We present a spatially-diverse optical vector network analyzer which is capable of measuring the partial or complete mode transfer matrix of a system as a function of wavelength in an arbitrary mode basis using single or multiple sweeps.
© 2014 Optical Society of America
Measuring the mode transfer matrix of a multi-moded device or component as a function of wavelength completely describes its linear behavior and yields important information about the device-under-test such as bandwidth, insertion loss (IL) and mode dependent loss (MDL). An optical vector network analyzer (OVNA) based on swept wavelength interferometry (SWI)  can be used to measure the 2x2 Jones matrix of a single-moded device as a function of wavelength. Recently, this technique was extended to include spatial diversity (SDM-OVNA)  to measure the entire NxN transfer matrix of a multi-moded device in a single-sweep and has previously been demonstrated in transmission mode using fixed phase plates. Within the context of Mode Division Multiplexing (MDM) or fiber manufacturing, many different types of components often need to be characterized which can have quite different modal properties. For traditional single-moded SWI, regardless of the device-under-test, the basis is always the same; a Gaussian-like mode and two orthogonal polarizations. For multi-moded components there is more variation. For example, there are step-index, graded-index, microstructured , photonic bandgap  , and ring-core   fibers as well as photonic lanterns , just to name few. All of which may have a different number of total modes, as well as different modal profiles for the modes themselves. Building or modifying SDM-OVNAs with a fixed mode basis for each desired fiber or device type is unlikely to be practical and is certainly inconvenient.
In this paper we present an SDM-OVNA where the spatial-diversity is implemented using spatial light modulators (SLMs). This creates a system which is highly reconfigurable and is not bound to any particular mode basis or number of modes at the time of its construction. Rather, the mode basis is defined by what phase mask is programmed onto the surface of the SLMs. As the SLM can generate arbitrary spatial distributions of amplitude, phase and polarization, it can be used to emulate other components and devices which may not be readily available or where many design parameters wish to be tested in rapid succession. For example, the SLM can emulate butt-coupling of the device-under-test to a fiber with the desired properties. As another example, it can be used to emulate a spot-based launch system  of arbitrary geometry as will be discussed in this paper. The type of mode basis need not be the same at both ends of the device-under-test neither in terms of the spatial distribution of the modes themselves nor the total number of modes. If the entire transfer matrix is not of interest, the system can be programmed to measure only the relevant portion per wavelength sweep, such as the portion of the transfer matrix corresponding to coupling within a degenerate mode-group, as well as piecing multiple sweeps together to construct some or all of the transfer matrix.
2. Principle of operation
In Fig. 1(a) it can be seen that the SDM-OVNA consists of a single-mode swept wavelength interferometer attached to a pair of SLM based mode-multiplexers, where each SLM mode-multiplexer block of Fig. 1(a) is the system illustrated in Fig. 1(b)  . For clarity, the SLM of Fig. 1(b) has been illustrated as if it were transmissive. However in reality, the SLM is a reflective device (HoloEye PLUTO) and the two sets of polarization diversity optics are positioned on top of one another on the same side of the SLM. The versatility of the SLM allows the system to be reasonably compact as shown in Fig. 1(c). This is because fine alignment of the system is performed in software, eliminating the need for precision translation stages, but also because all non-polarizing beamsplitter and phase plate operations required of the mode multiplexer are implemented by the SLM . In a traditional phase plate based mode multiplexer , phase plates are used to generate the modes and non-polarizing beamsplitter are used to combine them all into a single beam. In this case, a single phase mask on the SLM performs the same function. This means there are less optical components overall and the same number of optical components regardless of the number of modes.
The single-moded SWI of Fig. 1(a) used in this case is a Luna Technologies OVA 5000, which performs wavelength sweeps from 1525 to 1566.7nm at a rate of approximately 0.4Hz, which includes the laser sweep time itself, but also all data processing and transfer to the control software. Polarization diversity is handled within the OVA 5000 as per traditional single-mode operation . Light from the laser is split into two arms; a signal and a reference. The signal is to be transmitted through the device-under-test before being interfered with the reference at the receiver. At the transmitter, the signal is further split into two paths; one path is aligned with the horizontally polarized port of a PBS and the other passes through a delay line, before being aligned with the vertically polarized port of a PBS. Both paths are recombined using the PBS and coupled into the device-under-test. This allows the system to differentiate between the responses of the different transmitted polarizations on the basis of their different delays. At the receiver, another PBS splits the signal exiting the device-under-test and interferes each received polarization with each polarization of the reference.
This principle can be extended to support multiple spatial modes by adding paths with unique delays for each of the desired transmitted and received spatial modes as per Fig. 1(a). Light is coupled into and out of the SLM mode multiplexer and demultiplexer using single-mode fibers with different delays which allows the system to temporally filter out the respective elements of the mode transfer matrix [1,2]. Each input/output single-mode fiber can simultaneously be assigned an arbitrary mode defined by the phase mask programmed onto the surface of the input/output SLMs. The principle is identical to that discussed in , expect in this case the bulk-optic beamsplitters and phase masks have been replaced with multiplexed phase masks programmed onto the SLMs . An example impulse response is shown in Fig. 1(d) for a single polarization, where the delays have been color-coded to correspond with the relevant delay lines of Fig. 1(a). In this case, the large delays correspond to the transmitted mode (TX1 to TX3), i.e. a particular column of the mode transfer matrix, and the smaller delays are used to differentiate between the different receive modes (RX1 to RX3) indicated by their spatial profiles in Fig. 1(d), which represent the elements along a particular row of the transfer matrix. Another convenience of an SLM based SDM-OVNA is that it can easily add and remove transmit and receive modes in software. This allows the system to turn on and off each of the transmit/receive modes one at a time to identify which peaks/delays of Fig. 1(d) correspond with which transfer matrix element.
As mentioned above, in contrast to a traditional phase plate based mode multiplexer , where each mode is implemented using individual phase plates and beamsplitters, this SLM based approach implements all these optical components in a single static phase mask . Figure 2(a) is an example of a phase mask that has been designed to generate the entire orbital angular momentum (OAM) basis of a three spatial mode fiber (LP0,1, OAM + and OAM-) to emulate butt-coupling to a piece of step-index fiber. The phase masks have been calculated using techniques from computational holography  to generate the most efficient mode possible under the constraint of 99% beam quality in the a 50μm radius circle around the fiber core. The left and right sides of the phase mask correspond to the horizontal and vertical polarizations respectively as per Fig. 1(b). The phase patterns for the horizontal and vertical polarizations are slightly different to compensate for the slight differences between the optical paths of the two polarizations in terms of tilt and to a lesser extent aberrations, which the SLM compensates for . Figure 2(b) is the corresponding intensity simulated in the Fourier plane (plane of the multimode core) when the mask is illuminated by a Gaussian beam. When the angle of incidence of the illuminating beam is correct, corresponding to one of the input/output SMFs of Fig. 1(b), one of the three modes shown will align with the core of the fiber-under-test and the desired mode will be excited. The two other modes that do not align with the core are not coupled and incur loss, as per a beamsplitter. When illuminated by three independent beams at the correct angles, each will excite a separate mode of the fiber. Figure 2(c) and 2(d) show another example of a different basis used to emulate an offset spot coupler . For any basis it is also possible to adjust the mode field diameter (MFD), relative amplitude, relative phase and spatial position of the beams. Hence when a new device-under-test is attached to the system, the phase masks can self-align the positions of the beams to minimize the MDL. The system can also attempt to find the appropriate MFD for the basis. An example of this is shown in Fig. 2(e) where the appropriate MFD for a three spatial mode photonic lantern is being found by sweeping the scaling of the mode basis whilst keeping the power of the beams incident on the core of the device-under-test constant and measuring the coupling efficiency. The device-under-test in this case was a photonic lantern  with an NA of 0.06 and a core diameter of 30μm. Approximating the lantern as a step-index fiber theoretically yields an MFD of 27.8μm which is consistent with the measured values of Fig. 2(e) and hence this was selected as the appropriate MFD for the photonic lantern which is to be characterized in the following section.
Given enough input/output single-mode fibers with the appropriate delay lines attached, the SLM can also add or remove modes from the measurement basis. This allows the same system to be used regardless of the number of modes the device-under-test supports but also allows modes that are not of interest for a particular measurement to be ignored. As can be seen in Fig. 1(a), the SDM-OVNA  has splitters and combiners that feed into the swept wavelength interferometer, these introduce a loss which scales with the number of splitter/combiner ports which is equal to the number of spatial modes. In addition to this, the SLM multiplexers are effectively performing as an array of non-polarizing beamsplitters and phase plates and as such, also incur a splitting loss which scales with the number of spatial modes and hence the dynamic range of the system decreases as the number of spatial modes increase. The spectral resolution of the SWI also decreases with the number of simultaneously measured modes as a given window of time captured by the analogue-to-digital converters is divided up amongst more modes. Hence the performance of the system is best when fewer modes are measured in a single-sweep. For many purposes, particularly where the mode transfer matrix is known to be sparse a priori, little information is lost by not measuring the entire matrix . For example, in a length of fiber supporting many modes it could be expected that the coupling between the highest and lowest order modes would be small and have negligible impact on the measured parameters of interest and hence not worth the dynamic range and spectral resolution penalty required to measure both modes simultaneously.
It is also possible to exploit the fact that an absolute phase offset between the columns or rows of the mode transfer matrix has no effect on the measured IL or MDL, by launching a single spatial mode in both polarizations at the transmitter and measuring all spatial modes in both polarizations at the receiver. That is, the SLM is configured to select just a single input delay line and its corresponding mode, and a sweep of the SWI is performed. Thus, piecing together the entire matrix in N wavelength sweeps rather than a single-sweep. Where N is the number of spatial modes. Measuring the matrix in this fashion takes longer, but no beamsplitter splitting losses associated with multiplexing multiple modes are incurred at the transmitter SLM as only a single spatial mode is transmitted at a time. As the system can arbitrarily define which modes need to be measured for a given sweep, it is possible to reprogram the measurement apparatus in various ways. Either to maximize measurement speed using a single-sweep configuration as per traditional SWI, at the expense of dynamic range and spectral resolution, through various possible multiple sweep configurations which could consist of as many as N2 sweeps where dynamic range and spectral resolution is maximized at the expense of measurement time. However piecing together multiple sweeps requires the mode transfer matrix of the device-under-test, as well as the SDM-OVNA itself to be stable on the time scale of the total measurement. Hence the multiple sweep approach is more appropriate for devices of short length and many modes, where the splitting losses incurred by attempting to measure the entire matrix in a single sweep could be prohibitive. A single-sweep would likely be required for the measurement of rapidly evolving components such as kilometres of fiber.
A further extension of the system is a hybrid of that discussed in reference  and , where the phase relationship between the modes is measured at a reference wavelength by interference of the phase masks for the different modes on the SLM using a CW source and a power meter , with the SWI being used to measure the wavelength dependence for a given mode . As this uses only a single input/output fiber at a time it is immune from phase drift between the input single-mode fiber arms of Fig. 1(a) and 1(b), but also increases the time taken to perform the measurement.
3. Measurement of a three spatial mode photonic lantern
As an example, the system is used to characterize a photonic lantern  supporting three spatial modes. A photonic lantern is an adiabatic transition between N single-mode fibers and an N spatial mode multimode core. Theoretically it is a lossless device with no MDL.
When the photonic lantern is first attached to the SLM system, before the transfer matrix is measured using the SWI, a single spot is scanned across the multi-moded end using the SLM to probe the structure and geometry of the device-under-test, as well as verify the alignment of the lantern and the optics. Figure 3 is a measurement taken using the system of Fig. 1(c) configured to scan a horizontally polarized 1545.54nm 7.75μm MFD Gaussian spot across the multi-moded core of the lantern and the power coupled out of each single-mode port have been measured on separate power meters. Figure 3(a) is the sum of the power out of all the single-mode ports as a function of position of the spot scanned across the multi-mode core at the other end of the lantern. Figure 3(b)-3(d) are the powers coupled out each of the single-mode fiber ports which when summed become Fig. 3(a). Although in practice the test is easiest to perform by coupling light into the multi-mode port and measuring the power out the single-mode ports, the device is reciprocal and it is perhaps more intuitive to interpret the meaning of such a test as if the light were propagating in the opposite direction. Light coupled into each of the single-mode ports and observed at the multi-mode end will have the intensity patterns shown in Fig. 3(b-d). This corresponds to what is observed on a camera at the multi-moded end convolved with the 7.75μm MFD Gaussian sampling spot, when coupling power into each of the three respective single-mode ports. The scan reveals that the three single-mode cores have not completely collapsed into a homogeneous multi-mode core, with the three-core structure still identifiable in Fig. 3(a). Although theoretically, a perfect photonic lantern would have no MDL, from inspection of the image of Fig. 3(a) at least some MDL for the three spot launch can be expected, due to the different coupling efficiencies for each of the three residual core structures, which have not fully merged and are not equal.
With the SWI attached and the system configured as per Fig. 1, the photonic lantern is characterized first in the OAM basis using a single-sweep configuration where all elements of the mode transfer matrix U are measured in a single wavelength sweep. The OAM basis consists of LP0,1 with zero topological charge and OAM modes of topological charge ± 1 in both polarizations. Figure 4 illustrates the measured squared magnitude of U as a function of wavelength. The input and output polarization axes in which the SWI defines its measurement and the polarization axis of the SLM system of Fig. 1(a) have been approximately aligned with one another such that the two polarizations are nearly independent. It can be seen that power launched into any particular SMF port produces a mixture of modes at the multi-mode end with proportions that are fairly consistent across the measured wavelength range.
An example of the measured mode transfer matrix U at a center wavelength (1545.54nm) is shown in Fig. 5(a) and 5(b), where the x-axis represents power transmitted through a particular single-mode port in a particular polarization and the y-axis represents the corresponding value for each spatial mode in each polarization at the multi-mode core end. Figure 5(a) is the amplitude of the transfer matrix and the chequerboard-like appearance of the matrix amplitude is a consequence of the polarization axes alignment as mentioned above. When the transfer matrix U is multiplied by its complex conjugate transpose the matrix of Fig. 5(c) is produced which is approximately the identity, confirming that the matrix U is close to unitary and hence has low MDL. Each of the 6 diagonal elements of Fig. 5(c) represent each of the 6 orthogonal spatial/polarization channels the photonic lantern supports. Performing the singular-value decomposition (SVD) of the matrix U yields the singular values of Fig. 5(d) which represent the relative loss of each of the channels. Each channel in this case is an eigenvector of the transfer matrix rather than the eigenmodes of the fiber. These singular values can be seen as a function of wavelength in Fig. 6(a). The ratio between the maximum and minimum value is the MDL and the average value is the IL.
In addition to characterizing the lantern in the OAM basis in a single-sweep as above, it was also characterized using the three-spot offset launch basis  as well as in the OAM basis using multiple wavelength sweeps (one transmitted spatial mode at a time), to construct the matrix two columns at a time. For the offset spot basis, single-sweeps were performed using three different MFD values; 10.4, 13.9 and 26.0μm. Given the 4:3 focal length ratio of the SLM demultiplexer system the 13.9μm spot size corresponds to imaging the input single-mode fibers directly onto the core of the lantern. That is, the SLM performs no resizing and the pattern programmed onto its surface is simply that of superimposed blazed gratings as per Fig. 2(b). For the 10.4 and 26.0μm cases the principle is the same, however the phase patterns have been designed to decrease/increase the spot size at the expense of overall efficiency. For each different MFD, the system automatically adjusts the spot positions to minimize the MDL. The measured MDL values as a function of wavelength for each basis and method are plotted in Fig. 6(b). The N sweep results shown was the best of several trials, as this technique is sensitive to phase drift between the single-mode fiber ports of the photonic lantern, as well as the couplers and delay lines during the course of the N scans. For a more reliable N sweep approach it would be necessary to isolate the different optical paths from their environment to stabilize their relative phases. It can be seen that all basis choices perform similarly, but not identically. Theoretically, for a perfect photonic lantern perfectly measured, all bases shown would have zero MDL. In practice, the measured MDL has contributions from both the photonic lantern itself, but also the SLM system being used to measure it, which can only be partially calibrated out. For example, the different losses of the different optical paths corresponding with the different modes (coupler port, delay line, SLM path) can be easily measured and offset as a function of wavelength. However what is not calibrated out is the free-space alignment of the photonic lantern in the SLM system itself, which cannot be distinguished from the performance of the photonic lantern in isolation. For example, coupling light into the multi-mode core of the lantern at a slight angle will have negligible impact on the coupling efficiency but can have a more noticeable effect on the MDL. That is, even though the total power coupled into the multi-mode core might be very similar for all modes using the SLM, the orthogonality between the different modal excitations will be degraded by the tilted wavefront, increasing MDL. It is also easier to align the OAM basis as there is effectively only two variables x-position and y-position, whereas the offset spot basis has independent positions for each spot. The approximately 1.4nm ripples across the responses of Fig. 4 and Fig. 6 correspond with the reflection off the cover-glass of the SLM (approximately 1.5mm path-length in glass). The same photonic lantern was also measured in the OAM basis at 1545.54nm using a separate SLM system of the same design as Fig. 1(c) using the technique outlined in . This alternate approach does not use an SWI and measures all input and output modes sequentially one at a time, at a single wavelength. It is simpler as there is only a single optical path through the system, however the measurement is much slower and has no spectral information. This alternate approach measured an MDL of 1.45dB, which is consistent with the value of Fig. 6(b).
A spatial light modulator based spatially-diverse swept wavelength interferometer has been presented which allows various multi-moded devices or components to be measured using the same system in a reconfigurable choice of mode basis. This reconfigurable basis can be selected to emulate different fibers or couplers to which the device-under-test could be attached, such as a particular type of fiber, or an offset spot coupler. This system allows the number or type of modes to be selected for each sweep to construct some or all of the mode transfer matrix using single or multiple sweeps.
The authors would like to thank Sergio Leon-Saval and Joel R. Salazar-Gil for providing the first generation experimental lanterns used as a study example for this work. We also acknowledge the Linkage (LP120100661), Laureate Fellowship (FL120100029), Centre of Excellence (CUDOS, CE110001018), and DECRA (DE120101329) programs of the Australian Research Council.
References and links
1. G. D. VanWiggeren, A. R. Motamedi, and D. M. Barley, “Single-scan interferometric component analyzer,” IEEE Photon. Technol. Lett. 15(2), 263–265 (2003). [CrossRef]
2. N. K. Fontaine, R. Ryf, M. A. Mestre, B. Guan, X. Palou, S. Randel, Y. Sun, L. Gruner-Nielsen, R. V. Jensen, and R. Lingle, “Characterization of space-division multiplexing systems using a swept-wavelength interferometer,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW1K.2. [CrossRef]
3. R. Ryf, R. Essiambre, A. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “Space-division multiplexed transmission over 4200 km 3-core microstructured fiber,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper PDP5C.2. [CrossRef]
4. T. G. Euser, G. Whyte, M. Scharrer, J. S. Y. Chen, A. Abdolvand, J. Nold, C. F. Kaminski, and P. St. J. Russell, “Dynamic control of higher-order modes in hollow-core photonic crystal fibers,” Opt. Express 16(22), 17972–17981 (2008). [CrossRef] [PubMed]
5. Y. Jung, V. Sleiffer, N. Baddela, M. Petrovich, J. R. Hayes, N. Wheeler, D. Gray, E. R. Numkam Fokoua, J. Wooler, N. Wong, F. Parmigiani, S. Alam, J. Surof, M. Kuschnerov, V. Veljanovski, and H. Waardt, de, F. Poletti, and D. J. Richardson, “First demonstration of a broadband 37-cell hollow core photonic bandgap fiber and its application to high capacity mode division multiplexing,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper PDP5A.3.
6. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013). [CrossRef] [PubMed]
7. C. R. Doerr, N. Fontaine, M. Hirano, T. Sasaki, L. Buhl, and P. Winzer, “Silicon photonic integrated circuit for coupling to a ring-core multimode fiber for space-division multiplexing,” in 37th European Conference and Exposition on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Th.13.A.3. [CrossRef]
8. S. G. Leon-Saval, A. Argyros, and J. Bland-Hawthorn, “Photonic lanterns: a study of light propagation in multimode to single-mode converters,” Opt. Express 18(8), 8430–8439 (2010). [CrossRef] [PubMed]
9. R. Ryf, N. K. Fontaine, and R.-J. Essiambre, “Spot-based mode coupler for mode-multiplexed transmission in few-mode fiber,” IEEE Photon. Technol. Lett. 24(21), 1973–1976 (2012). [CrossRef]
10. J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Degenerate Mode-Group Division Multiplexing,” J. Lightwave Technol. 30(24), 3946–3952 (2012). [CrossRef]
11. J. Carpenter, B. J. Eggleton, and J. Schröder, “110x110 optical mode transfer matrix inversion,” Opt. Express, Opt. Express 22(1), 96–101 (2014). [CrossRef]
12. J. Carpenter and T. D. Wilkinson, “Holographic mode generation for mode division multiplexing,” in National Fiber Optic Engineers Conference, OSA Technical Digest (Optical Society of America, 2012), paper JW2A.42. [CrossRef]
13. J. Carpenter and T. D. Wilkinson, “Aberration correction in Spatial Light Modulator based mode multiplexers,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper JW2A.27. [CrossRef]