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Multimode fibers are attractive for a variety of applications such as communication engineering and biophotonics. However, a major hurdle for the optical transmission through multimode fibers is the inherent mode mixing. Although an image transmission was successfully accomplished using wavefront shaping, the image information was not transmitted individually for each of the independent pixels. We demonstrate a transmission of independent signals using individually shaped wavefronts employing a single segmented spatial light modulator for optical phase conjugation regarding each light signal. Our findings pave the way towards transferring independent signals through strongly scattering media.
© 2016 Optical Society of America
1. Introduction
Since spatially structured light can be transmitted through a multimode fiber (MMF), a MMF is advantageous in many applications compared to a single-mode fiber. In contrast to a bundle of multiple single-mode fibers, no pixelation occurs and a significantly higher spatial density of light information is possible when using a MMF [1], providing a minimum instrumentation footprint, e.g., for endoscopy [2].
However, light launched into a MMF excites multiple modes mixing with each other, which results in a scrambling of the spatial light distribution at the fiber output. Although the output pattern looks random, it is resulting from deterministic scattering processes in the fiber, meaning that the scrambling is systematic and can be corrected. There were early correction attempts in order to see through multimode fibers and other strongly scattering media [3–5]. Pioneering work with modern optoelectronic devices was accomplished by Vellekoop and Mosk [6], who applied wavefront shaping to focus through scattering media using an iterative optimization method employing a spatial light modulator (SLM). This iteration method uses a feedback from the signal intensity at the output facet of the MMF, which is compared with the desired intensity pattern [7,8]. In [8], the transmission of multiple focal spots through a MMF has been shown, but the spot signals were depending on each other, i.e., the signals could not be turned on or off individually. A second method is to describe the linear and deterministic light propagation through scattering media. This can be accomplished by measuring the transmission matrix between incident and outgoing waves or orthogonal modes, respectively, which can be achieved, e.g., using digital holography. As an alternative, efforts for the prediction of the transmission matrix for a MMF have been made [9]. After obtaining the complex matrix elements, the inverse of the matrix is displayed on an SLM for wavefront shaping [10–13]. Recently, the transmission of two independent signals using this matrix based method employing two SLMs was demonstrated [13], which allows the signals to be individually turned on or off. The third method is the digital optical phase conjugation (DOPC) [14–18]. DOPC is sensor based, i.e. first the optical phase of light passing the fiber is measured, when the desired intensity pattern, e.g. a focal spot (so-called beacon), is launched at the output side of the scattering medium. Finally, the received signal with inverse (conjugated) phase is playing back from the input side of the medium. In general, DOPC is superior to the methods mentioned above because of the potentially shorter response time, since there is no need to iteratively optimize the shaped wavefront or to successively determine the elements of the transmission matrix. Note that the latency of the scrambling correction has to be shorter than the decorrelation time of the scattering processes, which is given in MMF due to changing temperature and stress caused by bending the fiber. Although the transmission of multiple signals through a MMF using DOPC has been shown before, the signals were not independent [19].
Apart from pure information transmission through MMFs, which is of interest in communications engineering [20,21], there are scientific applications of wavefront shaping, e.g. the focusing and scanning inside biological tissue at deep penetration depth as well as micromanipulation and several further applications in biology and medicine [22–24]. An innovative application of wavefront shaping for light transmission through a MMF is optogenetics combining optoelectronics and genetics to control the behavior of cells [25–27]. Optogenetics allows the activation or the inhibition of genetically modified neurons. This offers the ability to elucidate the functions of neural circuitry, as well as new therapy approaches for neurological dysfunctions [28], e.g. drug addiction, chronic pain, Parkinson disease, schizophrenia or epilepsy. Using wavefront shaping techniques in optogenetics enables addressing single cells at deep penetration depths by suppressing turbidity of biological tissue [29–31]. So far, only single light signals have been transmitted for addressing the cells. In contrast, the transmission of independent light signals using wavefront shaping bears a high potential to boost the performance in the field of optogenetics, as it would enable individual activation or inhibition of various neurons.
In this paper, we present the transmission of independent signals through a MMF based on DOPC employing a single SLM only. To this aim, the SLM is segmented into regions as depicted schematically in Fig. 1, here for two signals a1(t) and a2(t). At first, the experimental setup used for the novel transmission method as well as the procedure of the transmission is described. Then, the transmission of these two independent signals is demonstrated. Assuming an ideal transmission, the relations for the received signals b1(t) = a1(t) and b2(t) = a2(t) hold. The actual performance of the transmission is evaluated including the analysis of the signal crosstalk. Finally, perspectives and challenges regarding the proposed method for optogenetics and other applications as well as its potential for more than two signals are discussed.
Fig. 1 Concept for transmission of two independent light signals through a multimode fiber (MMF) using a single spatial light modulator (SLM). The transmission of the light signals a1 and a2 is accomplished by shaping the incident light at the proximal side of the MMF on multiple regions of the SLM, which is done individually for each signal. In the ideal case, the spatially separated light signals b1 = a1 and b2 = a2 are received at the distal side of the MMF.
2. Experimental setup and procedure
The experimental setup schematically depicted in Fig. 2 is used for the transmission of two independent light signals a1(t) and a2(t) from the proximal side of a step-index MMF (Thorlabs M14L02, 2 m length, 50 µm core diameter, numerical aperture 0.22) to the distal side, where two individual spots are generated, corresponding to the received signals b1 and b2. We call the proposed method a two-signal DOPC employing a single phase-only SLM for phase modulation and one CMOS camera for the holographic phase measurement.
Fig. 2 Experimental setup of the transmission of two independent signals using digital optical phase conjugation: a) The calibration step is performed by setting the target light pattern (focal spots) for the output signals b1 and b2 at the distal side of the MMF and measure the phase of the inversely propagating light at the proximal side. b) The transmission step is performed by manipulating the light phase of the input signals using an SLM. Abbreviations: APD = avalanche photodiode; BS = beam splitter; CCD = charge-coupled device; CH = optical chopper; CMOS = complementary metal-oxide semiconductor; HWP = half wave plate; L = lens (achromatic); LP = linear polarizer; MMF = multimode fiber; OBJ = microscope objective; SLM = spatial light modulator.
2.1 Setup
The light source is a single-frequency solid state laser (Spectra-Physics, Excelsior) which emits a continuous-wave at a wavelength of λ = 532 nm with 0.15 W output power. The laser light is divided by a polarizing beam splitter (PBS) into two beams: The object beam serving as beacon light on the distal side is needed for the phase measurement in Fig. 2(a) only. The reference beam is needed in Fig. 2(a) for the phase measurement using off-axis holography and in Fig. 2(b) to provide light for the transmitted signals a1(t) and a2(t) having a variation of intensity in time, which is accomplished by two optical choppers CH1 and CH2. A half-wave plate (HWP) between the laser and the PBS is used to adjust the intensity ratio between object beam and reference beam in order to achieve a high contrast in the hologram. The second HWP at the object beam can be utilized for adjusting the polarization of the beacons at the distal end. In order to obtain two independent light signals, both object beam and reference beam are split each into two separate beams using the beam splitters BS1 and BS7, respectively. As an alternative, a single laser could be used for each signal. The separated beams are directed to separate regions 1 and 2 on the SLM and CMOS camera, since an individual measurement and modulation of two independent light signals is required. Each of the regions on the CMOS camera (Mikrotron MC4082, 2336 x 1728 pixel) and the SLM (liquid crystal on silicon, Holoeye Pluto VIS, 1920 x 1080 pixel) has a size of 400 x 400 pixel, i.e, the regions do not fill the CMOS and the SLM entirely. Since only one polarization can be modulated by the SLM, a linear polarizer (LP) is used for filtering the light accordingly.
In order to achieve a high performance of the DOPC, an accurate matching of the pixels of camera and SLM is necessary [32]. Since there is a mismatch of the pixel pitch between SLM (8 µm) and camera (7 µm), a correction is made by using a Keplerian telescope employing the lenses L1 and L2 with a resulting magnification factor of 7/8. In order to check the DOPC performance, a CCD camera is employed at the distal side. Moreover, two fiber-coupled avalanche photodiodes (APD) are employed for receiving the signals b1(t) and b2(t) separately. The SLM, the CMOS camera, and the fiber facet on the proximal side represent optically conjugated planes as well as the CCD camera and the fiber facet at the distal side do, which is accomplished by two microscope objectives OBJ1 and OBJ2 (both with a magnification factor of 20 and a numerical aperture of 0.4).
2.2 Procedure
The procedure for the transmission using the proposed multi-signal DOPC is based on a time reversal operation [33]. In the first step, the calibration is performed in Fig. 2(a) by sequentially switching on each of the two signals b1 and b2 at the distal side of the MMF. These signals are used as independent beacons, similar to the laser guide star concept at earth-bounded telescopes in astronomy [34]. The complex phasor of the scattered light obtained at the proximal side is measured using off-axis holography employing the CMOS camera, where the intersection angle between reference and object beam amounts to approximately 2°. A hologram is acquired for each of the beacon signals corresponding to b1 and b2. Since off-axis holography is used here, a single acquisition is sufficient for the reconstruction of the phase, which is performed using the angular spectrum method [35]. According to [35], a filtering in the spatial frequency domain omits the zeroth and minus first diffraction order being irrelevant here. The hologram for each beacon is captured in a separate region of the CCD camera which is mapped to a separate region on the SLM. Note that the SLM is inactive in this first step, i.e. it behaves like a (reflective) mirror.
In a second step, the independent signals a1(t) and a2(t) are transmitted in Fig. 2(b) by projecting the inverse of the measured phase on separate regions SLM and directing independent light signals a1(t) and a2(t) on these regions in order to be individually modulated in phase there. Using this procedure, the phase-conjugated light is played back through the MMF, such that two spots for each of the signals b1(t) and b2(t) are generated at the distal side.
3. Analysis of the performance
The intensity at the distal fiber facet is depicted in Fig. 3, measured by the CCD camera. Two beacons are shown in Fig. 3(a), being generated by two laser beams focused at the distal side. In Fig. 3(b), the speckle field of the MMF without DOPC is shown. The average radius of the speckle was determined to 1.94 µm using the autocorrelation function. This value is related to the radius of the diffraction limited spot of λ / (2 NA) ≈1.21 µm with the laser wavelength λ = 532 nm and the numerical aperture NA = 0.22 of the fiber. In Fig. 3(c), the unscrambling of the light field of the DOPC is outlined: As intended, two light spots are generated at the positions of the beacons. The corresponding intensity profiles are displayed in Fig. 4. Using DOPC, two laser spots appear, which are enhanced over the speckle granulation, see Fig. 4(c). The diameter of the spots amounts to 1.5 µm (full width at half modulation), which is close to the diffraction limit. However, the laser spot is accompanied by a speckle background pattern, which is also visible in Fig. 4(c) and results from the finite performance of the DOPC.
Fig. 3 Light intensity at the distal fiber facet measured by a CCD camera. a) Generation of two beacons as references for the calibration procedure of the digital optical phase conjugation (DOPC). b) Speckle pattern resulting from the scattering process in the multimode fiber. c) Generation of two foci at the positions of the beacons by two-signal DOPC.
Fig. 4 Intensity profiles along the x-axis of the distal fiber facet, cf. Figure 3: a) Beacon with a full-width half maximum (FWHM) size of 1.7 µm. b) Random speckle pattern due to coherent scattering inside the multimode fiber. c) Phase conjugated light with an unscrambled spot which corresponds to the beacon.
In order to quantify the performance of the multi-signal DOPC, the peak-to-background ratio (PBR) is used as a figure of merit [15, 36], which is defined as focus peak value over the arithmetic mean value of the grainy intensity pattern.
The analytically calculated PBR reads:
where “N is the number of optical modes intercepted and time-reversed by the phase conjugate mirror” [15]. The number of modes is strongly depending on the degree of freedom of both the complex scattering media, i.e. the number Nmodes of mesoscopic scattering channels or modes and the DOPC, i.e. the pixel number Npixels of the modulator (assuming each mode can be represented by one pixel). As a consequence, N equals the smaller value of both quantities. The factor k depends on the light modulation process. Owing to the SLM employed here, the phase is modulated only, i.e. the amplitude remains unchanged, which yields k = π / 4 [15]. In Eq. (1), C is usually defined as the number of targets, which equals one if only one focal spot is desired. For the calculation of the PBR according to Eq. (1), we assume that the number of modes is the limiting degree of freedom N. This is valid because the number of modes reads Nmodes = 16 R2 NA2 / λ2 ≈1710, according to [17], using the fiber’s core radius of R = 25 µm and NA = 0.22. In contrast, the number Npixels of effective pixels of one region on the SLM is much larger: Npixels = 400 ∙ 400 ∙ π / 4 ≈1.26 ∙ 105, considering the circular geometry of the fiber core. Using Eq. (1), for a degree of freedom N = Nmodes ≈1710 and C = 1, a theoretical maximum PBR of 1343 results. We have measured a PBR of 36 and 54 for the signals b1 and b2, respectively, if only one of the signals is transmitted at a time. The deviation from the theoretical value is expected to result from insufficient excitation of the modes within the fiber, i.e. not all available modes were employed for the transmission. Moreover, the non-perfect DOPC may also reduce the PBR, which is caused, e.g., by a limited alignment accuracy, a limited fill factor of the SLM pixels, the crosstalk between the SLM pixels, decorrelation effects and optical aberrations.In the following, we like to address the PBR in case of the transmission of multiple independent signals. Since an individual wavefront shaping for the independent signals is necessary, each corresponding focus spot at the distal end is accompanied by an individual speckle background pattern. These multiple background patterns originating from C independent signals overlap and the resulting background intensities can be calculated as the sum of the first order moments, i.e. the arithmetic mean values of the speckle pattern [37]. Assuming approximately the same background levels for each of the signals, this yields:
where it has been assumed that the number of freedom N from Eq. (1) is still limited by the number Nmodes of modes of the fiber and not by the number of pixels of the SLM. We have measured a PBRmultiple of 17 and 20 for the signals b1 and b2, respectively, if both signals are transmitted at the same time, i.e. C = 2. This is roughly half of the PBR measured for the transmission of a single signal, i.e. C = 1, which agrees well with Eq. (2).The question arises, how many independent signals can be transmitted through a MMF at a given PBRmultiple. Based on the Eq. (2) the maximum number Cmax of signals reads Cmax = kNmodes/PBRmultiple ≈67, assuming PBRmultiple ≈20 achieved here.
4. Signal crosstalk
The parasitic background speckle pattern results in signal crosstalk which is considered now for the transmission of two signals through the MMF. The transmitted signals are amplitude modulated by optical choppers, see Fig. 2 and can be written (only regarding the fundamental harmonic components) as a1(t) = A sin (ω1t) and a2(t) = A sin (ω2t) with the amplitude A and the angular frequencies ω1 = 2 π 220 s−1 and ω2 = 2 π 170 s−1. At the distal side, two signals are received when using the multi-signal DOPC. Figure 5 shows the spectra of the received signals b1(t) and b2(t) at the distal side. They are related to the fundamental harmonic components of the transmitted signals a1(t) and a2(t) at the proximal side, respectively. However, there is a crosstalk at the received signals, which is typically defined as the amplitude ratio of the undesired signal originating from the adjacent desired signal to the adjacent signal itself.
Fig. 5 Crosstalk between two received signals. Two independent signals a1(t) and a2(t) are transmitted through one multimode fiber, compare Fig. 1. The light transmission is performed using a multiple DOPC, which provides two signals b1(t) and b2(t) at the distal side. Left: Measured amplitude spectrum of the signal b1(t) at the spot 1. The signal crosstalk in terms of an amplitude ratio is 1:28.5 ( = −29 dB). Right: Measured amplitude spectrum of the signal b2(t) at the spot 2. The signal crosstalk in terms of an amplitude ratio is 1:15.8 ( = −24 dB).
The received signals can be written as b1(t) = B11 sin (ω1t) + B12 sin (ω2t) at spot 1 and as b2(t) = B21 sin (ω1t) + B22 sin (ω2t) at spot 2, see Fig. 5. At the signal b1, the desired signal amplitude B11 corresponding to the signal a1 is disturbed by the crosstalk signal amplitude B12 originating from the signal a2. The crosstalk from the signal 2 to the signal 1 was determined to B12:B22 = 1:28.5 ( = −29 dB). At the signal b2, the desired signal amplitude B22 corresponding to the signal a2 is disturbed by the crosstalk signal amplitude B21 originating from the signal a1. The crosstalk from the signal 1 to the signal 2 was determined to B21:B11 = 1:15.8 ( = −24 dB).
Now we perform a theoretical modelling of the crosstalk assuming the same size of the spots 1 and 2. In addition, the PBR is large compared to 1 for both spots so that the amplitudes of both received signals are approximately identical, having the value B. Then, the signal amplitudes originating from signal 1 are B11 ≈B and B21 ≈B/PBR1, resulting in a crosstalk of B21:B11 ≈1/PBR1 = 1:17. The signal amplitudes originating from signal 2 are B22 ≈B and B12 ≈B/PBR2, resulting in a crosstalk of B12:B22 ≈1/PBR2 = 1:20. A qualitative agreement with the direct determination of the crosstalk can be observed, both for signal 1 having a crosstalk of 1:17 (from model) and 1:15.8 (from direct determination) and for signal 2 having a crosstalk of 1:20 and 1:28.5, respectively. The quantitative deviation is due to the fact that the average intensity (as basis for the direct crosstalk determination using the amplitude spectrum) measured by one APD is generally lower than the peak value (as basis for the crosstalk model using the calculated PBR) existing in the corresponding spot. To conclude, the crosstalk of the amplitudes is about 5% on average for both signals. As a consequence, the crosstalk can be reduced by increasing the PBR according to Eq. (2).
5. Discussion
To realize the transmission for a large number C of independent signals, many beam splitters would be necessary to provide separate beams for each of the C signals at C regions of the SLM and the CMOS camera, respectively. Alternatively, integrated optical multiplexers such as diffractive optical elements (DOEs) are promising here. DOEs have already shown convincing performance at coherently combing a huge number of beams [38].
For in-vivo applications in biomedicine and real-world optical communication techniques, the DOPC latency has to be shorter than the decorrelation time of the scattering process. A fast response of the SLM can be achieved by using ferroelectric modulators (e.g. Forth Dimension Displays) [39]. Furthermore, adaptive elements like steering mirrors [40] and electrically tunable lenses [41] as well as deformable mirrors [42] or digital micromirror devices [43] also offer small response times. In the future, the response time (less than a second on a standard PC) has to be reduced further, e.g., by performing an on-line signal processing at the camera using field programmable gate arrays (FPGA) for parallelization. This is possible on commercially available cameras and enables the wavefront shaping to be calculated individually for the independent signals. Taking this into account, a small response time in the milliseconds range is realistic and perspective advantage of this transmission method. For comparison, a response time of 33.8 ms is needed for the successive determination of the 256 x 256 elements of the complete transmission matrix [44].
The requirement for optical access at the distal end of the MMF is challenging, when a device has to be placed in biological tissue. One approach uses a coherent beacon source, which is placed at the distal tip of the MMF [45]. However, a second waveguide for the illumination of the beacon is needed at the distal end, which can be realized using a double-core fiber. Alternatively, a partial reflector can be used at the distal end of the MMF [46].
6. Summary
We have demonstrated the transmission of two independent signals through one MMF waveguide using DOPC employing a single SLM. The crosstalk of the signal transmission is below −20 dB and depends on the performance of the proposed multi-signal DOPC. The expected advancements of the digital equipment in optoelectronics will foster the performance of the method especially regarding a short response time and offer perspectives for the application at strongly scattering media. Transmission of independent signals is demanded for photostimulation in biological tissue as well as, e.g., in internet data transfer. In optogenetics, a targeted illumination with both spatially and temporally controlling is of paramount importance for neural networks, since multiple independent light spots are desired for an individual stimulation of distinguishable cells with diverse temporal pattern.
Acknowledgment
Support by a Reinhart Koselleck project (CZ 55/30) of German Research Foundation (DFG) is gratefully acknowledged.
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