Optical transducer exploiting time-reversal symmetry - arXiv

Reference-free single-point holographic imaging exploiting time-reversal symmetry of light scattering Seungwoo Shin1,2, KyeoReh Lee1,2, YoonSeok Baek1,2, and YongKeun Park1,2,3* 1Department

of Physics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Republic of Korea; 2KAIST Institute for Health Science and Technology, KAIST, Daejeon 34141, Republic of Korea; 3Tomocube, Daejeon 34051, Republic of Korea. *Email: [email protected]

One of the fundamental limitations in photonics is the lack of a transducer that can convert optical information into electronic signals or vice versa. In acoustics or at microwave frequencies, wave signals can be simultaneously measured and modulated by a single transducer. In optics, however, optical fields are generally measured via reference-based interferometry or holography using silicone-based image sensors, whereas they are modulated using spatial light modulators (SLMs). Here, we demonstrate a scheme for measuring optical field images using an SLM. By exploiting the principle of time-reversal symmetry and optical phase conjugation, two-dimensional reference-free holographic imaging is realized using an SLM combined with a single-point detector. We present wide-field holographic imaging consisting of 128128 spatial modes at visible and short-wave infrared wavelengths. The ability to measure both the amplitude and phase information of a light field, or holography, is central to optical metrology, with potential applications in materials science, nanotechnology, and biophotonics1-3. Optical phase information can be obtained indirectly by recording the patterns that are formed as a result of the interference of a sample beam with a well-defined reference beam (Fig. 1a). From the measured interference pattern, an optical field image can be retrieved using a phase retrieval algorithm4. Interferencebased holographic imaging techniques5-7 have led to the emergence of various research disciplines, and their applications have been further expanded with recent advances in the development of silicon image sensors, such as charge-coupled devices and complementary metal-oxide-semiconductor devices. However, conventional holographic imaging techniques require the use of an interferometer and an image sensor, and this requirement substantially restricts broader realization and application of holographic imaging, particularly at wavelengths where the applicability of high-quality image sensors is limited. Moreover, the measurement and modulation of optical fields are achieved using separate principles: siliconbased image sensors are used to measure the interference patterns of the sample and reference beams that are generated in reference-based holography, whereas spatial light modulators (SLMs) are used to modulate optical fields through the reorientation of liquid crystal molecules or the actuation of deformable surfaces or micro reflective elements. The simultaneous measurement and modulation of optical light fields using a single electronic device or an optical transducer have not yet been demonstrated. Here, we propose and experimentally demonstrate a method of reference-free single-point holographic imaging using an SLM. Instead of employing interferometric imaging, which requires a reference beam and an image sensor, the proposed method exploits the time-reversal nature of optical phase conjugation to retrieve the incident optical field with high fidelity. The principle of the approach is based on time-reversal symmetry and reciprocity of light scattering8, 9. After a light-matter interaction, plane-wave illumination results in a scattered field S, which serves as the sample information in conventional imaging (Fig. 1a). When the scattered field is propagated back to the sample in a time-reversed manner, the beam will become a plane wave after a light-matter interaction8, 9. To utilize this time-reversal symmetry for holographic imaging, we exploited time-reversal nature of optical phase conjugation. For a monochromatic wave E(r, t)  real[ A(x, y) e i(kz – t + (x, y))], the optical phase conjugation of the wave is identical to the time reversal of the wave, E r (r , t )  real  A( x, y )ei   kz t  ( x , y )    real  A( x, y)ei  kz t  ( x , y )    E (r ,  t ) , 

(1)

where r denotes the complex conjugation in spatial domain9, 10. Thereby, we can rewind a scattered field S to a plane wave by modulating the scattered field using the phase conjugation of the field S*, which can be confirmed by a single-point detector to monitor the intensity of the focused light after a lens (Fig. 1b).

Figure 1 | Comparison of conventional digital holography and single-point holography. a, Schematic illustration of conventional digital holography: a sample beam interferes with a reference beam on an image sensor, and the image sensor records the interference pattern to enable retrieval of the incident field. b, Schematic illustration of single-point holography: all optical power in the incident field S converges to a single point when the incident field is modulated using its phase conjugation S by a spatial light modulator. Without the need for a reference field and an image sensor, the incident field can be retrieved using a point detector. The yellow and green arrows represent the beam paths for conventional digital holography and single-point holography, respectively. Exploiting time-reversal symmetry, an unknown field can be retrieved. The unknown field is sequentially modulated by complex-valued patterns displayed on a SLM, and the intensities of the modulated fields are correspondingly measured by a point detector. Then, we can find a displayed pattern whose corresponding intensity shows the maximum intensity or equal to the intensity of the plane-wave illumination. From the previous time-reversal discussion, the pattern displayed at that time is identical to the optical phase conjugation of the unknown field, from which both amplitude and phase images of the unknown field can be produced. To realize our idea, the key point is a way to find an optical phase conjugation pattern of an unknown field which makes the maximum intensity of focus at a point. Recently, several algorithms are developed to find modulating patterns for focusing light at a point11, 12. In this work, we utilized the basis transformation to systematically find a phase conjugation pattern of an unknown field. Using an SLM, an incident field S is modulated by a complex-valued map Dn. The intensity measured by a single-point detector located behind 2

the lens at a distance equal to the focal length is expressed as I n   Dn S  k 0   Dn S da , where ◦ 2

represents element-wise multiplication or the Hadamard product, k is a spatial frequency vector, and

 da

i

denotes a surface integral. To effectively control the wavefront, we use a phase shift e p and an arbitrary i basis H to construct the complex-valued map as follows: Dn  e p H q  H1 , where i  1 , p  1, 2,3 , and q  1–N, where N represents number of basis vectors in the basis H. Then, the intensity measured by the single-point detector is i p

In  e

 Hq

S da   H1 S da 

2

i

 e p sq  r

2

,

(2)

where sq   H q S da represents the decomposition coefficient of S calculated on a basis vector Hq and r   H1 S da is a constant. Interestingly, the intensity measured by the single-point detector has a form

analogous 2

to

that ip

I n  sq  r  e sq r  e 2



measured i  p

in

conventional

reference-based

holographic

imaging:

sq r . Thus, complex values sq r can be obtained from the phase shifts 

modulated by the SLM (see the Methods). By regarding the constant value r as a global phase, we can obtain the sq values for q  1–N as the representation of the incident field in the basis defined by the vectors Hq. The incident field in the standard basis can be obtained via a basis transformation. Note that the modulation patterns are combinations of the reference and basis vectors; therefore, our method inherently has the geometry of common-path interferometry. For an experimental demonstration of single-point holography, we used three optical components: a digital micromirror device (DMD), a lens, and a single-point detector comprising a single-mode fibre and a photodiode (Fig. 2a). A DMD consists of up to a few million micromirrors, which are individually switchable between the on and off states at a speed of tens of kHz. We utilized a DMD rather than a liquid-crystal SLM to achieve fast modulation of the light field and broadband operation. However, the proposed method is not limited only to this specific type of light modulator; any type of SLM, regardless of its amplitude or phase modulation, can be used. The working principle and workflow of single-point holography are presented in Fig. 2. An incident field is sequentially modulated by multiple binary patterns displayed on a DMD (Figs. 2ab). The intensities of the modulated fields are then measured by a single-point detector (Fig. 2c). To achieve effective modulation, a Hadamard basis, an orthogonal basis whose elements are either 1 or 1, was used to construct the modulation patterns (Fig. 2d). Since all elements of a Hadamard basis have nonzero values, the entire region of an incident field can be modulated, thereby ensuring a high signal-to-noise ratio (SNR)13, 14. To display the complex-valued map on the DMD, we utilized the superpixel method15 (see the Methods). From the intensity responses measured by the single-point detector, the incident field can be retrieved (Fig. 2e).

Figure 2 | Working principle of single-point holography. a, The field S that is diffracted by a sample is modulated by a binary pattern Dn displayed on a digital micromirror device (DMD). The intensity of the modulated field at a single point is measured by a photodiode after passing through a lens and a single

mode fibre. b, Multiple binary patterns that are sequentially displayed on the DMD. c, Measured intensities corresponding to the displayed pattern index. d, Construction of binary patterns using phase shifts and a i Hadamard basis. e p and Hq denote the phase shift ( p  1, 2,3 ) and the qth basis vector of the Hadamard basis (q  1–N), respectively. The superpixel method of displaying a complex-valued map using a DMD is illustrated in the inset. e, The retrieved incident field. The narrow arrows indicate the entire sequence of single-point holography used to retrieve the incident field. We first experimentally validated the proposed method at a visible wavelength. An interconvertible setup was used for the direct comparison of single-point holography and conventional digital holography using an off-axis Mach-Zehnder interferometer (see the Methods and Supplementary Information). To demonstrate the feasibility of our method for various types of samples, we measured the fields diffracted by a phase object (a polystyrene bead with a diameter of 10 m, Figs. 3ab) and an amplitude object (the number 7 representing group 7 from the United States Air Force (USAF) resolution test chart, Figs. 3cd). Both the amplitude and phase images of the samples that were measured using the proposed method were well consistent with those obtained via conventional holographic imaging, thereby serving as a proof of principle for single-point holography.

Figure 3 | Experimental demonstrations using various samples. a-b, The field diffracted by a phase object, namely, a 10-m-diameter polystyrene bead immersed in oil, as retrieved via single-point holography (a) and conventional holography (b). c-d, The field diffracted by an amplitude object, namely, the numeral “7” representing group 7 in the United States Air Force (USAF) resolution test chart, as retrieved via singlepoint holography (c) and conventional digital holography (d). The amplitude and phase images of the retrieved fields are labeled with the symbols A and , respectively, in the bottom right corner of each figure. The green and yellow colours of the symbols indicate the methods used for field retrieval, namely, singlepoint holography and conventional digital holography, respectively.

Single-point holography offers several advantages compared with conventional methods. Because it does not require the use of a reference-based interferometer, single-point holography allows robust measurements to be performed using a simple instrument. In addition, the high SNR achieved by means of the Hadamard basis ensures high-quality field measurements by significantly reducing the image deterioration caused by dark-current and read-out noise. The quantum efficiency of a photodiode and the fill factor of a DMD are greater than those of typical image sensors14. More importantly, the principle presented in this work can be readily applied to electromagnetic waves of other wavelengths, ranging from X-ray and deep ultraviolet wavelengths to infrared and terahertz wavelengths. To demonstrate the applicability of the proposed method at other wavelengths, we demonstrated holographic imaging in the short-wave infrared (SWIR) at a wavelength of 1.55 m. At SWIR wavelengths, silicon image sensors are blind; as an alternative, indium gallium arsenide (InGaAs) image sensors can be utilized, but their applications are highly limited because of their limited response and pixel resolution as 

well as their high price. A silicon wafer etched with patterns was used as the phase object for the SWIR experiment (Fig. 4a). A laser with a wavelength of 1.55 m and an InGaAs photodiode were employed (Fig. 4b), and the field diffracted from the wafer was measured using single-point holography (Figs. 4c and 4d). The height map d of the wafer was calculated from the measured phase image (x,y) (Fig. 4d) as d ( x, y)      x, y  /  2 n  , where n represents the difference between the refractive indices of silicon and air. To verify the success of the SWIR holographic imaging process, a topographic map of the letter “P” was measured via atomic force microscopy (AFM). For direct comparison, a magnified image of the height map obtained using the proposed method is shown alongside the topographic map measured via AFM (Fig. 4e). To systematically compare these images, the measured profiles are presented on the same plot (Fig. 4f); the results serve as validation of the holographic image measured in the SWIR.

Figure 4 | Single-point holography in the short-wave infrared, where conventional silicon detectors are blind. a, A photograph of a silicon wafer, which acts as a phase object at a wavelength of 1.55 m. On the wafer, repetitive patterns are etched to a depth of 200 nm in the form of the word “PHASE.” b, The beam transmitted through the wafer is projected onto a DMD. The field diffracted by the pattern is retrieved via single-point holography. For visualization purposes, the DMD, which is in fact a reflective modulator, is depicted as a transmissive modulator. c, The amplitude image of the retrieved field. d, The depth map of the wafer produced from the phase image of the retrieved field. e, A magnified view of the portion of the depth map indicated by the dotted red box in d (upper) and a topographic map of the letter “P” (lower) as measured via atomic force microscopy. f, The depths along the profiles indicated in e.

It should be emphasized that the proposed method of reference-free single-point holography is fundamentally different from previously presented techniques based on intensity correlation or illumination engineering16-22. Previously, holographic measurements have been demonstrated using both a single-pixel camera and reference-based interferometry16, 17, but the use of a reference beam results in the difficulties described above (See the Supplementary Information). Unlike existing methods of wavefront shaping12, 23, 24 and digital optical phase conjugation25-27, our method does not require a reference field and an image sensor, and thus, its applicability is significantly broader. From a technical perspective, the proposed single-point holography technique can be combined with illumination engineering methods to realize holographic imaging with sub-diffraction resolution using the synthetic aperture method28, 29 or 3-D resistive index (RI) tomography exploiting optical diffraction tomography30, 31. Moreover, single-point holography can be extended to spectroscopic holography by means of the linear dispersion of a DMD and a spectrometer. Although the presented method has limitations with regard to dynamic studies because of the need for sequential measurements, compressive imaging approaches14, 17 can be adopted to improve the image acquisition rate. 

In conclusion, we have proposed and experimentally demonstrated a reference-free single-point holographic imaging method that exploits phase conjugation. By exploiting the time-reversal nature of optical phase conjugation, the field diffracted from a sample can be precisely retrieved without the need for an interferometer and an image sensor. In the present work, we used an SLM, which can control optical fields; holographic measurements obtained using the SLM were presented to demonstrate an optical transducer that can both measure and modulate an arbitrary optical field. Furthermore, the applicability of the proposed method in the SWIR was verified by measuring complex field images. We expect that our method may offer a solution for high-fidelity holographic imaging at wavelengths where the applicability of high-quality image sensors is limited. Methods Field retrieval using a matrix representation In terms of the three different phase shifts, equation (2) can be represented in matrix form as [I]  [e i]  [S],  I1   I2 I  3

1 ei1 e  i1 I3N 2    I 3 N 1   1 ei2 e  i2 1 ei3 e  i3 I 3 N  

 s1 2  r 2        s1r   s r  1  

sN  r sN r  sN  r 2

2

   ,  

where [I] and [S] are 3N matrices constructed in the orders of the phase shifts and the basis vectors. The phase-shifting matrix [e i] is a 33 invertible matrix. By multiplying the inverse of the phase-shifting matrix by the measured intensity matrix, the matrix [S] can be obtained as follows: [S]  [e i]–1  [I]. Then, from the second row of the obtained matrix [S], the complex values sq r can be obtained for q  1–N. Since the constant complex value r can be regarded as a global phase factor, we can then obtain the values sq for q  1–N, which correspond to the incident field S represented in the Hadamard basis. Through a basis transformation, the incident optical field S can be represented in the standard basis (Fig. 2e). Superpixel method and the construction of binary patterns To display a complex-valued map on the DMD, we utilized the superpixel method proposed by the Mosk group15. With the appropriate placement of the single lens, linearly varying phase shifts can be assigned to each micromirror of the DMD, increasing by  and /2 along the horizontal and vertical directions, respectively (inset of Fig. 2d). Thus, the phase shifts of the 4 neighbouring micromirrors in a rectangular region will be equally divided between 0 and 2, allowing superpixels to be defined as 22 arrays of DMD micromirrors. To allow the phase shifts in a superpixel to be combined, the set-up should be designed such that the micromirrors in each superpixel are unable to be resolved by optics. Thus, by turning on different combinations of the micromirrors in a superpixel, 9 different complex values can be displayed. The addition of e ip  Hq, the phase-shifted q-th basis vector of the N-dimensional Hadamard basis, to H1, the first basis vector, generates a complex-valued map, e ip  Hq + H1, where p  1, 2,3 and q  1–N. Then, a binary pattern Dn is generated from the complex-valued map via the superpixel method. Since a superpixel must be able to modulate 9 different complex values, the phase shifts are set to 0, /2, and  for p = 1, 2, and 3, respectively. For the construction of the binary patterns, we used the Hadamard basis method with N  128. With three phase shifts, 49152 patterns were displayed on the DMD with a display rate of 10 kHz, from which we acquired an optical field in 4.9 seconds. The image acquisition rate depends on the number of spatial modes N and the display rate f of the DMD as f/3N2. Superpixels consisting of 22 arrays of DMD micromirrors were chosen; altogether, 256256 micromirrors were used to display binary patterns in the visible wavelength range. The numerical aperture of OL1 and the focal length of TL1 were appropriately selected to ensure that all micromirrors were unresolvable. In the SWIR range, because of the larger diffraction limit, each superpixel was chosen to consist of 44 DMD micromirrors, and in total, an array of 512512 micromirrors was used to display the binary patterns. Experimental set-up To experimentally validate the proposed concept, we constructed an interconvertible set-up for the direct comparison of two imaging methods: single-point holography and conventional digital holography using a Mach-Zehnder interferometer (Figs. S1a-S1c). For a systematic comparison, a flip mirror and a 22 singlemode fibre optic coupler (22 SMFC; FC532-90B-FC, Thorlabs Inc.) were used to share the optical set-up 

between the two methods. To allow both conjugate planes of the sample plane to be used as the image planes for the two different imaging methods, the optical set-up was constructed symmetrically. The same visible laser (  532 nm; LSS-0532, Laserglow Inc.) was used as the source for both imaging methods. For the measurement of an optical field via single-point holography, a plane wave was introduced into the optical set-up by the flip mirror (Fig. S1b). The plane wave impinged on the sample, and the beam diffracted by the sample was projected onto the DMD (maximum switching rate = 22.7 kHz; V-7001, Vialux Inc.) by an objective lens (OL1, 0.7 NA; LUCPLFLN60X, Olympus) and a tube lens (TL1, f  250 mm). Then, the optical field of the diffracted beam could be retrieved via single-point holography as described. The optical field was sequentially modulated by multiple binary patterns, and an avalanche photodiode then measured the intensity of the light passing through a single lens and one arm of the 22 SMFC. Finally, by multiplying the inverse phase-shifting matrix by the measured intensity matrix, the optical field was retrieved. For comparison, the optical field diffracted by the same sample was measured via Mach-Zehnder interferometry (Fig. S1c). The laser beam was split into two arms (sample and reference arms) by the 22 SMFC. To illuminate the sample with a plane wave, the DMD was set to display a planar wavefront. This guaranteed conditions equivalent to those for the single-point holography measurement. An objective lens (OL2, 0.7 NA; LUCPLFLN60X, Olympus) and a tube lens (TL2, f  180 mm) were used to project the beam diffracted by the sample onto a camera (FL3-U3-13Y3M-C, FLIR Inc.), where the sample beam interfered with a reference beam at a slightly tilted angle to generate an off-axis hologram. From the measured hologram, the optical field diffracted by the sample was retrieved via the Fourier transform method32. To demonstrate the broadband capability of our method, a SWIR laser beam (  1.55 m; SFL1550P, Thorlabs Inc.) was directed onto an etched wafer sample by a 4-f telescopic system comprising a tube lens (TL2) and an objective lens (OL2, 0.3 NA; LCPNL10XIR, Olympus). The beam diffracted from the wafer was collected and projected onto the DMD by another 4-f telescopic system comprising a tube lens (TL1) and an objective lens (OL1, 0.65 NA; LCPNL50XIR, Olympus). The intensity of the light passing through a single-mode fibre (P3-SMF28E-FC-2, Thorlabs Inc.) was measured by an InGaAs switchable gainamplified detector (PDA10CS-EC, Thorlabs Inc.). Sample preparation We used polystyrene beads with a diameter of 10 m (n  1.5983 at   532 nm, Sigma-Aldrich Inc.) immersed in index-matching oil (n  1.5660 at   532 nm, Cargille Laboratories). To separate aggregated beads, the beads in the immersion oil were sandwiched between coverslips before measurement. To measure the complex field diffracted by a phase object in the SWIR, we used a double-sided polished wafer on which repetitive patterns were dry etched. Data availability The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Acknowledgements The authors acknowledge H. S. Yeo in Korea Advanced Institute of Science and Technology for help in using AFM. This work was supported by KAIST, BK21+ program, Tomocube, and National Research Foundation of Korea (2015R1A3A2066550, 2014M3C1A3052567, 2014K1A3A1A09063027). Author contributions S.S. performed the experiments and analysed the data. K.L. and Y.B. contributed analytic tools. Y.P. conceived and supervised the project. S.S and Y.P wrote the manuscript, which was revised by all authors. Correspondence and requests for materials should be addressed to Y.P. Competing financial interests The authors declare no competing financial interests.

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