Non-interferometric phase retrieval using refractive index manipulation

We present a novel, inexpensive and non-interferometric technique to retrieve phase images by using a liquid crystal phase shifter without including any physically moving parts. First, we derive a new equation of the intensity-phase relation with respect to the change of refractive index, which is similar to the transport of the intensity equation. The equation indicates that this technique is unneeded to consider the variation of magnifications between optical images. For proof of the concept, we use a liquid crystal mixture MLC 2144 to manufacture a phase shifter and to capture the optical images in a rapid succession by electrically tuning the applied voltage of the phase shifter. Experimental results demonstrate that this technique is capable of reconstructing high-resolution phase images and to realize the thickness profile of a microlens array quantitatively.

i x y 0 ( , ) where u 0 (x, y, z) is a slowly varying field amplitude and ϕ (x, y) carries the phase distribution of the object, x and y represent the positions transverse to the z (optic) axis. Let the corresponding angular spectrum of u 0 (x, y, z) be U 0 (k x , k y , z) with the angular frequencies k x and k y . As this beam propagates through the medium with the refractive index of n, the resultant angular spectrum becomes x y x y kx ky L ik nL 1 0 ( ) i k n 2 0 where k 0 is the wavenumber. Whereas, this beam propagates through this medium having its refractive index tuned to n + Δ n, the resultant angular spectrum is x y x y kx ky L ik n n L 2 0 ( ) ( ) 2 ( , , ) By applying the inverse Fourier transform of (4), we obtain the differential equation of where ∇ 2 represents the 2D Laplacian operator over x and y. We observe that Eq. (5) has a similar form of the paraxial wave equation. Thus, we can derive an equation, similar to the TIE, to describe the intensity change due to the refractive index variation in the medium. Let with the real-valued intensity distribution I(x, y, n) and spatially-varying phase φ (x, y) and substitute into Eq. (5), we obtain the new intensity-phase equation This equation states that the phase information can be retrieved in the variation of the intensity with respect to the refractive index change. An estimate of the intensity derivative with respect to n can be obtained by measuring different intensity images at a sufficiently close variation of the refractive index. As a result, we propose a practical approach to electrically tune the extraordinary refractive index of the LC by using a LC phase shifter as the medium. After acquiring a series of intensity image stacks with different applied voltages, the phase φ (x, y) is quantitatively computed by solving this new intensity-phase equation numerically based on a TIE solver by using Hilbert transform technique or by using fast Fourier transform (FFT) 22 , where the derivative of I with respective to n is approximately calculated by using the finite difference (FD) method, i.e., Experiments. Fabrication of LC phase shifter. Figure 1 shows the configuration of the LC phase shifter. A LC mixture MLC 2144 (Merck) is injected into an empty cell consisting of two glass substrates where a thin conductive indium-tin-oxide (ITO) layer and the polyimide (PI) layer are coated on the inner surfaces. The homogeneous alignment of the LC layer is obtained by gently rubbing the surface of polyimide layers. The thicknesses of the glass substrates are approximately 700 μ m, and that of the LC layer is 50 μ m. A thick LC layer is chosen to obtain large enough phase shift because later we would investigate the accuracy of the acquired phase images with respect to the refractive index variation. As a result, the response time in our system is not the issue of the current concern. The ordinary and extraordinary refractive indices of MLC 2144 are 1.5119 and 1.7612, respectively 23 , at the wavelength of 589.3 nm. Before turning on the voltage, the MLC 2144 molecules are parallel to the substrates in an initially stable state. The phase shift dependence on the applied voltage is characterized by sandwiching the LC phase shifter between two crossed polarizers at an incident wavelength of 633 nm. The polarizers are rotated at 45° with respect to the axes of ordinary and extraordinary rays. The driven voltage is a square-wave voltage at a frequency of f = 1 kHz. The phase retardation between the ordinary and extraordinary rays is calculated directly from the measured transmission 24 . Then, the calculated phase shift and the corresponding refractive index variation of the extraordinary ray as a function of the applied voltage is shown in Fig. 2(a). We observe that the threshold voltage for LC molecules essentially aligned with the applied electric field is 0.5 V rms . The voltage to start saturating the LC response is roughly 1.4 V rms . This device has approximately a sharp linear response at the bias voltage within a range between 0.55 V rms and 1.4 V rms . By taking into account the precision of the voltage setting and the mapping from the transmittance to the refractive index, the refractive index variation has an accuracy of 10 -3 . The dynamic response of this phase shifter is measured by applying a square voltage with a voltage of 2.344 V rms at f = 1 kHz. The measured rise time is roughly 1 second (sec) and the fall time is 9.2 sec. We can roughly record 5 optical images within a minute by tuning the applied voltage of the phase shifter. Large intensity stack might be realized by replacing LC materials with fast response time, such as blue-phase LCs, or by reducing the cell thickness.
In order to examine the uniformity of refractive index of our fabricated liquid crystal phase shifter, we measured the wavefront phase in use of a wavefront sensor (WFS150-7AR, Thorlabs). The result shown in Fig. 2(b) reveals that a reasonable flat wavefront (having the phase variation less than 0.2 radians) within an area of 400 μm × 400 μ m is found, which is larger than the sizes of the concerned specimens that is less than 250 μm × 250 μ m. The incident wave impinges upon this region, and then we can assume that the refractive index variation is uniformly distributed for the whole images.
Experimental apparatus. The experimental arrangement is exhibited in Fig. 3. A red LED with a low temporal coherence at a wavelength of 633 nm is used to uniformly illuminate the object. The incident LED light is polarized by a polarizer along the axis of the extraordinary ray (e-ray) axis of the LC. A microscope objective (Olympus, MPLAPON 50X, NA = 0.5) placed behind the object is used to focus the image field onto the camera (Nikon-D600) with 6016 × 4016 pixels and pixel pitch sizes of 6 μ m. The LC phase shifter is put in between the lens and the camera. The image stack is obtained by tuning the applied voltage of the phase shifter. For measuring  a birefringent object, a setup similar to the polarized optical microscope could be used. An analyzer is inserted between the object and the phase shifter and set parallel to the axis of the extraordinary ray. Then, the phase images are measured by rotating the polarizer prior to the object.

Results
In this section, we provide a discussion of the effect of changing the refractive index on the image quality numerically and measure two test objects, a positive 1951USAF resolution test target (R3L3S1P, Thorlabs) and a microlens array (made by VisEra Co.) as a phase array object, to demonstrate the resolution performance and quantitatively retrieve the phase difference (height) of the microlens array, respectively. In addition, these specimens are also measured by using conventional TIE method for a comparison. In order to suppress noise effect in the conventional TIE method, we use five images to calculate the intensity derivatives by using five-point stencil. On the other hand, we used only two images to calculate intensity derivatives in our proposed approach and the noise was clearly mitigated.  Effect of the refractive index variation on the image quality. Equation (6) is similar to the TIE, and we use FD methods to approximate the derivative by using two images. As a result, an optimal choice of Δ n is necessary in order to avoid unwanted nonlinearity error and the noisy phase reconstructions 13,25 . In simulations, a phase specimen with a Gaussian profile is used, as shown in Fig. 4(a). The incident wavelength is 633 nm and the length of the phase shifter is 50 μ m. A 1% white Gaussian noise is added to each intensity image, as shown in Fig. 4(b). Figure 4(c) shows the root-mean-squared (RMS) error in phase construction as a function of Δ n. We can see that minimal RMS is obtained as Δ n is approximately 0.2. The corresponding optical path difference for this refractive index variation is 10 μ m. This optical path difference nearly equals to the optimal Δ z for the conventional TIE method using mechanical translation 22 . Furthermore, to have a signal to noise larger than 10, the cell thickness may be further reduced to several μ m based on the results in Fig. 2(b). Figures 4(d) and (e) show the phase reconstruction at the optimal Δ n = 0.2 and the corresponding phase error. We can see that the reconstructed phase profile is similar to the phase specimen except that small amount of noise is residual. When Δ n increases, the restored phase image become enlarged and blurred due to the error of the FD approximation, as shown in Fig. 4(f) and (g). As a result, in our method there is an optimal Δ n to achieve the best phase result.
Characterization of an USAF target. Figure 5 presents the optical image with different applied voltages where the groups 6 and 7 of the USAF target are directly imaged onto the CCD camera. These optical images are approximately the same as the applied voltage is varied mainly due to the small refractive index variation. The smallest lines are in the group 7, element 6 with a line width of 2.19 μ m. Furthermore, it shows this target is roughly magnified 100 times, thus the corresponding calculated pixel pitch size is 0.06 μ m. Figure 6 displays the retrieved phase images using Eq. (6) with Δ n of 0.205 and the conventional TIE method with 10 μ m optical path difference. As can be seen in Fig. 6, all lines in both phase images are completely resolved as a result of the Hilbert  transform which provides enhanced phase changes with a large phase gradient. That is, the smallest resolvable element is group 7, element 6, defining a lateral resolution of 2.19 μ m. In addition, our approach provides a better image quality with better noise suppression compared to the conventional TIE technique. There are some horizontal lines in both images because we apply one-dimensional Hilbert transform horizontally to solve the TIE equation. Figure 7 exhibits the normalized phase distributions (i.e., each phase distribution is divided by the maximal phase value) for the group 7, element 6 obtained by our proposed approach and the conventional TIE method and the corresponding optical image. The resolved line patterns by using both approaches are in a close agreement with that obtained from the optical image. In addition, the curve by this approach is smoother than that by the conventional TIE method. These results show that we can reproduce the phase image of the USAF target with a comparable resolution successfully by using the LC phase shifter.
Characterization of a microlens array as a phase object. Figure 8(a) and (b) show the optical image and tomography measured by the atomic force microscope (AFM) of a plano-convex microlens array, respectively. By using the aforementioned pixel pitch size, the calculated pitch of microlens array is roughly 14.31 μ m, which is slightly larger than 14.04 μ m obtained by AFM. In addition, we observe interference patterns in each microlens element as a result of multiple beam interference. Here, we use the TIE solver by FFT because these interference patterns might result in rippled phase images after Hilbert transform. Figure 9 reveals the retrieved phase images of the microlens array obtained by using Eq. (6) with Δ n of 0.205 and the conventional TIE method. We can see that each microlens is clearly resolved without noise fluctuation. Figure 10 illustrates the resolved thickness distribution of the microlens and the tomography measured by the AFM. The resolved profiles closely agree with that measured by the AFM.

Discussion
In summary, we have demonstrated a new quantitative phase retrieval system by introducing a thick LC phase shifter with an electrically-tuned refractive index between the object and the camera to rapidly record optical images with no moving elements. We carried out a new equation for the intensity-phase relation with respect to the change of refractive index, which is similar to the TIE, and then used the available TIE solvers to recover the phase images without considering the variations of the magnifications and locations between successive optical images. We used a resolution test target and a microlens array as the objects and showed this system can be successfully applied to reproduce the test target image with high resolution and to measure the thickness distributions of the microlens array accurately. Therefore, this method can not only recover the intensity object with high imaging quality but also retrieve the phase object quantitatively. Although the response time of the used LC phase shifter is slow, it can be accelerated by reducing the cell thickness of several μ m as the imaging system maintains a signal to noise ratio greater than 10. Furthermore, this imaging system has a great potential to accomplish the phase images of birefringent objects by introducing an analyzer in front of the CCD camera.