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Abstract
Transmission matrix (TM) is an ideal theoretical model describing light propagation through scattering media. Until now, most of the present TMs utilize the eigenstates of spatial position as input and output bases. Thus, they describe the relationship between the spatial distributions of two light fields. Here, we demonstrate that wider relationships between the light fields could be described by a TM. As a significant example, we propose a generalized TM with the eigenstates of OAM as input bases – OAM-basis TM. With the measured OAM-basis TM, we achieved single-spot and multiple-spot focusing, verifying its availability in light propagation manipulation. The distinct eigenchannels property was also discussed. The OAM-basis TM has broadened the definition of TM. Meanwhile, it will open new perspectives for OAM-based communication, as well as the applications of wavefront shaping technology in biomedical photonics and optical communication.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Light is strongly scattered by microscopic-scale inhomogeneity of refractive index in a wide class of media including biological tissues, turbid atmosphere, white paint and so on [1–3]. This poses a fundamental limitation to numerous research fields such as biomedical imaging, photodynamic therapy, and telecommunications. Fortunately, the past 20 years have witnessed the rapid development of the wavefront shaping technology, which is proven to be an effective method to minimize the influence of light scattering [4–9]. The principle underlying the wavefront shaping technology is the concept that light scattering is a deterministic process in static media. This signifies that once the mathematical model of the scattering medium is known, any desired output could be obtained by well designing the input light field. As the matrix form of the Green’s function, transmission matrix (TM) is an ideal model of the wave propagation through complex media [10,11]. Since TM was first introduced to the field of optics by Popoff et al. in 2010 [7], it has attracted much attention and emerged to be a main approach of wavefront shaping. TM is generally measured on the basis of holographic interferometry, which is performed in a Mach-Zehnder interferometer [12–14] or a co-propagation interferometric setup [7,15]. Then recently, the reference-less configuration using phase retrieval algorithms has also been demonstrated to be an effective measuring method [16–19]. With the retrieved TM, researchers have achieved deep focusing [14,15], image detection [20], multispectral control [21–23] and communication [2] through scattering media. Moreover, the transmission eigenchannels of scattering media could be obtained from the TM [12,24]. Their wonderful natures, such as transverse localization, are of increasing interests in manipulating the optical energy distribution inside the media [25–28].
The elements of the TM describe the amplitude and phase variations when the light beam propagates from the input plane to the output plane. Generally, to perform the digital control of the wavefront, the input light field is divided into N individual segments, such that the light field could be represented by a N-dimension vector. Likewise, the output field is represented by a M-dimension vector, as it is measured by recording M individual modes. Hence, the M × N TM, which transforms the input vector to the output one, could be abstractly explained as a linear transformation between two Hilbert spaces. It can be seen that the basis vectors of both the two spaces are the eigenstates of spatial position, because the spatial distributions of light fields are relatively easy to obtain and widely used in general. However, in some cases, other physical quantities are more concerned, such as orbital angular momentum (OAM) [29,30]. Nowadays, light beams that carry OAMs have become versatile tools for optical manipulation [31], super-resolution imaging [32], and high-capacity optical communication [33,34]. Especially, the applications of OAM beams are urgent in scattering environment [2]. In such a condition, although the conventional TM is used to model the scattering process, it fails to directly describe the influence of the input OAM spectrum on the output speckles. Thus it is significant to construct a TM with the eigenstates of OAM as input bases.
In this article, we have proposed a novel type of TM – OAM-basis TM, which links the input OAM spectrum and the spatial distribution of the output field. We first detailed the theoretical framework of the OAM-basis TM. Then, to verify its availability, we measured the OAM-basis TM experimentally and achieved single-spot and multiple-spot focusing with time reversal operation. In addition, we also demonstrated the convenience of OAM-basis TM in generating doughnut-shaped foci, which is characterized by carrying OAM and is potential in multi-degree-of-freedom optical tweezers and super-resolution imaging through biological tissues [35,36]. In the end, we analyzed the singular values distribution of OAM-basis TM and the contribution of different bases to the eigenchannels. In our OAM-basis TM, the input field is represented by the eigenstates of OAM, while the output one is represented by the eigenstates of spatial position. Therefore, it brings much convenience to the applications of OAM beams in scattering environment. Moreover, in consideration of its non-spatial-basis characteristic, the OAM-basis TM will inspire new ideas for the applications of wavefront shaping technology in biomedical photonics and optical communication.
2. Model
The schematic diagram of the OAM-basis TM is depicted in Fig. 1. The eigenstates of OAM is given by $\exp (il\phi )$, where $ \phi $ is the azimuthal angle and l is the quantum number of OAM. An important realization with this phase structure is Laguerre-Gauss (LG) mode, which is expressed as the factor $\exp (il\phi )$ multiplied by a radial distribution function [37]. Although the Hilbert space expanded by LG modes is infinite dimensional, but a finite number of low-order modes are enough to precisely represent a local light field, which makes it possible to experimentally realize the control of the light field. In this model, a total number of N LG modes are utilized as the input bases, and we mark the space expanded by the N modes as OAM space. Each mode is indicated by a N-dimension vector in OAM space. The blue square in the diagram donates an amplitude of one, while the white square donates an amplitude of zero. Then on the output plane, the light fields at M different positions are recorded as the output modes, which constitute an output vector. Different colors of the squares in the diagram represent different complex amplitudes. The OAM-basis TM is a linear operator that transforms each input vector to the corresponding output one,
Fig. 1. The schematic diagram of the OAM-basis TM. Each input vector represents an LG mode. The blue square donates an amplitude of one, and the white square donates an amplitude of zero. The output vectors represent the spatial distributions of the output light field. Different colors of the squares represent different complex amplitudes.
Once the TM is calculated, one can manipulate the light propagation inside the scattering and achieve the deep focusing with time reversal operation. The input vector for achieving a target output vector Eout is written as $T_{\textrm{OAM}}^\dagger {E^{out}}$, where $\dagger$ donates the conjugate transpose. For instance, the m-th column of $T_{\textrm{OAM}}^\dagger$ will act as the input vector for focusing light to the m-th output mode. The input vector is converted to a spatial distribution function by adding up all the N input modes with its elements serving as coefficients,
where $t_{mn}^ \ast $ represents the conjugation of the TM elements.In the following sections, we will demonstrate the experimental measurement of the OAM-basis TM and the realization of deep focusing with the input field described by Eq. (3).
3. Experimental setup
The experimental setup for measuring the OAM-basis TM is sketched in Fig. 2. The light source was a 532 nm solid-state CW laser (CNI Laser, MGL-III-532) with a maximum output power of 200 mW and a M2 factor of 1.1. After being 8 × expanded, the beam was split to the signal part and reference part by a polarization beam splitter (PBS). The half-wave plate (HWP1) before the PBS was utilized to adjust the power ratio between the signal light and reference light. The horizontally polarized signal light was modulated by a liquid-crystal spatial light modulator (SLM, Hamamatsu, X13138-04, pixel size: 12.5 µm × 12.5 µm, resolution: 1280 × 1024). The method proposed by Arrizón et al. was applied to achieve the amplitude-phase encoding of the LG modes with the phase-only SLM [38]. After leaving the SLM, the signal light was reflected by a 50:50 beam splitter (BS1) and projected to the entrance pupil of the objective lens (Obj1, Olympus, MPLN 10 ×, NA = 0.25) by the telescope system (f1 = 150 mm, f2 = 75 mm). Then the focused signal light illuminated the scattering medium that was a ground glass diffuser (S, Edmund, 220 grits). The transmitted speckle pattern, which was 7 mm away from the back surface of the diffuser, was imaged to the CMOS camera (AVT, Mako G-131B, pixel size: 5.3 µm × 5.3 µm) by an objective lens (Obj2, Olympus, MPLN 10 ×, NA = 0.25) and an imaging lens (f3 = 180 mm). To accurately record the intensity of each speckle, the speckle size should be larger than the pixel size of the camera. According to the diffraction theory of the speckle field, the speckle size is inversely proportional to the illumination area on the scattering medium [39]. That is also the reason why Obj1 was utilized to shrink the beam size. Moreover, the illumination size could be fine adjusted by moving the scattering medium between Obj1 and its front focal plane (the front focal length is 10.6 mm). In our case, the scattering medium was placed 5 mm away from Obj1. Consequently, each speckle covers an area of at least 8 pixels × 8 pixels. As for the reference light, a half-wave plate (HWP2) and a polarizer (P) adjusted its polarization state to be the same as that of the signal light. Then the reference light passed through a telescope system (f4 = 150 mm, f5 = 150 mm) and was combined with the signal light by a 50:50 beam splitter (BS2). An extra phase shift $\Delta \varphi$ was applied to the signal light to perform holographic interferometry. For $\Delta \varphi = 0$, ${\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}$, $\pi$ and ${{3\pi } \mathord{\left/ {\vphantom {{3\pi } 2}} \right.} 2}$, four interference intensity patterns were recorded as ${I^0}$, ${I^{{\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}}}$, ${I^\pi }$ and ${I^{{{3\pi } \mathord{\left/ {\vphantom {{3\pi } 2}} \right.} 2}}}$, respectively. Then the signal field was calculated to be [7]:
Fig. 2. The experimental setup for measuring the OAM-basis TM. SLM, spatial light modulator; BS, beam splitter; PBS, polarization beam splitter; HWP, half-wave plate; BE, beam expander; A, aperture; f, focusing lens; Obj, objective lens; S, scattering medium; P, polarizer; M, reflecting mirror. Part of the input LG modes used in the experiment are displayed in the dashed box.
To perform the measurement of the OAM-basis TM, we utilized 1000 input LG modes (i.e. N = 1000). We took the radial index p of the LG modes from 0 to 19, and the azimuthal index l from 0 to 49. Then the sequence number of each modes is given by $n = 50p + l + 1$. Part of the input modes are displayed in the dashed box in Fig. 2. On the other side, the signal light fields on an array of 32 × 32 camera pixels were selected as the output modes (i.e. M = 1024). The space interval between adjacent output modes was set to be large enough to ensure that there is no obvious correlation between them. The amplitudes and phases of all the output modes were recorded for each input modes, and the OAM-basis TM was obtained according to Eq. (2).
4. Results and discussion
4.1 Single-spot and multiple-spot focusing
With the measured OAM-basis TM, the deep focusing through the scattering medium could be performed. We first calculated the time reversal operator ${O^{TR}} = {T_{\textrm{OAM}}}T_{\textrm{OAM}}^\dagger$, which is often used to characterize the capacity of TM in focusing light to the output modes [7]. The rows of ${O^{TR}}$ predict the output patterns, and the elements on the diagonal describe the foci. The normalized amplitude profile of ${O^{TR}}$ in our case is shown in Fig. 3(a), and the part inside the dashed box is magnified. The diagonally-dominant matrix signifies the ability of generating sharp and bright foci. After modulating the laser beam as Eq. (3) and injecting it to the scattering medium, we have achieved single-spot focusing, as shown in Fig. 3(b). The FWHM of the focus was 4.8 µm, and the peak to background ratio (PBR) was calculated to be 310. The amplitude and phase profiles of the input field are illustrated in Figs. 3(c) and (d), respectively.
Fig. 3. (a) The normalized amplitude profile of the time reversal operator, the magnified view of the local part inside the dashed is displayed. (b) The focus generated after the wavefront shaping. (c) and (d) The amplitude and phase profiles of the optimized input light field. Scale bar: 10 µm.
In addition, with the knowledge of the measured OAM-basis TM, we can also generate multiple foci behind scattering media. The input vector for multiple-spot focusing is equal to the linear superposition of those for single-spot focusing:
Fig. 4. The optimized input light field and the experimental results of the multiple-spot focusing with the OAM-basis TM. (a) and (b) The amplitude and phase profiles for generating horizontal 3-spot focusing. (c) The horizontal 3 foci measured in the experiment. (d) and (e) The amplitude and phase profiles for generating vertical 3-spot focusing. (f) The vertical 3 foci measured in the experiment. Scale bar: 10 µm.
4.2 Generation of the doughnut-shaped focus
The doughnut-shaped focus that carries OAM is potential in multi-degree-of-freedom optical tweezers and super-resolution imaging through biological tissues [35,36]. The conventional real-space TM can be employed to generate this class of foci by designing the TM-based point spread function [40]. However, we demonstrate here that the OAM-basis TM provides a more convenient way to generate the doughnut-shaped focus.
We built an operator ${O^M}$ that is applied to the vectors in the OAM space,
Equation (7) is equivalent to add extra $1\hbar$OAM to most of the terms in Eq. (3). The extra $1\hbar$OAM is transferred to the focus during the focusing process, which leads to the formation of a doughnut-shaped focus. Figures 5(a) and (b) respectively illustrate the amplitude and phase profiles of $E_m^{in({mod} )}$, and the generated doughnut-shaped focus is shown in Fig. 5(c). Therefore, the operator ${O^M}$ could be phenomenologically defined as the creation operator of the OAM inside the focus.
Fig. 5. (a) and (b) The amplitude and phase profiles of the modified input field for generating the doughnut-shaped focus. (c) The doughnut-shaped focus measured in the experiment. Scale bar: 10 µm.
4.3 Eigenchannels of the OAM-basis TM
The eigenchannels of scattering media come from the singular value decomposition (SVD) of TM. Each of them is characterized by an individual transmittance decided by the eigenvalue $\tau$ (square of the singular value). The eigenchannels given by the conventional real-space TM have been well investigated [12,24,25]. As a new type of TM, the eigenchannels of OAM-basis TM need to be explored. Hence, we conducted SVD of the OAM-basis TM,
where the columns of the matrices V and U respectively give the input and output vectors of the eigenchannels, and the elements on the diagonal of the matrix S give the singular values. The normalized singular values are plotted in Fig. 6(a) with the linear-logarithm coordinate. As a contrast, we measured the real-space TM with 1024-order Hadamard bases with the same setup. Its singular values are plotted in Fig. 6(b). Obviously, there is a difference between the distribution properties of the OAM-basis TM and the real-space TM. This derives from the specific structures of OAM bases. Different OAM bases possess different spatial sizes and phase structures, so the contributions of different OAM bases to the output speckles are not uniform. As a result, the statistical distributions of the OAM-basis TM elements are not identical. However, all the elements of the real-space TM obey an identical distribution [10]. So the two TMs show different distribution properties of singular values.Generally, the high-transmittance channels (channels with high singular values) are attractive, as they provide optimized combinations of input bases to enhance the optical energy transmission. A legitimate question is which OAM bases contribute more to the high transmittance. To explore this question and draw a general conclusion, we took the input vectors of 50 highest-transmittance channels and calculated their average amplitude. The elements of the resulting vector are defined as the contribution factors of the corresponding OAM bases. The normalized contribution factors are plotted in Fig. 7(a) as a function of the OAM basis index. It can be seen that the curve shows a periodical variation, and each period corresponds a different radial index p. This phenomenon signifies that p has little influence on the contribution factor. In each period, the azimuthal index l varies from 0 to 49. We took the 20th period as an example and the magnified view of it is shown in Fig. 7(b). Obviously, the contribution factor increases with the increase of l. Therefore, it is concluded that the OAM bases with higher values of l contribute more to the high-transmittance channels. Then we also calculated the average amplitude of the input vectors of 50 lowest-transmittance channels, and the result is shown in Fig. 7(c). The contribution factor is also a nearly periodic function. According to the magnified view of the 20th period shown in Fig. 7(d), we can see that the contribution factor decreases with the increase of l. This suggests that the low-transmittance channels are dominated by the OAM bases with lower values of l. Moreover, from these results we could infer that the OAM modes with higher values of l have higher transmittance through the scattering medium, which is in accordance with the conclusion of Wang et al. [41].
Fig. 7. (a) Contribution factors of the OAM bases to the high-transmittance channels. (b) The magnified view of the 20th period in (a). (c) Contribution factors of the OAM bases to the low-transmittance channels. (d) The magnified view of the 20th period in (c).
5. Conclusion
In conclusion, we have proposed the OAM-basis TM as a new approach for controlling the light propagation through scattering media. We have introduced the theoretical model of the OAM-basis TM and measured it in the experiment. The availability of the OAM-basis TM for wavefront shaping was verified by achieving single-spot and multiple-spot focusing, and the focus with a FWHM of 4.8 µm and a PBR of 310 was realized. Then we showed that a doughnut-shaped focus that carries OAM could be generated by introducing a simple operator and modifying the input light field. At last, the properties of the eigenchannels was discussed. Compared with the conventional real-space TM, the OAM-basis TM shows a different distribution property of singular values, which derives from the specific structures of OAM bases. Moreover, we have also analyzed the contributions of the input bases to the eigenchannels. It revealed that the OAM bases with higher values of azimuthal index l contribute more to the high-transmittance channels. The OAM-basis TM has extended the definition of TM, and it will brifng new research perspectives for the technologies of wavefront shaping, deep focusing, and biomedical imaging. Additionally, the OAM-basis TM could guide the tailoring of the input OAM spectrum to obtain desired optical energy propagation in scattering media, and it is potential to serve as a theoretical foundation for the applications of OAM light in biological tissues and turbid atmosphere.
Funding
National Natural Science Foundation of China (61905128, 61875100).
Disclosures
The authors declare no conflicts of interest.
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