In the last few decades, optical fibers have seen wide applications in many fields, for example, fiber communication [1], fiber sensors [2], and optical imaging [3], due to the excellent characteristics of fibers such as ultrahigh bandwidth, anti-electromagnetic interference, low security risk, small diameter, and light weight. At the beginning, a single-mode fiber (SMF) can only support one optical mode. To improve the capacity while maintaining the compact size, few-mode fibers were developed via mode-division multiplexing [4], but the improvement of the capacity is still limited as only a few modes are supported. Comparably, multimode fiber (MMF) can support a large number of optical modes, which can be used to greatly improve the information capacity [5]. That said, when light transmits inside an MMF, the coupling of different optical modes makes the resultant signal hard to be demodulated; if coherent light is input, a randomly distributed speckle pattern is observed at the fiber output. The phenomenon is quite similar to the optical propagation inside or through a scattering medium such as ground glass and biological tissue. MMFs were initially developed to transmit digital signals in the time domain [6], and there were few attempts to transmit images through the MMF. But since 2007, the application scenarios of MMFs have been greatly extended after the proposal of wavefront shaping with the availability of spatial light modulators (SLMs) [7–10]. As the output light field from the MMF can be modulated arbitrarily by applying different phase and/or intensity patterns on the SLM or iteratively optimizing the input wavefront [11–14], some exciting applications based on a single MMF have been inspired, such as deep-brain imaging [15], 3D holographic optical tweezer [16], single neuron stimulation [17], spectrum analyzer [18], and nonlinear multimode interactions [19]. In these applications, all input modes on the SLM serve as a whole unit and contribute to a single output target with enhanced optical energy and/or shifted wavelength, showing a single-channel signal processing task. On top of that, multi-channel signal processing can be introduced by dividing the input and output into several segments, and some more complex functionalities have been demonstrated, such as programmable optical circuits [20], linear quantum networks [21], and linear operators [22]. In these works, the input/output segments are independent, which is restricted to processing separate signals.

Making the input/output segments interdependent can further extend this implementation to processing associated signals. To achieve this goal, in this work, a group of phase bias maps are superposed on the input segments to globally adjust the phase of the corresponding output fields, thus the coherent superposition of the desired group of output fields could make the input signals associated, and the superimposed output field can be used to recognize how the input signals are associated. That is, different superimposed output fields can be related to different association modes or associations of different input signals. In experiment, we utilize this strategy to demonstrate logic signal processing and transmission through a 15-m-long MMF simultaneously by defining the output states with binary values. To be specific, we employ a digital micromirror device (DMD, a type of SLM) loaded with precalculated wavefront patterns to modulate light before transmitting through an MMF. The DMD aperture is divided into several subregions, as illustrated in Fig. 2, that correspond to binary input digits (“0” and “1”), logic types (AND, OR, and NOT), and a common reference region, respectively. The sub-TM of the MMF corresponding to each subregion is determined by the TM-based WFS approach [23]. Two randomly selected small regions on the output plane (e.g., the camera sensor) serve as the output optical logic states “0” and “1”; light projected from each DMD subregion through the MMF can be refocused to these two regions, individually or simultaneously, with optical phases with respect to the speckle pattern projected from the common reference region on the DMD.

As a proof of concept, built upon constructive or destructive interference among the focuses, different combinations of the subregions on the DMD loaded with precalculated wavefronts can perform different logic functions. Cascaded optical logic gates can be implemented by linking a series of logic gates with logic types placed between any two adjacent logic gates. For bitwise operators, bit-by-bit operations are performed between two 6-bit operands with the same or different logic types in parallel. Simultaneous reconstruction of multiple optical logic gates in a single logic gate with polarization multiplexing is also explored, which can be used to improve the scalability and security. With the fast development of optical wavefront shaping and optical modulators, more complex and practically useful operations could be accomplished in the near future.

Figure 1 illustrates the schematic and physical model of the proposed optical logic operators via an MMF. As shown, three subregions on the input layer (i.e., the DMD) are selected and marked with different colors, carrying desired wavefront patterns that represent a logic operation “0+1” in combination. Light reflected and modulated by the input layer transmits through an MMF, and the corresponding light field on the output layer can be observed and recorded by a digital camera. Two circular regions on the output layer are preselected to represent logical states “0” and “1”, respectively. As an example, with logic operation “0+1” being activated on the input layer, the output is “1”, producing an optical focus at the region corresponding to “1” (the left circle at the output layer in Fig. 1). The physical model to demonstrate the above schematic is briefed herein. Ei (i=1,2,3) is the input light field from each of the three subregions on the input layer. The transmission matrix (Ti) used to build the relationship between each input light field (Ei) and the corresponding output field on the output layer can be calculated. The optical intensities in the preselected regions can be expressed by $$\begin{gathered} Itotal1=(∑Area“1”TiEi)2 \\ Itotal0=(∑Area“0”TiEi)2, \end{gathered}$$(1)where Itotal1 and Itotal0 are the accumulated optical intensities in the regions “1” and “0”, respectively, on the output layer. The logic output results (Q) can be judged by the rule Q={0,Itotal1≪Itotal01,Itotal1≫Itotal0.(2)

The TM of the MMF is measured by scanning a two-dimensional array of optical focal spots across the proximal facet of the fiber. Each spot position is addressed by a specific diffraction grating loaded on the DMD (serving as a phase modulator) using the computer-generated off-axis hologram method [24]. More details about measurement of the TM can be referred to Ref. [23].

The uniformity of focusing efficiency from all subregions on the DMD is critical to ensure sufficient intensity contrast for all logic operations. In experiment, however, the uneven optical illumination on the DMD and the optical coupling efficiency of different subregions into the MMF impair the uniformity. Thus, some measures need to be taken. For example, the beam pattern projected onto the DMD is adjusted to be twice the size of the DMD aperture to form a relatively even optical intensity distribution on the DMD; from any individual subregion on the DMD, the size of the scanned region on the proximal facet of the MMF is adjusted to make sure that all input channels (focal spots) within the scanned region are efficiently coupled into the MMF, which can be gauged by monitoring the output intensity from the MMF. With all these, the uniformity of focusing efficiency from all DMD subregions can be guaranteed. In the experiment, each subregion on the DMD contains 160×160 pixels. On the output plane of the MMF, a square grid consisting of 30×30 focal spots is used to achieve bright focuses of high enhancement ratio in the region of interest. Note that it is possible to obtain multiple optical focuses from different DMD subregions in the same region on the output plane, but there is no explicit phase difference among the phases of these focuses as we can only modulate the phase of the focus from individual DMD subregions separately. Since the eventual logic states are achieved based on constructive or destructive interference among different focuses generated by selected DMD subregions at the same preselected position on the output plane, bridging the phase relationship between different focuses is crucial. Therefore, the former TM measurement method is modified by introducing a common reference subregion on the DMD [the blue region in inset (a) of Fig. 2]. The reference region contains a lot more pixels (eight times larger than each working subregion) than each of the individual subregions to ensure that the reference contribution dominates in the interfered speckled field. All pixels in the reference subregion are switched “on” to project light into the MMF during the entire TM measurement process for all selected DMD subregions (control units). These pixels are switched “off” when the TM measurement is completed.

The optical setup of the proposed platform is illustrated in Fig. 2. A continuous wave laser of 532 nm wavelength (EXLSR-532-300-CDRH, Spectra Physics, USA) serves as the light source. The laser output is first expanded by a 4f system to illuminate the whole screen of the DMD (V-7001, Vialux GmbH, Germany). After being modulated and reflected by the DMD, light is shrunk by another 4f system, and then it transmits through a collimator to be focused into the MMF (length=15m, NA=0.22, corediameter=400μm, SUH400, XINRUI, China). At last, the output optical field from the MMF is recorded by a CMOS camera (BFS-U3-04S2M-CS, FLIR Integrated Imaging Solutions Inc., Canada), which is triggered by the DMD. In experiment, once the DMD pattern is updated, the camera will record an optical pattern.

An inverse design approach is adopted to arrange focuses on the output layer formed by the seven DMD subregions individually. First, the desired intensity and phase (0 or π) of each focus are calculated for a set of basic logic functions (AND, OR, and NOT) (the calculation process is provided in Appendix A). The solution is given in Fig. 3(a) and explained in detail in the caption. Note that two focuses of specific intensity and phase in the designated regions on the output plane should be generated synchronously from some DMD subregions with phase superposition of two corresponding phase patterns. These results are used to calculate the wavefront of each DMD subregion in reverse. The experimental verification of the design arrangement is shown in Figs. 3(b)–3(k). As mentioned earlier, two randomly picked regions on the output plane serve as logic states “0” and “1”. As an example [Fig. 3(b)], when only the subregion corresponding to “0” on the left column of the DMD (as shown in Fig. 2) is activated and loaded with the precalculated wavefront [as shown in Fig. 3(i)], two optical focuses can be observed in the selected regions [Fig. 3(b)]. Figures 3(c)–3(h) display the output images when other subregions are activated individually (one active subregion at one time) and loaded with the corresponding precalculated wavefront. As seen, all experimental results agree well with the simulated sets in Fig. 3(a). The enhancement ratio, defined as the ratio between the optimized intensity and the average intensity before optimization, is about 240 for cases with one focus [Figs. 3(d) and 3(f)]; for cases with two focuses, the enhancement ratio is in the range of 120–150. Figures 3(i)–3(k) show the precalculated binary patterns on three subregions corresponding to binary digit control units “0” (left column), “1” (right column), and logic type control unit “AND”, respectively. The intensity profiles along the central line of red circles in the upper region of Figs. 3(b), 3(c), 3(e), 3(g), and 3(h) indicate relatively balanced enhancement ratios for cases with two focuses.

The DMD subregions will display the corresponding calculated wavefronts when the subregions are selected for specific logic operations; the other subregions will be turned “off,” in which the DMD is used as a spatial switch. Figures 4(a)–4(c) show the DMD deployment for three basic optical logic gates. Inputs A and B, two binary operands, are chosen between two subregions, each representing one set of binary digits (“0” or “1”) in the leftmost column and the rightmost column, respectively. For AND and OR logic gates, Input A, Input B, and the subregion representing the corresponding logic type are activated, as shown in Figs. 4(a) and 4(b). For NOT logic gate, either Input A or Input B is activated besides the subregion representing NOT logic type, as shown in Fig. 4(c). Activated subregions will be loaded with the corresponding wavefronts that are precalculated based on the TM method discussed earlier. For a given logic operation, the logic output state (QAND, QOR, and QNOT) can be either 0 or 1.

Experimental results of the three basic logic operations (AND, OR, and NOT) in a single logic gate are shown in Fig. 5. Switching on a desired group of subregions will result in appropriate optical logic output. As defined, if a bright focus is formed at the region representing output logic state “1” on the output plane, the logic operation registers Q=1 and vice versa. As shown, all experimental outputs for each mode behavior in Fig. 5 agree well with the required logic outputs. According to the inverse design, the desired focus intensity contrast ratios of logic operations AND (0,0) and OR (1,1) are 6 dB. For other logic operations, the contrast ratios shall be infinity as long as the focal peak-to-background ratios (PBRs) are sufficiently large to suppress the speckle background and most diffusive photons are focused onto designated regions. In our experiment, the intensity contrast ratios are 5.2 dB and 4.7 dB for logic operations AND (0,0) and OR (1,1), respectively. These are slightly lower than the designed contrast ratios, which may attribute to two aspects. First, the focusing efficiency of two focuses from the same subregions is not well-balanced. Second, the enhancement ratios are also not completely unitary at the same region on the output plane for different DMD subregions. The intensity contrast ratios of other logic operations are all more than 10 dB in our experiment. The intensity profiles along the central line of red circles in the upper region of all these figures show high intensity contrast ratios for all operations.

After the demonstration of basic logic functions, the solution to realize cascaded optical logic gates is further pursued. Cascaded optical logic gates can be implemented by cascading a series of logic gates with logic types placed between any two adjacent logic gates; details can be referred to Appendix B. As a proof of concept, we experimentally demonstrate two logic operations by cascading two logic gates, as shown in Fig. 6. The logic output values are “0” and “1” for logic operations (0+1)⋅(1⋅0) and (0+1)+(1⋅0), respectively, which agree with the true values of these two operations. The intensity profiles along the central line of red circles are plotted in the upper region of the figures, whose intensity contrast ratios are 5.4 dB and 9.3 dB, respectively.

The proposed method can be used not only to implement logic gates for single-bit operands, but also to build other kinds of operators for multi-bit operands. For demonstration, we experimentally perform 6-bit operations (“bitwise AND” and “bitwise OR”), as shown in Fig. 7. Bitwise operators perform a bit-by-bit operation on two operands. They take each bit in one operand and perform the operation with the corresponding bit in the other operand. As shown in Fig. 7(a), “X” is a 6-bit operand, “Y” is the other 6-bit operand, “L” is the logic type, and “O” is the output result. The calculation of the wavefront pattern for each subregion is quite similar to that of single-bit logic gates, except that different sets of regions representing “0” and “1” should be preselected for different bits of the operand [Fig. 7(b)]. On the output plane, there are six regions in both the upper region (solid arrow) and the lower region (dashed arrow) to represent for “0” and “1” for each bit, and the numbers (0–5) correspond to the bit positions of the output. For example, Fig. 7(b) shows the experimental result of “010000” for “bitwise AND” operation between 011010 (X) and 110101 (Y), which agrees well with the true value; Fig. 7(c) shows the experimental result of “110111” for “bitwise OR” operation between 010101 (X) and 100010 (Y). Furthermore, it is possible to implement different functions for different bits by assigning them with different logic types. For example, an operand (X) “111100” can be combined with another operand (Y) “000011” via a group of logic types of “AND, OR, AND, OR, OR, AND,” respectively, to implement “bit-set” in the second bit, “bit-clear” in the fourth and sixth bits, and “bit-check” in other bits. As shown in Fig. 7(d), by recognizing the locations of the focuses, a result of “010110” is output, which is consistent with the desired value. The intensity profiles along the “0” and “1” regions between the solid and dashed arrows in Figs. 7(b)–7(d) are also plotted in Figs. 7(e)–7(g), respectively, showing high intensity contrast ratios for all bitwise operations.

The proposed method is also explored for simultaneous multiple optical logic operations with a single logic gate. As shown in Fig. 8(a), with the same set of input digits, the system can output two logic operations (OR and AND) in different output polarization channels. In experiment, a linear polarizer is put behind the MMF to change the polarization state of the speckles. Two TMs (v-TM and p-TM) are measured when the polarization is vertical (π/2) and parallel (0) to the horizontal plane, respectively. The wavefronts displayed on each subregion for OR and AND operations are calculated according to v-TM and p-TM, respectively. Two sets of wavefronts are superimposed and displayed on the corresponding subregions as shown in Fig. 8(b). The experimental results are demonstrated in Fig. 8(c) when the polarization angle takes different values. As shown, when the polarization angle is π/2, OR operations are performed; when the polarization angle is 0, AND operations are performed; and when the polarization angle is π/4, chaotic information is obtained with these two kinds of operations mixed. Such simultaneous construction of multiple logic operations in a single logic gate with polarization multiplexing can be used to extend the scalability and enhance the security of the system.

In this study, we have proposed the concept and working principle to achieve different optical logic operations, including basic logic gates and bitwise operations, with a single MMF empowered by TM-based wavefront shaping. In the proposed logic operators, the output states are recognized by the intensity contrast between two preselected regions that respectively represent logical states “0” and “1” on the output plane. With this MMF-based logic system, analog-to-digital and digital-to-analog conversions can be essentially avoided, especially when the camera is replaced with photoelectric detectors. Currently, the measurement of each sub-TM takes ∼7.2s, which is mainly restricted by the working frame rate (500 Hz) of the camera used in the system. One effective way to shorten this duration is to adopt a faster detector, such as a photoelectric detector [25], which could reduce the measurement time down to ∼160ms. This kind of task through MMF cannot be completed by traditional optical logic components based on waveguides [26–32] or metasurfaces [33,34] without reconfiguration, and in the meantime our proposed method exhibits some unique features or advantages. To begin with, different logic functions can be obtained by switching the modulating patterns on the SLM, as long as the TM of the MMF is calculated. In addition, under conditions when the MMF is considerably perturbed causing large variations to the TM of the MMF in real-world applications, TMs of MMF under plenty of perturbation conditions can be measured in advance and utilize the light field reflected by the distal end of the MMF in the proximal end to screen out the suitable TM [35]. Installing a series of controlled units on the MMF surface directly to locally adjust the MMF status with much faster speed will be a good alternative solution to improve the speed and in the meantime decrease the complexity of the optical setup [36]. As a proof of concept, in this paper, we demonstrate 6-bit bitwise operations; the bit length is mainly limited by the pixel number on the DMD (1024×768) used in experiment while guaranteeing the accuracy of each logic output. But our results show that the crosstalk among different parallel channels in bitwise operations is quite low, which means the method can be potentially extended to more bit operations when a DMD with a larger number of pixels is adopted. For example, if the merged pixel number in each subregion is kept at 160×160, which is the same as that in this paper, the maximum number of parallel channels (480×160 pixels for each parallel channel, which contains three subregions) can be as large as 104 using a new DMD that has a large number of pixels up to 3840×2160 (DLP660TE, Texas Instruments).

Energy consumption is another critical concern for a computing system. Since the sub-TMs are retrieved and all patterns to be loaded on the DMD subregions are determined in advance, the logic operations are actually performed in a single shot in a passive system, and no further calibration is needed as long as the optical system is moderately stable. Within this duration, the laser, DMD, and detectors should be in charge to complete the task. Notably, the energy consumption will not increase with the increase of logic operations if these logic operations can be performed within a DMD. In addition, in our work, different logic functions are obtained by switching on the corresponding subregions on the DMD. The speed is limited by the update rate of DMD, which can be further accelerated using other components such as an acousto-optic deflector (AOD) [23]. Stability over a long time can be obtained by packing the MMF carefully [37]. For example, high-speed image projection through a 200-m unfixed MMF with a duration of 5 days was recently reported by using single point feedback in the proximal end [35]. The work can also provide inspirations for our method to adapt to the environment perturbations.

It also should be noted that, although only basic functions such as AND, OR, and NOT are demonstrated in the study, the concept can be extended for the remaining logic gates, such as NAND, NOR, XAND, and XOR [38], and other operators including arithmetic, relational, equality, reduction, shift, concatenation, and conditional [39] with appropriate modifications. To achieve these extensions, in experiment, the amplitude of the focus is assigned with three values at 0, 0.5u, and u, and the phase of the focus is assigned with values of 0 and π. To extend the method to other logic operations, the amplitude and phase of the focus are needed to take more values. This procedure is feasible in existing optical systems with more accurate adjustment: the amplitude can be tuned accurately by controlling the “open” pixel number in each subregion; the phase of the focus can be set to an arbitrary value from 0 to 2π with respect to the common reference light field as we use a DMD to modulate the phase of the light field. More values of output states can be introduced with more precise modulation, and thus this method can be furtherly used for operations among operands of high bases such as quaternary and octonary operands. These logic operations are performed in the spatial domain, which can be potentially combined with other optical fiber multiplexing techniques, such as optical time division multiplexing (OTDM) [40] and wavelength division multiplexing (WDM) [41], for the purpose of signal transmission and processing.

Last but not least, it is also worthwhile to highlight that it is nearly impossible for hackers to steal the useful signals/information from our system due to the inherently anti-eavesdropping feature. Stealers cannot decipher the carried signals but only obtain unintelligible contents when intercepting the signals at a position between the proximal end and the distal end of the MMF as signals associated with the TMs vary dramatically at different positions along the MMF. This encryption method is inherent due to the optical transmission process inside MMF and can be enhanced by specifically modulating the polarization of the output field in the output terminal. Even hackers can obtain access to the MMF; they have to decipher the polarization first in order to decode the logic signals. The proposed method can be combined with information encryption means in the input terminal such as symmetric-key algorithm and asymmetric-key encryption. Thus, the proposed schematic may function as a device that is able to simultaneously transmit and process optical logic signals in a highly encrypted way. Also, the transmission matrix (TM) of an MMF is wavelength-dependent because optical modes inside MMF are wavelength sensitive. Adopting the procedure similar to the parallel computation with polarization multiplexing, different logic functions can be designed into different wavelength channels; individual wavefront patterns can be superimposed and displayed on the corresponding subregions to achieve simultaneous multiple logic operations with wavelength multiplexing. Besides, the strategy proposed in this work can be extended to fiber bundles, as controlling the phase of light of each core in the proximal end of fiber bundles via SLM will similarly induce variations to the output light field from the fiber bundle [42].

To conclude, optical logic signal processing and transmission through a 15-m-long MMF based on TM-based wavefront shaping are proposed and demonstrated in this work. Light is first modulated by a DMD loaded with predetermined modulation patterns and then transmits through an MMF to output logic states. By choosing a common reference speckle field for light reflected from seven DMD subregions, the study has experimentally demonstrated the feasibility and robustness of basic logic functions, cascaded logic operations, as well as bitwise operations for multiple bits of operands. Finally, simultaneous reconstruction of multiple optical logic gates in a single logic gate with polarization multiplexing is also explored, which provides a solution to the improvement of the integration and security of the system. Whilst the performance, such as the speed of precalibration, can be further engineered in the next phase, the proposed method empowers reconfigurable optical logic processing capability via MMFs, which has not been demonstrated by any existing implementations in the field. The capability may also find or benefit wide applications in long-distance optical information transmission and encryption.