Femtometer-resolved simultaneous measurement of multiple laser wavelengths in a speckle wavemeter
Graham D. Bruce, Laura O'Donnell, Mingzhou Chen, Morgan Facchin, Kishan Dholakia
FFemtometer-resolved simultaneous measurement of multiple laser wavelengths in aspeckle wavemeter
Graham D. Bruce, ∗ Laura O’Donnell, Mingzhou Chen, Morgan Facchin, and Kishan Dholakia
1, 2 SUPA School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK Department of Physics, College of Science, Yonsei University, Seoul 03722, South Korea
Many areas of optical science require an accurate measurement of optical spectra. Devices basedon laser speckle promise compact wavelength measurement, with attometer-level sensitivity demon-strated for single wavelength laser fields. The measurement of multimode spectra using this ap-proach would be attractive, yet this is currently limited to picometer resolution. Here, we presenta method to improve the resolution and precision of speckle-based multi-wavelength measurements.We measure multiple wavelengths simultaneously, in a device comprising a single 1 m-long step-index multimode fiber and a fast camera. Independent wavelengths separated by as little as 1 fmare retrieved with 0.2 fm precision using Principal Component Analysis. The method offers a viableway to measure sparse spectra containing multiple individual lines and is likely to find applica-tion in the tracking of multiple lasers in fields such as portable quantum technologies and opticaltelecommunications.
The speckle produced when coherent light is scatteredby a rough surface can provide a surprising method withwhich one can track the properties of the incoming light.The precise speckle pattern produced by this multiple-interference is uniquely determined by the beam param-eters, and can therefore be used as a fingerprint forlinewidth [1], polarization [2], beam position [3] or trans-verse mode characteristics [4]. Broadband spectrome-ters have been constructed which extract the spectrumof light from the speckle, by using either the transmissionmatrix method ([5–8]) or deep learning [9], achieving aspectral resolution limited by speckle correlation. Typi-cally, this speckle correlation limit is on the picometer-scale. For monochromatic light, speckle wavemetersutilizing Principal Component Analysis (PCA) [10–12],Poincar´e descriptors [13] and convolutional neural net-works [14] have greatly surpassed this limit, measuringan isolated wavelength with a resolution down to theattometer-scale. It remains an open challenge to simul-taneously measure multiple wavelengths or spectra atsuch high resolution using speckle. A successful methodpromises applicability in laser stabilization for portablecold atoms experiments, wavelength-division multiplexedtelecommunications and chemical sensing.In this letter, we demonstrate that the high resolu-tion achieved by using PCA to analyze speckle can beextended beyond a single laser-line, to measure sparsespectra composed of multiple laser wavelengths. We es-tablish that wavelength measurements of lasers separatedby 1 fm, five orders of magnitude less than the specklecorrelation limit, can be performed simultaneously andwith an accuracy of 0.2 fm. Simultaneous measurementof up to ten laser lines is demonstrated.The principle of measurement is outlined in Fig. 1.A single scattering element is illuminated by a beamcomposed of multiple wavelengths; each wavelength is ∗ [email protected] l + l FIG. 1. Principle of multi-wavelength measurement in aspeckle wavemeter. Laser beams are overlapped and illumi-nate a single scattering medium, generating a speckle pattern.The speckle pattern is uniquely determined by the precise val-ues of each wavelength, so can be used as a marker to recoverthe wavelengths. scattered to produce a unique speckle pattern. Providedthe wavelengths of the Components are sufficiently sep-arated, the resultant speckle patterns are a simple inten-sity sum of the speckles produced by each wavelength inisolation. A calibration dataset is acquired to train PCAto recognize how the speckle changes with wavelength.To demonstrate the training method, we simulate (us-ing paraxial wave theory, see [11] for details) the propaga-tion of two co-polarized, co-incident and co-propagatingGaussian laser beams of equal power and identical spatialdistribution. The light propagates through five equally-spaced planes (separated by one Rayleigh length) atwhich the phase is randomized. The refractive indexdifference to air is small (∆ n = 0 . ×
256 pixel grid with a bit-depth of 8,to approximate the acquisition by a camera. A seriesof 1200 speckle patterns are accumulated, where wave-lengths λ and λ of the two lasers are centered around780.000 nm and 780.014 nm. They are both sinusoidallymodulated with a 1 pm-amplitude but with different pe-riods of oscillation (such that they undergo three and tenoscillations in the measurement period, respectively), as a r X i v : . [ phy s i c s . op ti c s ] O c t P C -101 P C -5 frame number -202 P C -5 ( n m ) frame number ( n m ) (nm) ( n m ) -1 0 1 PC -5 -101 P C -5 (a) (b)(c) (d) FIG. 2. Principal Component Analysis of simulated specklepatterns produced by wavelength variations of two overlappedlasers. (a) Sinusoidal modulations applied to the wavelengthsof two lasers λ and λ . (b) Principal Components 1 - 3 ofthe image set. The first Principal Component (PC ) cap-tures a mixed signal of both wavelength modulations, whilePC and PC show responses dominated by λ and λ respec-tively. Parametric plots of (c) λ vs λ and (d) PC vs PC show that the combined modulations are faithfully recordedin PC-space. A small rotation angle between (c) and (d) high-lights mixing of the two wavelength Components across thetwo PCs. shown in Fig. 2(a). At each time interval, the multi-wavelength speckle pattern is obtained by summing theintensities of the speckle distribution of each wavelengthin isolation, i.e. neglecting interference between the twobeams. PCA is then performed on the time-series ofmulti-wavelength speckle patterns. The Principal Com-ponents (PCs) are the projections of the data onto theeigenbasis of the covariance matrix of the training set,i.e. by design they measure the maximal variations inthe dataset. The largest three PCs (PC , PC and PC ,shown in Fig. 2(b)), contain 96% of the variations in thedata. The non-commensurate modulation rates for thetwo beams ease the identification of the contribution fromeach wavelength. The first Principal Component, PC ,shows modulation of the speckle pattern at both of theapplied modulation rates. This is associated with inten-sity fluctuations due to speckles moving in and out of thefield of view of the camera. However, the separate modu-lations are dispersed across the next two Principal Com-ponents (PC and PC in Fig. 3(b)), in analogy with the Laser
Camera
Speckle wavemeter
MMF
SMF A O M A O M l /2 FIG. 3. Experimental setup. The output of a stabilized diodelaser is split into two beams, each of which undergoes sep-arate wavelength modulation using an acousto-optic modu-lator (AOM). The beams are recombined, co-polarized anddelivered to the speckle wavemeter via single-mode opticalfiber (SMF). The speckle wavemeter is a 1 m-long multi-modefiber (MMF) and a CMOS camera. (Inset) A typical multi-wavelength speckle pattern recorded in the speckle waveme-ter. wavelength-dependent dispersion produced in a grating-based spectrometer. Retrieval of these two PCs in iso-lation is sufficient to characterize the independent wave-lengths: the parametric relationship between λ and λ isillustrated in Fig. 2(c), and the same parametric relation-ship is shown to exist between PC and PC in Fig. 2(d).A small rotation angle between the two parametric plotssignifies cross-talk between the two measurement chan-nels, i.e. PC is strongly dependent on λ and weaklydependent on λ while PC is strongly dependent on λ and weakly dependent on λ . We find that this cross-talk can be minimized by using wavelength modulationsof equal amplitude, but regardless it does not effect theaccuracy of the PCA, as the two wavelengths are alwaysuniquely identified by the measurement of these two PCs.The link between PCs and wavelength is established by alinear fitting of this training set. A speckle pattern pro-duced by an unknown combination of wavelengths withinthe training range can subsequently be projected into thisPC-space to retrieve the wavelengths.We experimentally verify the method using the appara-tus shown in Fig. 3 to generate tunable, multi-wavelengthspectra. Light from an external cavity diode laser (Top-tica DL-100, LD-0785-P220), stabilized to the Rb D line ( F = 2 → F = 2 × time (s) (f m ) FIG. 4. Simultaneous measurement of two wavelengths,shown relative to λ = 780 . the same spatial profile, and delivered to a multi-modefiber (MMF) speckle wavemeter. Laser speckle is gener-ated by multiple scattering and modal interference in the1 m-long step-index MMF, which has 105 µ m core diame-ter and NA = 0.22 (ThorLabs FG105LCA). After exitingthe MMF, the light propagates for 5 cm and is capturedby a fast CMOS camera (Mikrotron EoSens 4CXP). Im-ages of 240 ×
240 pixels were recorded at 2,000 fps with anexposure time of 10 µ s and a power of 150 µ W per beam.The multi-wavelength speckle image at each time inter-val is independently normalized by the total intensity.The speckle correlation limit of this system is ∼
320 pm,which is determined as the HWHM of the Pearson corre-lation coefficient of the speckle patterns at different wave-lengths.Fig. 4 shows the measurement of two wavelengths withan average separation of 22 fm, which is four orders ofmagnitude below the speckle correlation limit and forwhich the speckle patterns acquired at each wavelengthhave a structural similarity index > .
97. Training wasperformed by acquiring the speckle patterns over a 1 s in-terval for a 8.5 fm-amplitude sinusoidal wavelength mod-ulation to each beam, with incommensurate periods of125 ms and 37.5 ms. After the training phase, an expo-nential decay of the amplitude of the wavelength mod-ulation is introduced. The standard deviation betweenthe set wavelength and that measured by the specklewavemeter is 0.37 fm for the slowly modulated beamand 0.29 fm for the fast modulated beam. The accu-racy of the measurement of each wavelength is limitedby high-frequency modulations of the laser wavelengthintroduced by the lock-in electronics for wavelength sta-bilization, and are in agreement with those reported in[12] for measurements of a single wavelength.When the wavelength separation is large, PCA accu-rately recovers the wavelength. However, if the wave-lengths converge, PCA is incapable of correctly analyz- ing the speckle pattern, giving large values of the PCswhich do not correspond to the expected wavelengths.The erratic values for close approach are due to interfer-ence between the beams causing the speckle pattern onthe camera to flicker. When the beat-note frequency ofthis interference-induced flicker is fast compared to theexposure time of the camera, PCA gives reliable wave-length estimation. Using the camera settings above, wemeasured the wavelengths of two beams to an accuracy(standard deviation of measured and set wavelength over1 s) of 0.21 fm and 0.19 fm when the wavelength separa-tion was 1.0 fm. In principle, decreasing the measurementrate and using longer exposure times should allow for animprovement in spectral resolution, while the issue maybe avoided in the measurement of separate laser sources.In addition to wavelength separation, we also inves-tigated the role of other potential issues with our ap-proach. For modest power ratio between the beams, thetwo wavelengths are always uniquely determined by PC and PC . However, when this power ratio is large, e.g.500 µ W and 50 µ W, the less intense beam is instead dis-persed into PC , therefore further PCs must be consid-ered to track multiple wavelengths in this regime. Whenthe lasers have different linewidth, the differing Rayleighdistributions of the resultant speckle patterns will aidthe discrimination of the contributions of each laser tothe speckle.Simultaneous measurements of more than two wave-lengths are also possible. Fig. 5(a) shows simulated wave-length modulation of three separate beams, with meanseparations of 14 pm, which are combined as before. Asin the two-wavelength case, the wavelength modulationsare dispersed across PC-space (97% of the variation isdescribed by the first four PCs). PC shows a mixtureof all three modulations, while PC to PC are respec-tively dominated by λ to λ (Fig. 5(b)). Mixing of thespectral channels is again observed: the 3-dimensionalparametric plot of wavelength (Fig. 5(c)) is related tothe parametric plot of PC , PC and PC (Fig. 5(d)) bya 3-dimensional rotation. The mixing of spectral chan-nels can also be seen in the transformation matrix T λ, PC which defines the linear transformation between PCs andwavelength, i.e. PC PC PC = T λ , PC T λ , PC T λ , PC T λ , PC T λ , PC T λ , PC T λ , PC T λ , PC T λ , PC λ λ λ . (1) T λ, PC is established in the training phase by multiplica-tion of the matrix containing the time series of the PCsand the inverse of the matrix containing the time seriesof the corresponding training wavelengths, and is plot-ted in Fig. 5(e). It shows that the mean dependence ofPC i +1 on λ i is 83.1%, with an average contribution of12.1% from the nearest neighboring wavelength(s). Thewavelengths present in any individual unknown specklepattern can be measured by multiplying the matrix in-verse of T λ, PC with the PCs extracted for that image.The transformation matrix representation is necessary -101 ( p m ) -101 ( p m ) frame number -101 ( p m ) P C - -4 -101 P C * -101 P C * frame number -101 P C * -0.500.51-1 P C * PC * -10 PC * ( p m ) (pm) (pm) PC | T , P C | PC FIG. 5. Tracking three wavelengths simultaneously via Prin-cipal Component Analysis. (a) Control wavelength modu-lations to the three lasers. δλ − are measured relative to780 nm, 780.986 nm and 779.014 nm respectively. (b) Princi-pal Components 1 - 4 of the resultant speckle patterns. PC i *denotes PC i × . Parametric plots (c) of wavelengths and(d) of PCs, showing that the PC-space representation is re-lated to the wavelength-space by a three-dimensional rotation.(e) The transformation matrix T λ, PC gives the relationshipbetween each wavelength and each PC. to examine the correlations between higher numbers ofbeams. As shown in Fig. 6, a similar transformationmatrix can be established for a system of 10 distinctlasers, where the ten wavelengths are dispersed acrossPC to PC . In this simulation, the ten wavelengthswere evenly separated by 1 pm, and undergo incommen-surate sinusoidal modulations of 200 fm amplitude over400 frames. The period of oscillation of λ i was set so thatit undergoes 2 p i oscillations in the training phase, where p i is the i th prime integer. The PCA finds a basis inwhich 74% of the variance is contained in the first elevenPCs. We note that the variance captured in higher PCsin this case follows a step-like trend in groups of ten PCs:continuously falling by 50% within the group but discon-tinuously dropping by 50% between the last PC of one PC PC PC PC | T , P C | PC PC PC PC PC FIG. 6. Transformation matrix T λ, PC showing the relation-ship between each wavelength and each PC in the specklepatterns produced by ten overlapped wavelengths. group and the first PC of the next. Ignoring these higherterms and projecting test data into the 10-dimensionalPC space comprising PC to PC recovers the wave-length to within 20 fm. As can be seen in Fig. 6, there isgreater mixing between the spectral channels in this ten-wavelength measurement, with the diagonal elements of T λ, PC having a mean value of 31.3% and a standard de-viation of 9.1%.In this letter, we have demonstrated that the wave-lengths of multiple lasers can be measured simultane-ously using a speckle wavemeter with Principal Compo-nent Analysis. The procedure projects a speckle patterngenerated by n wavelengths into an n -dimensional Prin-cipal Component space. In the experiment, we demon-strated simultaneous recovery of the wavelengths of twolasers, separated by as little as 1 fm with an accuracyof 0.2 fm, limited by the stabilization electronics of thelaser. The approach is limited in spectral range, re-quiring that the Principal Components vary monotoni-cally with wavelength. However, for single wavelengthmeasurements, PCA has been shown to be complimen-tary to the transmission matrix method, which operatesover a much larger range but with lower resolution [11].We suggest such a tandem approach will also be possi-ble for the measurement of multiple wavelengths. Themethod is likely to find application in the developmentof portable quantum technologies, where robust methodsto lock multiple lasers for atom cooling are sought. Cou-turier, et al, have shown that such stabilization can beachieved using a commercial (Fizeau) wavemeter and amulti-mode fiber switch, but report fluctuations of theatomic fluorescence due to the switching [15]. Stabiliza-tion of a single laser using speckle was demonstrated in[11], and we suggest that the simultaneity of measure-ments of multiple wavelengths with speckle may obviatethe switching limitation. In future work, the trainingphase could be extended to include variable powers ofthe beams, which would allow for the recovery of sparse spectra with variable mode intensities, which may be ap-plicable to areas such as chemical analysis.We acknowledge funding from Leverhulme Trust(RPG-2017-197)and EPSRC (EP/R004854/1). andthank D Cassettari and P Rodr´ıguez-Sevilla for technicalassistance and discussions. [1] W. Freude, C. Fritzsche, G. Grau, and L. Shan-da, Jour-nal of Lightwave Technology , 64 (1986).[2] T. W. Kohlgraf-Owens and A. Dogariu, Opt. Lett. ,2236 (2010).[3] I. Alexeev, J. Wu, M. Karg, Z. Zalevsky, andM. Schmidt, Appl. Opt. , 7413 (2017).[4] M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright,and K. Dholakia, Appl. Phys. Lett. , 231115 (2012).[5] B. Redding, S. F. Liew, R. Sarma, and H. Cao, NaturePhotonics , 746 (2013).[6] B. Redding, M. Alam, M. Seifert, and H. Cao, Optica , 175 (2014).[7] M. Chakrabarti, M. L. Jakobsen, and S. G. Hanson, Opt.Lett. , 3264 (2015).[8] H. Cao, J. Opt. , 060402 (2017).[9] U. K¨ur¨um, P. R. Wiecha, R. French, and O. L. Muskens, Opt. Express , 20965 (2019).[10] M. Mazilu, T. Vettenburg, A. Di Falco, and K. Dholakia,Opt. Lett. , 96 (2014).[11] N. K. Metzger, R. Spesyvtsev, G. D. Bruce, B. Miller,G. T. Maker, G. Malcolm, M. Mazilu, and K. Dholakia,Nature Communications , 15610 (2017).[12] G. D. Bruce, L. O’Donnell, M. Chen, and K. Dholakia,Opt. Lett. , 1367 (2019).[13] L. O’Donnell, K. Dholakia, and G. D. Bruce, arXiv:1909.00665 (2019).[14] R. K. Gupta, G. D. Bruce, S. J. Powis, and K. Dholakia,arXiv: 1910.10702 (2019).[15] L. Couturier, I. Nosske, F. Hu, C. Tan, C. Qiao, Y. Jiang,P. Chen, and M. Weidem¨uller, Rev. Sci. Instrum.89