Optical Waveguide Fingerprint Recognition Device with Two‐Stage Dual‐Function Gratings

In the under‐screen fingerprint recognition of smart cell phones, the tiny image signals are easily deteriorated by the protective membranes of screens. To solve this issue, an optical waveguide comprising two stage dual‐function gratings is proposed to supply extra lighting. The first stage is formed by two adjacent gratings possessing a cross angle of 120°, which serve as couplers and beam splitters to trap and split the incident light into two paths within the optical waveguide film, respectively. The second stage gratings, possessing a 60° included angle to the first stage, act as beam orientators and expanders by consecutively redirecting a portion of the light beams towards the sensing zone. By covering the waveguide film outside on an organic light emitting diode display screen and setting a camera and a laser under the screen, this slim construction shows high efficiency and large area under‐screen fingerprint recognition with high signal contrast. The design inspires a novel method for efficiently and flexibly controlling photons by arranging a series of gratings with the proper orientations and locations on a chip, which is highly demanded in displays and optical communication.

In the under-screen fingerprint recognition of smart cell phones, the tiny image signals are easily deteriorated by the protective membranes of screens. To solve this issue, an optical waveguide comprising two stage dual-function gratings is proposed to supply extra lighting. The first stage is formed by two adjacent gratings possessing a cross angle of 120°, which serve as couplers and beam splitters to trap and split the incident light into two paths within the optical waveguide film, respectively. The second stage gratings, possessing a 60° included angle to the first stage, act as beam orientators and expanders by consecutively redirecting a portion of the light beams towards the sensing zone. By covering the waveguide film outside on an organic light emitting diode display screen and setting a camera and a laser under the screen, this slim construction shows high efficiency and large area underscreen fingerprint recognition with high signal contrast. The design inspires a novel method for efficiently and flexibly controlling photons by arranging a series of gratings with the proper orientations and locations on a chip, which is highly demanded in displays and optical communication.

Introduction
Fingerprint recognition (FPR) has been widely used in access control systems and mobile payment identification owing to its convenience and high biometric security. [1][2][3][4] For decades, FPR techniques for portable devices have advanced from opaque capacitive to transparent optical due to the increasing demand www.advmatinterfaces.de

Design of TSG
As schematically depicted in Figure 1 for the TSG structure, two symmetric mirror gratings with period T 1 and a cross angle of 120° form the first stage component, while the grating with period T 2 and a cross angle of 60° with the first stage gratings acts as the second stage. All the gratings are fabricated on a transparent polyethylene terephthalate (PET) film to form an optical waveguide for the light fill of UDS-OFPR. A laser is set above the first stage gratings as a light source.

Design of TSG Parameters
A collimated green laser beam with wavelength λ of 532 nm is vertically incident upon the first stage gratings, where it is diffracted and coupled into the substrate. The incident and diffracted lights satisfy the wave vector equation of ± 1 order diffraction as follow: where 0 k and 1 k s are respectively the wave vectors in air and substrate, 1 G is the grating vector in x-y plane with magnitude G 1 = 2π/T 1 . The vector components of 1 k s in the x-y plane and along the z-direction are given by Where n s is the refractive index of the substrate. Under the conditions that k s1z is real and simultaneously 0 0 2 1 2 k k G z = − is imaginary, the diffracted light beams are trapped in the substrate in the waveguide mode satisfying the total internal reflection between air and substrate. For a waveguide with thickness h, the step p 1 between the two adjacent reflections on the first stage grating equals to 2h · k s1xy /k s1z . The waveguide mode condition can be eventually described by the inequality k 0 < G 1 < n s k 0 , which means: In the second stage gratings, the lights with the wave vector of 1 k s coming from the first stage are diffracted once again, where parts of the lights are consecutively redirected to the sensing zone, as shown in Figure 1. The redirected lights satisfy the following equation: The wave vector components in the x-y plane and z-direction can be written as: The steps p 2 between the two adjacent reflections on the second stage gratings equal to 2h · k s2xy /k s2z . Similarly, the condition should be satisfied for the redirected light beams to be still trapped in the substrate, which can be transformed to: where α is the cross angle between the gratings of the two stages. Considering the convenience of fabrication, the periods of the gratings are chosen to be equal, thus T 2 = T 1 = T. By combining Equations (3) and (6), we can get which limits the period of the gratings under certain working wavelengths.

Design of Beam Splitting and Expansion
As illustrated in Figure 1, the incident light is first coupled and split into two waveguiding beams by the first stage gratings and then propagates along the directions of 1 k s xy in the waveguide plane. Based on the symmetry of the structure and the consistency of the material, the lights of the left and right paths adopt the same parameters. Generally, there are three zones when the light propagates from the incident spot to the sensing zone, and the propagation details for the right-hand waveguide are depicted in Figure 2. The first one is inside the incident spot, as shown in Figure 2a; at each reflection step p 1 , additional free space incident light is trapped as a new waveguide light with a coupling rate η 1 . Meanwhile, the trapped waveguide lights are coupled to the free space, reducing the light intensity with a loss rate L 1 . The second zone is located on the first stage grating out of the incident spot, where the waveguide light only experiences the coupling-out loss to the free space. As shown in Figure 2b, the third zone is on the second stage grating, and a part of the light is coupled and redirected to the sensing zone with a coupling rate η 2 . Following the analysis presented previously, the intensity of the rays emanating from the first zone I 1 can be expressed as Equation (8): where I in is the intensity of the incident light, the expression in the summation represents the reflected light by the i th www.advmatinterfaces.de reflection step. While n 1 represents the total number of reflection steps for the lights passing through the first zone, as shown in Figure 2a. In the second zone, the lights coming out from the first zone consecutively experience the waveguide to free space out-coupling loss L 1 by the first stage grating at each reflection step, thus the light intensity out of the final step I 2 can be described by Equation (9).
where n 2 represents the total number of reflection steps for the waveguiding lights passing through the second zone. Similarly, in the third zone, where the lights are further diffracted by the second-stage gratings which are depicted in Figure 2b, the light intensity I 3 along the 1 k s xy direction can be described by Equation (10).
While the intensity of the rays that redirected to the sensing zone along the s2 k direction at the j th reflection step can be expressed as in Equation (11): Similarly, where L 2 denotes the loss of the redirected lights by the diffraction of the second stage gratings, although part of the above crossing lights may be coupled back to the direction of 1xy k s , as shown in Figure 2b, where new loss items should be added to Equations (10) and (11), here we omit them for simplification. In short, the intensity of the point (x 0 ,y 0 ) inside the sensing zone, which is shown in Figure 1, is the sum of the intensities I 4 for all the redirected light reaching to point (x 0 ,y 0 ).
According to the equations above, the intensity in the sensing region is affected by η 1 , η 2 , L 1 , L 2 , p 1 and p 2 . Generally, the higher the in-coupling efficiency η 1 and the lower the out-coupling efficiency L 1 , the higher the light intensity in the sensing zone is. For the larger η 2 , the light intensity of the sensing zone is increased, while the area and the uniformity of sensing zone are decreased. Similarly, the smaller the steps p 1 and p 2 , the stronger the intensity and the smaller the area of the sensing zone, due to the overlapping of the redirected beams away from the waveguide beam. Since η 1 and η 2 are determined by the gratings, L 1 and L 2 are controlled by both gratings and cross angle α, p 1 and p 2 are determined by grating period and waveguide thickness. Thus, it is flexible to design the light intensity and area of the sensing zone, as described in the subsequent numerical simulations. As analyzed above, by the cooperative diffraction of the two-stage gratings, the point like incident beam is trapped and split into two main waveguide beams at first, then both waveguide beams are consecutively split into multibranch beams sent to the sensing zone. In the sensing zone, the multi-branch beams intersect to create a uniform surface light source for lighting the fingerprint. When a finger is pressed on a PET film in the sensing zone, the fingerprint's ridges come into direct contact with the PET film and generate bright states by reflecting more light from the OLED than the valleys do. Since the total internal reflection (TIR) of waveguide lights is broken, additional light from the waveguide is also scattered back towards the camera by the finger ridges.

Numerical Simulations of TSG
To optimize the TSG diffraction efficiency, numerical simulations based on the finite difference time domain (FDTD) were utilized. [26] In the simulation, the refractive indices of the substrate, gratings, and ZrO 2 coating are 1.55, 1.5, and 2.3, respectively. Considering the state of the art, the height h of the grating lines is set to be 130 nm and the duty circle f of 0.5 is taken. As mentioned, only the optimization of grating period  www.advmatinterfaces.de T and the cross angle α between the two-stage gratings are performed, since they directly influence the coupling efficiency and redirecting efficiency. According to Equation (3), the first grating period T 1 is changed between 343.23 and 532 nm, and correspondingly, the included angle α between the two-stage gratings is changed between 38.41° and 61.31° to satisfy the TIR condition according to Equation (6). In the FDTD simulation, the smallest mesh step is 0.25 nm and the time steps are 2 × 10 −11 s to assure the accuracy of the results. The perfect matched layer boundary condition is taken both along the x-and y-directions. As indicated in Figure 3a, when the incident wavelength is 532 nm, the best diffraction efficiency is found for T 1 = 350 nm has in the range of periods from 343.23 to 532 nm. As illustrated in Figure 3b, the decrease of the redirecting efficiency η 2 roughly follows the increase of the included angle α. The parameters given above will be utilized as references in the following light tracing simulations. By taking the simulated results of coupling and redirection efficiency η 1 and η 2 of FDTD, the macroscopic processes of the consecutive coupling, splitting, waveguiding redirecting,

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and crossing of the light in the waveguide are simulated by using the Monte Carlo light tracing method (LightTools). [27] The periods of both the first stage and the second stage gratings are 350 nm. The cross angle between the two mirror symmetric gratings of the first stage is 60° in their grating directions. The cross angles between the second stage and the first stage gratings are −61.31° ≈ −38.41°(left) and +38.41° ≈ +61.31°(right), respectively. The diameter of the vertical incident laser beam is 13 mm. Since the incident areas on the left and right side of the first stage gratings are the same, the incident beam is evenly divided into two paths, as V-shaped light paths illustrated in Figure 4. From Figure 4c-e, it is clear that, for the same substrate thickness, the effective sensing area enlarges as the crossing angle increases, because the rate of light is redirected by the second stage gratings decreases at each step. To verify the effects of substrate thickness, as presented in Figure 4f-h, at the same crossing angle α, the reduction of substrate thickness aids in achieving more uniform light in the sensing zone, due to the shortening of the step between every two adjacent reflection points. However, on the other hand, the thinner of the substrate leads to more times of reflections at the same propagation length and directly decreases the area of sensing zone. The difference between the two main V-shaped optical paths results in a nonuniform intensity distribution in the sensing area, as shown in Figure 4i-k, because the width and power of the two main waveguide beams are directly proportional to the occupied incident area. By comparing the simulation results, a cross angle of α = 60° and a substrate thickness of 0.5 mm are found to be the optimal configuration for TSG. It should be emphasized that when cross angle α = 60°, the second stage grating only changes the propagating directions of the expanded light beams while keeping the modulus of waveguide mode vectors constant.

Fabrication and Characterization of the TSG
According to the design scheme and simulation results, the TSG is fabricated on a flexible PET film using UV-cured Nano-imprinting. In practical applications, the grating side of the PET film is pasted on a cover glass with a thickness of 0.55 mm. As illustrated in Figure 5a, the whole fabrication process consists of the following steps: 1) Two-beam laser interference lithography fabricating photoresist (PR) relief gratings as initial templates; 2) Electroforming the PR gratings into nickel metal templates; 3) Via TSG shadow masks, UV cured imprinting grating structures onto the PET film basing on nickel template. 4) Electron beam evaporating high-refract index ZrO 2 on the grating to increase the diffraction efficiency. The fabricated PR gratings are shown in Figure 5b of the SEM (Scanning electron microscopy) images, where period T = 351.7 nm, height H = 109 nm, and duty circle f = 0.5 are chosen to be close to the simulations. The three UV-cured resin gratings with mutual cross angles of 60° that constitute the TSG are shown in Figure 5c. Figure 5d shows the snapshot of the entire device, with the three grating directions distinguished by the diffraction colors. Additionally, a bending state is depicted in Figure 5e, showing the possible applications in flexible displays.

Coupling Efficiency Measurement
To validate the performance of the TSG device, the outcoming light intensities of both stage gratings are used to estimate the diffraction efficiency of each stage gratings. As shown in Figure 6a, the incident light spot across the two adjacent gratings in the 1 st stage is divided and redirected to the sensing zone at the second stage. While, as illustrated in Figure 6b, the cutting along one of the boundaries between the two-stage gratings is used to respectively measure the coupling efficiency of the first stage gratings and the redirection efficiency of the second stage. Figure 6c,d demonstrated how the fiber detectors are vertically aligned with the cutting line to collect the outcoming light.
The wavelength-dependent intensities of incident light and waveguiding light diffracted sequentially by the two-stage gratings are depicted in Figure 6e. By comparing the peak values to the incident light, we ascertain that the 1 st stage gratings have a working efficiency of around 12.72% corresponding to Equation (9), whereas the 2 nd stage gratings have a substantially higher working efficiency of roughly 18.79% corresponding to Equation (11).

Fingerprint Acquirement Measurement
The fingerprint acquisition experimental setup is depicted in Figure 7a and Figure 7b, in which the grating structure surface of TSG and the cover glass surface were bonded by an acrylic adhesion film (AB adhesion). As a finger is pressed on the surface of the TSG without grating structure, a legible and complete fingerprint image appears on the waveguide surface and is captured by a charge-coupled device imaging sensor. To demonstrate the feasibility of TSG in fingerprint acquisition, Figure [28,29] with the contrast of 124.06 in Figure 7c 2 and 75.33 in Figure 7d 2 , respectively. The image contrast calculation method is to calculate the variance between each pixel of the image and its surrounding pixel. The quadratic sum of the gray value difference between the center pixel and eight surrounding pixels is obtained, which is divided by the number of above differences. The detailed Matlab program is provided in the supporting information. By evaluating two sets of images, the multi-stage grating combination in our TSG device, which contains multiple light compensation, significantly improves the quality of the fingerprint image. In comparison to other approaches, such as using infrared light as an additional light source and a micro lens to compensate for light from free space, [24,25,22] the TSG structure is more effective in compensating for the fingerprint acquisition region for light from both waveguide coupling and free space scattering. Despite that the overall coupling efficiency of the device is not ideal to some extent, the outcome images www.advmatinterfaces.de indicate the TSG's advantages in fingerprint acquirement. In other words, the TSG provides a potential solution for FPR on OLED displays, as the primary limitation of current FPR on OLED displays is the lack of efficient and slim light compensation techniques.

Verification Experiment For OLED Displays
To clarify and validate the capability of FPR for TSG embedded in OLED displays, a controlled experimental setup similar to the fingerprint acquirement experiment discussed above was    Figure 8a. The TSG film that created a crossing light in the sensing zone was attached to the surface of an activated OLED display. As a control group, Figure 8b shows the same setup but with an attached equal-thickness PET film that lacked the TSG structure. Additionally, Figure 8e and Figure 8f indicate the fingerprint acquisition results for the setups depicted in Figure 8a and Figure 8b. Despite the waveguide transparent parts of the light entering the sensing area, the OLED screen scattered the light from the light source in Figure 8f, resulting in a fuzzy image of a fingerprint with a contrast of 3.00. In comparison, the fingerprint image in Figure 8e obtained with the TSG structure has an 11-fold increased contrast of 34.90 and is clearer. More specifically, it further proves that the grating assistance waveguide has a higher light compensation capability than the single waveguide. In other words, the findings suggest that the ultra-thin and flexible architecture of TSG not only provides a favorable high-efficiency light compensation for large area fingerprint identification, but also can be integrated into the screen protector of OLED displays.

Conclusion
We have proposed a two-stage dual-function grating system for light compensation in fingerprint recognition. In operation, the first-stage gratings act as beam couplers and splitters, and the second gratings function as beam orientators and expanders. By delicate geometric design, the ratio of beam splitting can be well regulated by the area of the incident light on the first stage gratings, while the width and intensity of the expansion beam can be flexibly adjusted by the redirection rate. The large area and high uniformity of fingerprint sensing are verified both by numerical simulation and experimental demonstration. Moreover, the implementation of the device in an OLED display demonstrates the enhancement of image intensity and contrast, indicating that the device is ready for practical applications. With this proposed device, we offer a novel approach for flexibly controlling light in an ultra-thin dimension by combining a series of gratings, which is applicable to a wide range of areas, including optical communication, augmented reality displays, and sensors.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.