to inspect the defects in those electro-optic components, which are highly integrated into the IT products. Thus, critical issues on the defective IT products have been occurred persistently. For solving these problems, various techniques for detecting the microscopic defects of the IT products have been researched [1–4].

Particularly, most camera lenses embedded on the mobile devices including the smart phones, tablet PCs and notebooks, now have almost the same imaging performances as those of the digital cameras. However, unlike the digital cameras employing the large-scale imaging lenses, micrometer-scale lenses are mounted on the mobile cameras. Thus, in the mobile IT products, it is very important to make those microscopic optical lenses to capture the high-resolution object images without any image degradation and distortion. In fact, a mobile camera module is composed of several micrometer-scale lenses with different sizes depending on their functions. It is recently reported that the LG Innotek developed a very slim-structured mobile camera module whose size is 5.7mm. However, the detailed specifications of those micrometer-scale lenses employed in the mobile camera were not released since those data are usually classified [5].

Basically, micrometer-scale lenses for the mobile cameras are manufactured based on the injection process [6,7]. With this process, the shape of an optical lens can be precisely manufactured, whereas its refractive-index profile tends to be made unevenly in case the injection pressures happen to be unequally imposed on the local surfaces of the optical lens in the fabrication process. This has been one of the critical issues in the injection-based lens manufacturing process since it causes the camera imaging system to be optically deformed and distorted [8].

Therefore, many attempts have been done for measuring the surface and refractive-index profiles of optical lenses [9–14]. Most of them, however, have been tried on relatively large-scale optical lenses. In those trials, three kinds of methods have been used for measuring the refractive-index profiles, which includes two beam-based optical interferometers [9], single beam-based optical interferometers [10] and other optical systems without employing the interferometers including the phase retrieval-based optical system [11,14]. Here, it is noted that a liquid-immersion method has been used for the effective detection of the refractive-index profiles of optical components [9–13]. A target object is immersed in a liquid whose refractive-index is almost same with that of the target. Then, both of the target and liquid are matched in the refractive-index, which allows effective detection of 3-D data of the target object.

Thus far, the single beam-based lateral shearing interferometer (LSI) system has been developed in various types of optical configurations for measuring the shapes and refractive-index profiles of objects and for diagnosing the density and flow of multiple small objects, and for efficient usage of the available bandwidth of any sampling devices due to its simple, robust and cost-effective configuration [15-23].

Because in the LSI system, the holographic interference pattern is formed between two sheared object beams generated from the single object beam passing through the transparent target object unlike the conventional two-arm holographic interferometer, the holographic interference pattern of the LSI system inevitably contains the unwanted duplicated object image, which limits the measurable size of the object because the duplicated image is largely overlapped with the real image, causing the real object image to be distorted. For this, LSI systems using the phase-only spatial light modulators (SLMs) were proposed [21], where duplicated images were tried to be removed based on the double refraction by taking advantage of the birefringent property of the SLM. In addition, a computational shear interferometry (CoSI) using the phase shifting method was also proposed to overcome the duplicate image problem [22].

Furthermore, to alleviate this duplicate image problem of the conventional LSI system, another type of the LSI system called a modified lateral shearing interferometer (MLSI), was also proposed [23]. In this system, to remove the unwanted duplicated image, a concept of the subdivided two-beam interference (STBI) to separate the single object beam into one half-object beam with object information and the other half-reference beam without object information was proposed. The MLSI system can partially generate the same holographic interference patterns as those of the Michelson and Mach-Zehnder interferometer systems. Thus, with this MLSI system, the duplicated image problem can be solved. However, in this system, the lateral shearing distance (LSD), which represents the distance between the center-points of two sheared object beams, becomes fixed by the thickness of an optical window glass, so that its effective interference area (EIA) representing the overlapped area between the two half-object and half-reference beams, is limited as well as the measurable object size and its position in the object beam are restricted.

Thus, as an alternative, we propose another type of the single-arm off-axis holographic interferometer (SA-OHI) system for the robust detection of 3-D shapes and refractive-index profiles of the micrometer-scale lenses employed in most mobile camera modules. In the proposed system, the optical window glass is replaced with a couple of pellicle beam splitter (BS) and optical mirror, where the pellicle BS is fixed, but the optical mirror can be controlled to be tilted. With these pellicle BS and optical mirror, two off-axis beams can be generated from the single object beam. Each object beam is set to be composed of two areas with and without object information, which are called half-object and half-reference beams, respectively. Then, a subset of half-object and off-axis half-reference beams makes a holographic interference pattern just like the conventional two-arm holographic interferometer. This type of the holographic interferometer system called SA-OHI, can solve the DC bias, virtual and duplicated image problems occurred in most conventional systems.

The operational performance of the proposed system is analyzed based on ray-optics. In addition, to confirm the feasibility of the proposed system, experiments with test micrometer-scale lenses are performed and the results are compared to those of the conventional system.

2. Conventional MLSI system

Figure 1 shows an optical configuration of the conventional modified lateral shearing interferometer (MLSI) system [23]. As seen in Fig. 1, the object beam passing through the target object is subdivided into two areas with and without object information, which are called half-object and half-reference beams, respectively, by controlling the object’s location in the object beam and the incident angle of the object beam into the optical window glass. This single object beam is then reflected from both of the front and back surfaces of the optical window glass to generate two sheared object beams, which are referred to as OB₁ and OB₂. Then, a subset of half-object and half-reference beams of those beams can form an interference pattern without having a duplicated object image. In this system, the duplicated image problem can be solved by locating the target object on the somewhat displaced location from the optic axis, not right on the optic axis, as well as by controlling the incident angle of the object beam into the optical window glass considering the thickness of the optical window glass.

 

Fig. 1 Formation of the holographic interference pattern in the MLSI system based on a concept of the sub-divided two beam interference (STBI)

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As seen in Fig. 1, one object beam reflected from the front surface of the optical window glass, OB₁ is composed of two half-object and half-reference beam areas of O₁ and R₁ with and without object information, respectively. In addition, the other object beam reflected from the back surface of the optical window glass, OB₂ is also composed of two object and reference beam areas of O₂ and R₂, respectively. Thus, a hologram pattern can be partially formed between the half-object beam area of one object beam, O₁ and the half-reference beam area of the other object beam, R₂ just like the conventional two-arm holographic interferometer system. Thus, in the conventional MLSI system, the duplicated image problem seems to be solved. Here, the lateral shearing distance (LSD), two-beam interaction angle (TIA) and effective interference area (EIA) are the key parameters closely related to the duplicated image issue in the conventional MLSI system. As mentioned above, the LSD derived as Eq. (1), represents the distance between the center-points of two sheared object beams, where d, t, n₁, n₂ and θ_i denote the LSD, thickness of the optical window glass, refractive indices of the air and optical window glass, and angle between the object beam and optical window glass. The TIA expressed as Eq. (2), is defined as the two-beam intersection angle to make an interference pattern between two sheared object beams without a duplicated image, where r represents the radius of each of the two-sheared object beams. The EIA derived as Eq. (3), represents the overlapped area between two sheared object beams, where θ_t means TIA.

This MLSI system, however, has several problems in its practical application. Since the LSD is fixed just by the thickness of the optical window glass, the resultant EIA of this system is also limited. It means that the measurable object size and allowable object location in the object beam have to be very restricted in this system. Moreover, it cannot evade from the duplicated image problem in case the object size becomes bigger than the radius of the object beam. Thus, for being free from the duplicated image problem, the target object must be located on the side area displaced from the optic axis, not around the optic axis, as well as the LSD is set to be equal to the radius of the object beam.

3. Proposed system

3.1 Optical configuration of the proposed system

Figure 2 shows an optical configuration of the proposed SA-OHI system. As seen in Fig. 2, the optical window glass used in the conventional MLSI system is replaced with a couple of pellicle BS and optical mirror. Thus, by using them, two sheared beams can be generated from the single object beam passing through the target object. That is, one beam acting as the object beam is reflected from the front pellicle BS, and the other beam acting as the off-axis reference beam is reflected from the back optical mirror.

 

Fig. 2 Optical configuration of the proposed SA-OHI system

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Basically, the proposed system generates the holographic interference pattern based on a concept of the STBI just like the conventional MLSI system. The bottom line of the STBI scheme in the conventional system is that the target object should be located on the side location displaced from the optic axis. Moreover, the key parameters such as the LSD, TIA and EIA in the conventional system must be controlled to avoid the duplicate image problem. Among them, the LSD is the most important parameter because it can decide the resultant parameters of the EIA and TIA, as well as the maximum measurable object size. Here, the LSD is fixed, so the corresponding TIA, EIA and maximum measurable object size are also fixed.

On the other hand, in the proposed system, the optical window glass is replaced with a couple of the pellicle BS and optical mirror, where the pellicle BS is fixed, but the optical mirror can be controlled to be tilted. Thus, key parameters of the proposed system including the LSD, TIA and EIA can be optimally controlled depending on the size and location of the target object.

3.2 Operational principle of the proposed system

3.2.1 Interference angle

Figure 3 shows an operational concept of the proposed system to generate the holographic interference pattern from two sheared object beams reflected from both of the pellicle BS and tilted optical mirror. As seen in Fig. 3, based on a couple of pellicle BS and optical mirror, two sheared object beams acting as the object beam and off-axis reference beam are generated from the single object beam. Then, an interference pattern is formed between them based on the STBI concept. Thus, the proposed system can be viewed as an off-axis holographic interferometer just like the conventional two-arm holographic interferometer system.

 

Fig. 3 Operational principle of the proposed system

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Unlike the conventional system, the LSD can be controlled just by tilting the angle of the optical mirror. Thus, instead of the LSD, a new parameter called an interference angle (IA), which represents the angle between one object beam reflected from the pellicle BS and the other off-axis object beam reflected from the tilted optical mirror, seems to be an important parameter in the proposed system since two-beam interference is closely related to this IA.

As seen in Fig. (3), the IA can be defined by the angle difference of θ₁-θ₂, where θ₁ and θ_2, respectively, represent the angle between the pellicle BS and one object beam, and the angle between the optical mirror and the other object beam. Moreover, the distance of the CCD camera from the pellicle BS, which is denoted as the interference distance l, can be represented from the trigonometric relationship among the incident beam angle to the pellicle BS θ₁, the angle between the pellicle BS and one object beam θ₂, the angle between the optical mirror and the other sheared object beam t₁, and the radius of the object beam r as shown in Eq. (4).

Thus, the IA representing the angle between two sheared object beams of θ₁-θ₂, can be derived as Eq. (5), where d′ denotes the LSD of the proposed system, representing the distance between the center points of one object beam and the other off-axis object beam.

In the conventional system, the LSD is given by the fixed value of d, whereas the corresponding value of the proposed system, d′ can be given by d·cos(θ₁-θ₂). That is, d′ is ranged from 0 to d depending on the tilted angle of the optical mirror. For the case that the radius of the object beam and the distance between the pellicle BS and optical mirror are set to be r = 10mm, t₁ = 10mm, n₁ = 1.0 and n₂ = 1.5 respectively, the LSD dependences on the IA and thickness of the window glass in each of the proposed and conventional systems are shown in Figs. 4(a) and 4(b), respectively.

 

Fig. 4 LSD variations depending on the IA and thickness of the window glass in the (a) proposed and (b) conventional systems, respectively

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As seen in Fig. 4(a), the LSD of the proposed system can be controlled depending on the tilted-angle of the optical mirror. Here, for the angle variation of the IA from 0 to 0.066rad (3.8°), the corresponding LSD can be ranged from 0 to 20mm. However, in the conventional system, the LSD is fixed to about 7.55mm as seen in Fig. 4(b).

3.2.2 Allowable location of the target object

The LSD value may determine the allowable location of the target object as seen in Fig. 5. In the side views of Fig. 5, the LSD can be changed by the IA in the proposed system, whereas it is fixed in the conventional system. Moreover, the allowable location of the target object is given by 2r-d′-s in the proposed system, but it is fixed to ‘2r-d-s’ in the conventional system, where s denotes the object size. That is, in the proposed system, the allowable location of the object 2r-d’-s, can be changed depending on d’ just by controlling the IA. Here, the allowable location ranges from the centers of the object beams in the conventional and proposed system can be derived as r-d/2-s/2 ~3r-3d/2-3s/2 and r-d’/2-s/2 ~3r-3d’/2-3s/2, respectively.

 

Fig. 5 Schematic diagrams of the allowable locations of the target object in the (a) conventional and (b) proposed systems

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If the object size gets bigger than 2r-d′-s, the object would be located out of the EIA in the conventional system. On the other hand, in the proposed system, the location of the object can be controlled by tilting the angle of the optical mirror depending on the object size and its location in the object beam. For the case that the radius of the object beam and the distance between the pellicle BS and optical mirror are set to be r = 10mm and t₁ = 10mm, respectively, the allowable location of the target object in the object beam can be given by 0 ~20mm-d’-s in the proposed system, whereas its corresponding value is fixed to 12.45mm-s in the conventional system. Moreover, when the object is located on the center of the object beam, the object image is to be distorted since part of the object image gets away from the interference area in the conventional system regardless of the object size is. However, in the proposed system, this image distortion problem can be solved just by controlling the LSD in the proposed system.

3.2.3 Two-beam intersection angle and effective interference area

As mentioned above, the EIA represents the overlapped area between two sheared object beams. In the conventional system, two sheared object beams are shifted just along the lateral direction, thus the resultant EIA can be determined by the LSD. In the proposed system, the IA between two sheared object beams determines the EIA. Figure 6 shows the EIA, representing the overlapped area between two half-object and half-reference beams, whose area depends on the location and size of the target object as well as the IA.

 

Fig. 6 EIAs between two half-object and half-reference beams and its dependence on the location and size of the target object

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As seen in Fig. 6, the angle to make an interference pattern between two sheared object beams without the duplicated image problem can be defined as the two-beam intersection angle (TIA) of θ_t, which is given by Eq. (6) [23].

By using the TIA of Eq. (6), the size of the EIA, S_EIA, where a meaningful area of the interference pattern is formed, can be calculated by Eq. (7) [23].

Actually, unlike the conventional system, the LSD, TIA and EIA of the proposed system can be optimized depending on the size and location of the target object just by controlling the IA. Figure 7 shows the EIA dependence on the LSD, where the EIA increases as the LSD decreases due to the increased interference area. When the radius of the object beam and the distance between the pellicle BS and optical mirror are set to be r = 10mm and t₁ = 10mm, respectively, the EIA of the proposed system is ranged from 0 to 20cm², whereas the corresponding value of the conventional system is fixed to 13.78cm².

 

Fig. 7 EIA dependence on the LSD in the proposed system

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3.2.4 Maximum measurable object size

Figure 8 shows the maximum measurable object size depending on the EIA. That is, it can be defined as two kinds of values such as 2r-d’ and d’ according to the LSD.

 

Fig. 8 Maximum measurable object size depending on the LSD in the proposed system

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If the LSD becomes bigger than the radius of the object beam, the maximum measurable object size can be given by 2r-d’. Thus, in case the object size gets bigger than 2r-d’, part of the object happen to be distorted. On the other hand, if the LSD gets smaller than the radius of the object beam, the maximum measurable object size can be given by d’. In this case, if the object size becomes bigger than d’, the object happens to be overlapped with its duplicated object image. As seen in Fig. 9, the allowable location of the object increases as the EIA gets bigger. However, the variation of the maximum measurable object size looks different from that of the allowable location of the target object depending on the EIA.

 

Fig. 9 Maximum measurable object size depending on the EIA in the proposed system

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As seen in Fig. 7, when the LSD decreases until it equals to the radius of the object beam, the corresponding EIA increases, and at the same time the measurable object size also increases as shown in Fig. 9. However, when the LSD gets shorter than the radius of the object beam, the corresponding EIA still increases but, the measurable object size gets decreased. That is, the measurable object size can be controlled by the LSD. The duplicated image problem happens to occur depending on the object size as shown in Fig. 10. For the case that the radius of the object beam and the distance between the pellicle BS and optical mirror are given by r = 10mm and t₁ = 10mm, respectively, the maximum measurable object size is equal to the radius of the object beam. Thus, under this situation, the measurable object size becomes equal to, or smaller than the radius of the object beam.

 

Fig. 10 Occurrence of the duplicate image problem depending on the relationship between the LSD, and the size and location of the target object

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3.3 Optical capturing and digital reconstruction of the hologram patterns of a target object

Figure 11 shows a block-diagram of the proposed system, which is largely composed of four processes such as optical detection of the holographic interference pattern of a target object, digital reconstruction of the detected holograms, and extraction of 3-D shape and refractive-index data of the object from the reconstructed image.

 

Fig. 11 Overall block-diagram of the proposed system composed of three processes such as optical detection, digital reconstruction of the hologram patterns, and extraction of 3-D shape and refractive-index profiles

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Now, depth data of the target object can be extracted from the captured hologram pattern by using either of the Fresnel diffraction approximation (FDA) and angular spectrum (AS) methods. That is, the hologram pattern of the object on the hologram plane is transformed into the image plane by using one of those methods mentioned above. Then, the complex amplitude of the object on the image plane is converted into the phase data, and from which 3-D metrological object image can be reconstructed. Here it must be noted that for the FDA method case, the distance between the object and hologram planes must be sufficiently long enough compared to the size of the object or the hologram to meet the Fresnel approximation condition. Thus, this method cannot reconstruct the near fields for the diffractive objects. On the other hand, the AS method can reconstruct the wave fields of the objects at any distances from the hologram plane [24]. Therefore, in this paper, 3-D object images and refractive index distribution are numerically reconstructed from the phase difference calculated based on the AS method at the short distance from the hologram plane.

3.3.1 Optical detection of the hologram patterns of an object

The optical setup of the proposed system for capturing the hologram patterns of the target object is optimized for being free from the duplicate image problem depending on the size and location of the object to be measured. After that, the hologram patterns with and without the target object are obtained by using the CCD camera as seen in Fig. 11. For the effective detection of the holographic interference patterns of the target object, the liquid immersion method must be employed. This method can reduce the divergent angle of the object beam passing through the target object due to the abrupt difference in the refractive-index between the target object and ambient air, which enables the object beam to be involved in interference much more.

3.3.2 Digital reconstruction of the detected hologram patterns

Principally, the hologram pattern of an object is formed by the interference between two half-object and half-reference beams as shown in Fig. 1. Thus, the intensity of the interference pattern on the hologram plane is given by Eq. (8).

For the numerical reconstruction, another reference beam R₂ is illuminated on the hologram pattern of Eq. (8). The complex amplitude on the hologram plane is then given by Eq. (9).

Here, U(x, y, 0) represents the complex amplitude on the hologram plane. O₁ denotes the half-object beam with object information, which is reflected from the pellicle BS. In addition, R₂ means the half-reference beam without object information, which is reflected from the tilted optical mirror. Now, the angular spectrum of the complex amplitude is given by the Fourier transform as follows.

Where f_x and f_y denote the spatial frequencies in the x- and y-direction, respectively. By applying the low-pass filtering operation to the resultant angular spectrum, the un-diffracted reference beam called DC bias and the conjugate virtual image can be removed.

Where $f_{x_n}$and $f_{y_m}$mean the modified spatial frequency ranges in the x- and y-direction for removing the DC bias and conjugate virtual image. That is, using the delta-function, a filtered real object image can be obtained from the modified spatial frequency ranges. Thus, ${\widehat{U}}_L(f_x,f_y,0)$represents the filtered complex amplitude in the spatial frequency domain only with the object image. Thus, by using the AS method, the object image can be reconstructed and given by Eq. (12), where k, λ and z denote the wave number, wavelength of the light source and reconstruction distance between the hologram and image plane, respectively.

The filtered complex amplitude in the frequency domain is finally converted into the complex amplitude in the space domain through the inverse-Fourier transformation as follows.

3.3.3 Computation of the 3-D shape of the object

Basically, the 3-D shape of an object can be computed from the numerically reconstructed complex amplitude of the object image of Eq. (13). Here, the intensity of the complex amplitude $U(x,y,z)$ on the image plane $I(x,y,z)$ is given by Eq. (14).

Also, the phase information of the object locating at the distance d can be obtained by using the real and imaginary parts of the complex amplitude on the image plane as follows.

Here it must be noted that phase data are separately computed for each of the complex amplitudes of${\varphi }_O(x,y,d)$ with object information and ${\varphi }_R(x,y,z)$without object information. Thus, $U_O(x,y,z)$and$U_R(x,y,z)$represents the complex amplitudes with and without object information on the image plane, respectively. Phase data for each of them can be then given by Eqs. (16) and (17), respectively.

Here, the difference between the phases with and without object information on the image plane represents the phase data only related to the object, which can be given by Eq. (18).

Now, 3-D metrological information of the object can be computed based on the relationship between the optical path length and phase difference of Eq. (18), which is given by Eq. (19).

Where $\Delta n(x,y,z)$ and $\Delta L(x,y,z)$ represents the changes of the refractive-index and thickness of the object, respectively. Therefore, the thickness of the object along the depth direction can be derived as Eq. (20), where O and P represent the number of pixels in vertical and horizontal directions, respectively.

3.3.4 Computation of the 3-D refractive-index profile of the object

Based on the phase difference of Eq. (19), the refractive-index profile of the object can be computed. In Eq. (20), Δn is given by Eq. (21), which represents the difference between the refractive-indices of the object (n_obj) and index-matching oil (n_oil).

In case the object locally has errors in the refractive-index profile, the refractive-index of the object can be considered to be composed of two components of the designed and erroneous ones as follows.

Here, n_{design_onj} and σ_{error_obj} denote the designed and erroneous refractive-index values of the object, respectively. Thus, the erroneous refractive-index distribution can be given by Eq. (23).

For the case of n_obj = n_{design_obj}, no error occurs in the refractive-index distribution of the object, which means the object has been manufactured without a defect in terms of the refractive-index distribution.

4. Experiments and the results

Figure 12 shows an optical setup of the proposed system. A linearly polarized He-Ne laser providing the CW output at the wavelength of 632.8nm with the maximum output power of 2mW is used as the light source. As the test objects, two kinds of optical lens such as the half ball lens whose diameter and thickness are 500μm and 250μm, and the aspheric lens whose diameter and thickness are 1.6mm and 1.0mm, respectively, are used [25–27].

 

Fig. 12 Optical experimental setup of the proposed system

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As seen in Fig. 12, the optical beam coming from the He-Ne laser passes through the test lens, and the object beam with 3-D information of the test lens is magnified with the objective lens (5 × ). Then, its diameter is tailored to be 20mm using a control lens, and input to the pellicle BS (Model: BP233, Thorlabs), where one object beam is reflected from the pellicle BS, and the other is reflected from the tilted optical mirror. These two sheared object beams make a holographic interference pattern and captured on the CCD camera with 2,048 × 2,048 pixels (Model: B2020, IMPERX), where each pixel has a resolution of 7.4μm × 7.4μm. Since the lateral accuracy of the proposed system can be determined by the number of pixels of the CCD camera and magnification of the objective lens, it is calculated to be 1.28μm here. In addition, the axial accuracy of the proposed system is also estimated to be 2.47nm basing on the wavelength of the laser and the number of bits of the CCD camera.

In the experiments, the radius of the object beam and the distance between the pellicle BS and optical mirror are set to be r = 10mm and t₁ = 10mm, respectively. Under this condition, the system parameters such as the IA, LSD and EIA are set to have the ranges of 0~0.66rad (3.8°), 0~20mm and 0~20cm², respectively. In addition, the allowable location of the test lens and the measurable lens size are set to be 0~20mm-s and 0~10mm, respectively. For the comparative performance analysis, the radius of the object beam and thickness of the optical window glass are also set to be 10mm in the conventional system Thus, the corresponding LSD and EIA values are given by 7.55mm and 13.78cm², respectively, as well as the location and maximum measurable size of the test lens are set to be12.45mm-s and 7.55mm, respectively.

4.1 Detection of holographic interference patterns using a liquid immersion method

For effective detection of the holographic interference patterns of the test lenses, the liquid immersion method is employed in the experiments, where the immersion oil with a refractive-index of 1.5174 is used. Figures 13(a)-13(d) and 13(e)-13(h) show the captured holographic interference patterns with and without test lenses from the conventional and proposed systems. That is, Figs. 13(a)-13(d) show the hologram patterns containing the lens data, whereas those of Figs. 13(e)-13(h) are used as the reference hologram patterns. Here, the test aspheric lens whose diameter and thickness are 1.6mm and 1.0mm, respectively, is set to be located at about 2mm from the optic axis. The object beam passing through this test lens is then magnified to 8mm with the objective lens (5 × ). In fact, to avoid the duplicated image problem, it should have been located in the range of 2.225mm to 6.675mm, apart from the optic axis when the LSD is 7.55mm. Under this circumstance, the duplicated image problem inevitably occurs in the conventional system as seen in Fig. 13(a). Moreover, the magnified object image also exceeds the maximum measurable object size of 7.55mm. Thus, both real and its duplicated object images of the test aspheric lens are simultaneously generated and overlapped as seen in Fig. 13(a).

 

Fig. 13 Captured object and reference hologram patterns (a), (e) and (c), (g) from the conventional system for the case of the aspheric and half ball lenses, and (b), (f) and (d), (h) from the proposed systems for the case of the aspheric and half ball lenses, respectively

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On the other hand, in the proposed system, this location limitation of the test aspheric lens can be solved just by tilting the angle of the optical mirror even though it was originally located at the same location of the conventional system. Here, the LSD can be increased from 7.55mm to 10mm by controlling the IA from 0.042rad (2.4°) to 0.033rad (1.9°). Under this condition, the object can be located in the range of 1mm to 3mm from the optic axis for being free from the duplicated image problem if case the LSD is extended to 10mm. Moreover, the magnified object image does not exceed the maximum measurable object size of 10mm. Thus, as seen in Fig. 13(b), there exists no hologram data related to the duplicated image in the captured holographic pattern in the proposed system.

In addition, Figs. 13(c) and 13(d) shows the hologram patterns for the case of the test half ball lens whose diameter and thickness are 500μm and 250μm, respectively. Here, the half-ball lens with the much smaller size than the aspheric lens, is set to be located at about 3mm from the optic axis. In this situation, no image distortions occur in both of the conventional proposed systems since the distances from the center of the object beam to the object do not exceed their respective LSDs of 3.775mm and 5mm, as well as the magnified object size of 2.5mm is also smaller than their LSDs.

4.2 Digital reconstruction of captured holographic interference patterns

For determination of the 3-D shapes and refractive-index profiles of the test aspheric and half ball lenses from the captured holographic interference patterns of Fig. 13, at the first step, phase information of each hologram pattern are calculated based on the digital reconstruction processes of Eqs. (10)-(15), which are shown in Fig. 14.

 

Fig. 14 Block-diagram of the digital reconstruction process of the captured hologram patterns for extraction of 3-D shapes and refractive-index profiles

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Of course, before those phase extraction processes are performed, DC bias and complex conjugate image terms have been eliminated from the captured hologram patterns just by filtering processes. After reconstructing the hologram patterns by using the AS method, phase information of the test lenses are calculated from the reconstructed object images using Eqs. (16) and (17). Then, phase differences between the reconstructed images with and without object information can be calculated using Eqs. (18) and (19), where λ and Δn are given by 632.8nm, and 1.600(aspheric lens) - 1.517(index-matching oil) and 1.515(half ball lens) - 1.517(index-matching oil), respectively.

As seen in Fig. 15(a), the real object image of the aspheric lens is reconstructed together with its duplicated image and they are overlapped as expected from the captured hologram pattern of Fig. 13(a). Thus, the 3-D surface and refractive-index distributions of the aspheric lens can’t be extracted from those overlapped area. On the other hand, for the case of the proposed system, as seen in Fig. 15(b), only the real object image is reconstructed without the duplicated image as expected from the captured hologram pattern of Fig. 13(b). In addition, for the case of the half ball lens, in both conventional and proposed systems no duplicated images have been occurred as seen in Figs. 16(c) and 16(d).

 

Fig. 15 Phase information of the aspheric and half ball lenses extracted from the captured hologram patterns in the (a), (c) Conventional and (b), (d) Proposed systems

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Fig. 16 3-D surface profiles of the aspheric and half ball lenses calculated from their phase data of Figs. 15(a)-15(d), respectively

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4.3 Determination of the 3-D shapes of the test lenses

For determination of the 3-D surface profiles of the test lenses, thickness data of the test lenses are computed from their phase data extracted from the captured hologram patterns with Eq. (20). Figure 16 shows the 3-D shapes of the test aspheric and half ball lenses generated from the calculated thickness data.

As seen in Fig. 16(a), the duplicated image causes the real object image to be severely distorted in the conventional system, which results in an inaccurate determination of the 3-D shape as well as the 3-D refractive-index distribution of the aspheric lens. On the other hand, as seen in Fig. 16(b), the 3-D shape of the aspheric lens has been clearly reconstructed without the duplicated image problem in the proposed system. In addition, for the case of the half ball lens, as seen in Figs. 16(c) and 16(d), there exist no 3-D shape distortions since the test lenses were positioned at the appropriate positions with the proper object sizes in both systems as we expected from their hologram patterns of Figs. 13(c) and 13(d).

To confirm the measurement accuracy of the 3-D surface of the aspheric and half ball lenses in the proposed system, experiments have been performed with four samples for each of the test lenses. From them, the averaged values of the diameter and thickness of the aspheric and half ball lenses have been measured to be 1,599.945μm, 1,000.043μm and 499.994μm, 250.010μm, respectively. Thus, the diameter and thickness errors of each of the aspheric and half ball lens are estimated to be 0.055μm, 0.043μm and 0.006μm, 0.010μm, respectively. Since the diameter and center thickness tolerance of the aspheric and half ball lenses provided by the companies are given by ± 15μm, 4μm and ± 5μm, respectively [25,26], those measurement errors in the 3-D shape appear to be within their tolerance ranges, which means that the aspheric and half ball lenses used in the experiments have been found to have no defects in terms of the 3-D shape.

4.4 Determination of the 3-D refractive-index profiles of the test lenses

Defects such as cracks, bumps and scratches on the lens surface can cause the critical problems such as image distortions or aberrations. However, even though the lens has no surface defects, its uneven refractive-index distribution happened to occurred in the manufacturing process, may cause much more critical problems than the shape deformation when it is used for optical imaging. Thus, the refractive-index profile of the optical lens must be accurately inspected and those lenses with the refractive-index profiles varying within the specific error tolerance ranges must be selectively used in the practical optical imaging systems.

For this, in this paper, the difference between the designed and measured 3-D refractive-index distributions of the aspheric and half ball lenses are numerically calculated by Eq. (23) and the results are shown in Fig. 17. Here, the refractive indices of the aspheric and half ball lenses are given by 1.515 and 1.600, respectively. In addition, the refractive-index of the matching oil employed in the experiments is given by 1.517.

 

Fig. 17 Differences between the designed and measured refractive-index distributions of the aspheric lens in the (a) 1-D, (b) 2-D and (c) 3-D forms, as well as those of the half ball lens in the (d) 1-D, (e) 2-D and (f) 3-D forms, respectively

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As seen in Figs. 17(a)-17(c), the difference between the designed and measured refractive-index distributions of the aspheric lens has been measured to be in the range of −4.278 × 10⁻⁹ to 9.329 × 10⁻⁸. According to the data sheet of the test aspheric lens provided by the company, its refractive-index tolerance is given by ± 0.002 [25]. Thus, the test aspheric lens employed in the experiments has been found to have a very good quality in terms of the refractive-index profile since the test aspheric lens has the much lower tolerance than the designed value provided by the company. In addition, as seen in Figs. 17(d)-17(f), the difference between the designed and measured refractive-index distributions of the test half ball lens has been measured to be in the range of 2.437 × 10⁻¹¹ to 3.701 × 10⁻⁹. Likewise, this experimental result also confirms that the difference between the designed and measured refractive-index profiles of the half ball lens would be too tiny to be neglected [27]. In short, the test aspheric and half ball lenses employed in the experiments have been found to have no errors in their refractive-index profiles.

4.5 Performance comparison between the conventional and proposed systems

As mentioned above, several key parameters including the LSD, TIA and EIA, have been fixed in the conventional system, which means that its allowable object location in the object beam and measurable object size have been fixed. Thus, the flexibility of the conventional system turns out to be very low in its practical application fields. On the other hand, in the proposed system, those parameters can be made to have some dynamic ranges just by controlling the IA. That is, in case an arbitrary target object is to be tested, key parameters of the proposed system can be optimized over their operating arranges for enhancing the measurement performance. Table 1 summarizes the comparison results between the key parameters of the conventional and proposed systems. As seen in Table 1, under the condition that the thickness of the optical window glass and distance between the pellicle BS and optical mirror are given by 10mm and 10mm, respectively, the values of the key parameters have been fixed in the conventional system, whereas those values have some dynamic ranges in the proposed systems, allowing its flexibility to be much extended in the practical applications.

Table 1. Comparison results of the key parameters between the conventional and proposed systems

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5. Conclusions

In this paper, a single-arm off-axis holographic interferometer (SA-OHI) system for visual inspection of 3-D surface and refractive-index profiles of a micrometer-scale optical lens has been proposed. In the proposed system, a couple of pellicle beam splitter and optical mirror has been employed for generation of two sheared off-axis beams from the single object beam by controlling the tilted angle of the optical mirror. Each sheared object beam have been divided into two areas with and without object data, and these subdivided object and reference beams have made interference patterns just like the conventional two-arm holographic interferometer. From the ray-optical performance analysis and experiments with test lenses confirm the feasibility of the proposed system in the practical application fields of 3-D visual inspection of microscopic defects.