Effect of a q-plate placed behind the objective lens in an imaging camera

In coaxial illumination telecentric imaging optical systems such as microscopes and imaging cameras, functions such as surface shape observation and wide-field confocal observation are realized by placing a spatial polarization control device on the pupil plane of the objective lens. The measurement principle is the detection of changes in polarization distribution between the incident light and the detected light. The slight change in the optical path that occurs with the reflection on the sample surface is converted into polarization information by the spatial polarization control device. Thus, changes in the light path caused by the tilt of the sample surface, scattering due to nano-unevenness, and focus out of the sample are projected as polarization images on the camera.


INTRODUCTION
Polarized structured beams are created by manipulating the polarization and phase distribution of the beams. Its typical examples include cylindrical vector beams that have radial polarization, azimuthal polarization, and optical vortices. The polarized structured beams have wide applications, including microscopy [1][2][3] laser processing [4], optical communications [5], optical trapping [6], and particle acceleration [7].
These beams can be created using liquid crystal on silicon and other liquid crystals devices and other devices. Some of these devices are electrically controllable and have a high degree of freedom, while some devices are optical elements that have a fixed polarization structure. A typical example is a half-wavelength plate with a distribution along the optical axis called a q-plate [8]. Such an optical element can be more easily installed in devices compared to the variable control type. In addition, it is a practical option for applications that do not require a large number of polarization structure patterns, products that require low cost, or compact optical systems that do not require a large insertion space. In optical imaging, some studies have attempted to adjust image characteristics by inserting a pupil filter behind the objective lens [9], which includes traditional observation techniques such as Schlieren observation, differential interferometry, phase contrast microscopy, and narrowing the aperture stop of a camera.
In this study, we propose a new optical imaging technique by installing optical elements that generate polarization structures in the pupil plane. This method detects slight deviations in the optical path of light due to nano-order changes in surface tilt and nanotopology on the surface of a sample with high sensitivity. In this paper, we present the principle, theoretical calculations, and measurement results of the proposed method.

PRINCIPLE
The proposed method sensitively detects slight deviations in the optical path due to minute variations in the surface of a sample. Figure 1 illustrates the detection principle used in the proposed method. A polarization element that generates a structured polarization is installed behind the objective lens in an imaging system that can measure reflected light with coaxial illumination. The optical element may have different configurations. This module consists of two types of compartments: Cell 1, which does not change the incident polarization, and Cell 2, which rotates the incident polarization by 90° with respect to the incident linearly polarized light shown in Figure 1. For example, Cells 1 and 2 correspond to half-wavelength plates with optical axes at 0° and 45°, respectively. Next, a single incident ray passes through Cell 1 and enters the sample while maintaining the incident polarization. The ray having the same polarization as the incident polarization returns to Cell 1 at an axisymmetric position with respect to the optical axis. Since it is assumed here that there is no polarization change due to reflection on the sample surface, the polarization after passing through Cell 1 in the opposite direction is also unchanged, so the same polarization as the incident polarization is returned. In the case the reflection direction shifts, diffracted light is generated by the slope and nanosteps, respectively. As a result, the reflected light returns to Cell 2 instead of Cell 1, and the polarization is rotated by 90°. In this way, the shift of optical pass is converted into polarization information, allowing the nanomorphology of the sample surface to be detected by polarization. This scheme applies to any ray passing through this optical system. As shown in Figure 2, the optical path and polarization of the beam passing through this optical system change. The figure assumes an optical polarizing element, that generates radial polarization, although other polarization-structured beams as described above are also applicable in the system. As shown in Figure 2(a), the linearly polarized incident light first enters the polarizing element, where the polarization distribution is converted to radial polarization ( Figure 2(b)). The light is then focused on the workpiece through an objective lens, and the reflected light is collimated again by the objective lens and transmitted through the polarizing element from the opposite direction. A polarization beam splitter (PBS) is then used to divide the polarization into two orthogonal polarizations. The polarization parallel to the incident polarization is detected by camera R, while the orthogonal polarization is detected by camera T, respectively. The light intensity detected by each camera is denoted as IR and IT, respectively. When the reflected light from the workpiece returns along the path opposite to that of the incident light, the reflected light is completely directed to camera R, resulting in an IT : IR ratio of zero. As shown in Figure 2(c), when the reflected beam passes through a different optical path other than the incident light, it does not return to its original linear polarization after passing through the polarization element, and the ratio IT : IR increases with the beam shift R relative to beam radius D/2.
In this experiment, q-plates whose principal axis direction rotates with the azimuth angle [9] are used as the polarizing element. The angle of the principal axis is denoted as where, , , and represent azimuth angle of polar coordinates in the q-plate, order, and position in rotational direction of the q-plate, respectively. The higher the value of m, the higher is the spatial frequency in azimuthal direction of the polarization structure generated. Figure 3 shows a geometric representation of the beam shift on the q-plate, viewed from the sample side. As shown in the figure, a beam passes through the spatial filter from the light source. The main line of the incident beam passes through the origin O of the x-y coordinate system. The axis of the return beam passes through the origin O' of the x'-y' coordinate system. When the incident and reflected beams have the same optical paths, the two coordinate systems coincide. However, if a beam shift occurs in the direction of distance R and Θ when the incident beam reflects off the sample and returns as the return beam, then the incident beam enters through point I whose polar coordinates are (r, θ). The returned light reflected from the workpiece passes through point E, which is represented as (x, θ'+X) in the polar coordinate system with the origin at point O. These values of x, θ', and X are represented by the parameters of (R, Θ) and (r, θ) as where, As described in Eq. 1, the polarization property of the q-plate changes only in the azimuth position regardless of r and x. Therefore, the polarization returning to E point from point I is described by the Jones formulation as (6) Ein and Eout denote the incident polarization at point I and the output polarization from point E, respectively, where Ein = [1, 0] T . M(θ) denotes the action of a q-plate at the azimuth position of θ as, Here, S and R represent the reflection on the sample and mirror image transformation, respectively, and are denoted as follows, where r0 denotes the geometrical average of reflectance of horizontal and vertical polarization, tanχ = rx/ry is the ratio of reflectance of horizontal and vertical polarization components, and Δ is the phase difference between horizontal and vertical polarization. From the x component of Eout, the intensity iR and iT of the ray detected on the camera R and camera T are where, and (12) The results show that iR and iT are not affected by Θ but are determined only by the beam shift R. They are also independent of the installation angle δ in the direction of element rotation of the q-plate. In addition, the sensitivity of IT and IR responses to R also increases with m. The light corresponding to the dark area is detected as iT at another side of PBS. Note that the change in the beam shift direction Θ affects the rotation of the beam pattern shape but exhibits no effects on the beam intensities IR, and IT. This indicates that this measurement method has isotropic sensitivity for intensity measurements, and the orientation is measured by detecting the beam pattern. The light intensity distributions iR and iT in the beam are converted to intensity at each pixel by focusing on the camera R and T, respectively. That is, the integrated values of the intensity distributions of iR and iT in the O' coordinate space, IR and IT, are detected at the pixel. Figure 5 shows the calculated dependence of IR and IT on the tilt angle α of the workpiece in Figure 2. After reaching the shift above the beam radius, the change stops at the point where both intensities are in balance. Therefore, the measurement has a finite dynamic range. Note that the polarization distribution in the beam continues to change even after the IR and IT variations are saturated, and it can be measured to extend the dynamic range. The order of the q-plate also affects the sensitivity and dynamic range. In Figure 5, the results for m=1, 2, and 4 show that the sensitivity of the beam shift detection increases with the increase in order. For orders higher than or equal to 2, overshooting and ringing occur near the end of the sensitivity region. If the

< l a t e x i t s h a 1 _ b a s e 6 4 = " 0 v R L 7 i 4 M W G w a N D 3 m l S P f v B T + i t g = " > A A A D d H i c h V H P a x N B F H 6 b W K 3 x R 6 N e B D 0 M h k h K a Z i E R k U o F E X w 2 K Z N W 8 i k Y X Y 6 S Y b u 7 i y z 0 2 h c 8 g / 4 D 3 j w I A o e x D / D i / + A h 5 4 8 i 8 e K X j z 4 d h K U W o y z 7 L z 3 v v d 9 b / h 4 f h y o x F J 6 5 O X y Z + b O n p s / X 7 h w 8 d L l h e K V q 9 u J P j R C t o Q O t N n 1 e S I D F c m W V T a Q u 7 G R P P Q D u e M f P M z 6 O 0 N p E q W j L T u K Z S f k / U j 1 l O A W o W 7 x l e q y k N u B C d P m m K y S 2 v I f Y G t M W G x 0 b D V 5 1 H 2 6 V 8 c 2 E z p h g e z Z d k h c Z C m z A 2 k 5 W S b M D D R Z I o w b Y X n k u h U m e o a L t E 4 M Y Y m K K i f I i + M U w X 7 I x 8 y o / s A u k k l k 0 7 q z V y 9 0 i y V a p e 6 Q 0 0 l t m p R g e t Z 1 8 T M w 2 A c N A g 4 h B A k R W M w D 4 J D g 1 4 Y a U I g R 6 0 C K m M F M u b 6 E M R R Q e 4 g s i Q y O 6 A H e f a z a r o 4 x M / A E + f u o G s A q N H A W x V d C N y 1 j a v e m R d 5 o 5 r w J G m G d 8 R O n F 6 g O 8 D e o J F C m n + g 7 e k w / 0 v f 0 C / 3 5 z 1 m p m 5 F 5 G 2 H 0 J 1 o Z d x e e X 9 / 8 8 V 9 V i D H z 8 l s 1 Q + E j e 7 Y n C z 2 4 5 7 w o 9 B Y 7 J H M p J v O H z 1 4 c b 9
5 v l t P b 9 A 3 9 i v 5 e 0 y P 6 A R 1 G w 2 / i 7 Y Z s v o R s 4 b W / 1 3 s 6 2 a 5 X a 3 e q j Y 2 V 0 t q D 6 e r n 4 Q b c g g r u 9 y 6 s w W N Y h x Y I b 8 5 b 8 l a 8 R u 5 7 / m a + l C 9 P q D l v q r k G J 0 6 + + g t I X t h Q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Y j r z k u 7 i b l h K T c J u a v V n P 9 W q c m Y = "  . The comparison of these transitions reveals a common feature that the low-intensity region expands as the beam shift increases, and both eventually balance out. However, the change is more rapid at higher orders because the high-order polarization distribution has a finer periodic structure, such that even a slight beam shift is reflected in the intensity. The dense periodic structure also causes overshooting. When the intensity distribution is generated by beam shifting, the transition starts from the beam center, and the stripe-like structure gradually spreads to the beam periphery. Therefore, the beam intensity increases or decreases as the stripes disappear out of the beam. In addition, the number of increases or decreases in ringing is the same as the order of the q-plate.
The structure of the workpiece surface can be analyzed using two methods: the highly sensitive detection of IT, or the measurement of the IT : IR ratio. The former method is suitable for measuring slight changes, and the latter for measuring changes quantitatively, since the reflectance of the sample itself is not affected. Figure 7 illustrates the coaxial epi-illumination camera observation system. The LED light source is limited to a wavelength of 532 ± 5 nm by filter F, converted to linear polarization through a non-polarized beam splitter (NPBS) and beam splitter, and coaxially illuminated by objective lens Obj. through q-plate. Lens L1 is adjusted so that the LED light emitting elements form the image on the q-plate. The workpiece is fixed to a precision stage that allows focusing and adjustment of the surface tilt relative to the optical axis. The reflected image from the workpiece is projected onto the camera through Obj. and lens L2. The polarization is converted by the inverse path of the q-plate and then divided into orthogonal polarizations by the PBS. IT is observed on the transmitted side, while IR is observed on the reflected side after reflecting through the NPBS. The insufficient extinction ratio of PBS was covered by inserting polarizers Pol. 1, Pol. 2, and Pol. 3 before the PBS incidence and before the camera for IT and IR detection, respectively. The working distance of the objective lens is 125mm, and NA is selectable between 0.04 and 0.02 by limiting the beam diameter of the illumination light. The prepared q-plates have m=1 and 2, the waveplates are zero-order, and the design wavelength is 532 nm. The specular reflection of an aluminum metal mirror was measured with m=1 and m=2 q-plates attached in order to evaluate the polarization measurement performance. The extinction ratios IR / IT were 62 and 56, respectively. Figure 8 shows the results of IT and IR measurements when the tilt of mirror-polished silicon wafer is varied. In Figure  8(a), the order of q-plate was set to m=1, and the differences were investigated when NA was set to 0.02 and 0.04. The measurement and numerical results were compared to confirm the agreement between the measurement and the principle.

RESULTS AND DISCUSSION
The measured values include the effects of optical attenuation and vignetting in addition to polarization, and these effects are also reflected in the calculations. The attenuation of the detected light intensity is obtained by performing the same measurement in advance without inserting the q-late. In addition, the attenuation of elements that only affect IR, such as NPBS, are also determined. The model considers these measurements, and the experimental and calculated values showed good agreement for both NA. Therefore, the surface tilt profile is measured by the distribution of the ratio of IR to IT once the characteristics of the optical system have been measured. Sensitivity is proportional to NA, and the width of the sensitive area is inversely proportional to NA. This is because, as with optical lever, it depends on the ratio of the optical path length between the sample and the objective lens to the beam diameter. Therefore, the wider the field of view of the image measurement system, the higher the sensitivity of the measurement. As shown in Figure 8(b), the influence of m value is investigated by fixing the value of NA at 0.04 and the m value at 1 and 2 and then comparing the obtained two sets of results. Sensitivity is proportional to m value and inversely proportional to dynamic range. As shown in Figure 8(a), the experimental and theoretical results agree in the sensitive and insensitive regions; however, the experimental overshoot is smaller than the theoretical prediction at the angles where the overshoot occurs. This is due to the intensity distribution of the illumination light projected onto the q-plate. In theoretical calculations, the light intensity distribution irradiated on the q-plate is assumed to be uniform. However, the electrode patterns of the light emitters are projected on the q-plate in the case of actual LEDs. In addition, diffraction reduces the light intensity near the edge of the aperture limiting NA. Therefore, changes in light intensity at the edge of the beam, which causes overshooting and ringing, are less likely to affect IR and IT. The luminance distribution of the illumination beam must also be taken into account in the calculation for better reproduction of simulations in experiments. NA and m have the same effect on sensitivity and dynamic range, but the limitation of NA reduces the amount of illumination light, resulting in lower image quality. Figure 9(a) shows the cross-sectional structure of a microscale (MBM13100, Nikon) of a 30-nm-thick metal film deposited on a silica substrate and patterned with a 10-µm groove width. As shown in the figure, the metal part is reflective, the groove part is transmissive, and diffraction and scattering appear at the groove boundaries. In Figure 9(b), the metal surface is observed brightly and the grooves are observed darkly, when bright-field epiillumination observation is performed. Figure 9(c) shows the image of IT when a q-plate is inserted. Specular reflections from metal are not detected since IT detects only light with a shifted optical path. However, scattered light from steps can be detected. Consequently, the scale grooves, which were detected as a single black line in bright field, are observed as two bright lines at the edges in the q-plate observation. In this image, the brightness of both the groove bottom and the metal surface is similarly lower, and only the surface roughness is specifically detected, making it easy to detect a step of 30 nm. Metallic thin films have further nanoscopic roughness on their surfaces due to morphology and damage on the film surface. Figure 9(d) shows an image of a part of the metal surface in Figure 9(c) taken with a higher sensitivity and shows the surface structure of a much smaller scale than the 30 nm steps.
In Eq. 10-14, the IR and IT images should not change when the q-plate is rotated because IR and IT is not related on the installation angle δ of the q-plate. In Figures 10(a)-10(d), The images of IT were acquired by rotating δ by 0°, 90°, 180°, and 270°, respectively. The radial polarization is produced when δ = 0° and 180° and azimuthal polarization is produced when δ = 90° and 270°. Here, the scale bar is 0.3 mm. As is expected in theory, these images have no change in contrast, despite different incident polarizations. In the absence of polarization variation due to reflection, the direction of rotation or the front and back of the q-plate are not required to be considered for installation.  For evaluating the detection limit of the step, the sample used was an edge of bump structure using a silicon on insulator wafer, which consists of a silicon device layer on top of the silica insulating layer, which is on top of a silicon support layer. After patterning the device layer, the surface of this sample was dry etched non-uniformly to create a bump structure with a height distribution. Figure 11(a) shows an image of the IT side of the bumps obtained by attaching a qplate with m=1. In this sample, rectangular shaped bumps are lined up, and the bump height gradually decreases toward the right side of the image. The white interferometer results indicate that the device layer disappears in the middle of the third bump. Figure 11(b) shows the results of bump height measurements along the arrow lines using a probe profiler. Three bumps were measured to be approximately 92.2 nm, 51.3 nm, and 11.1 nm. The insulating layer is also etched by approximately 9.8 nm. Therefore, the device layer of the bump has a maximum thickness less than a few nm. The camera incorporating a q-plate was found to provide the same sensitivity as a white-light interferometer. The intensity of IT increased with the increase in step size. This indicates that IT and ratiometry between IT and IR can be measured to determine step height. The method was also effective for transparent thin films, and steps in a 10-nm-thick ITO film were clearly observed as shown in Figure 12. The figures show the difference in brightness between IT and IR that depends on surface roughness. In addition, at Ra 0.2μm, the difference between the grinding and paper finishing methods has little effect. Interestingly, the ratio of IT increases for smooth surfaces, i.e., the optical path shift is larger. This is inconsistent with the previous finding that IT was almost zero for silicon wafers. To investigate this, the surface topography of these samples was measured with a probe surface profiler. The results shows that the surface is smooth in the nanoscale range at Ra 6.3 μm, although it is rough on the μm order. In addition, the roughness is on the submicron scale at Ra 0.2 μm, but periodic roughness of about several tens of nanometer were observed on the surface. Thus, nanoscale irregularities, which are smaller than the roughness scale, were detected and found to be related to the surface roughness of the metal. These findings were concluded to be the cause of the increase in IT.

CONCLUSION
Our findings showed that the proposed method allows simple observation of surface profiles. It is as sensitive as differential interferometry but different from dark-field imaging as it provides more quantitative and coaxial observations. The proposed system is also extremely simple and inexpensive compared to white-light interferometry and holography. Above all, it is suitable for large-area observation. In addition to q-plate, there are infinite variations of polarization elements, and the technology can be developed to identify surfaces and various surface properties.