Harnessing speckle for a sub-femtometre resolved broadband wavemeter and laser stabilization
The accurate determination and control of the wavelength of light is fundamental to many fields of science. Speckle patterns resulting from the interference of multiple reflections in disordered media are well-known to scramble the information content of light by complex but linear processes. However, these patterns are, in fact, exceptionally rich in information about the illuminating source. We use a fibre-coupled integrating sphere to generate wavelength-dependent speckle patterns, in combination with algorithms based on the transmission matrix method and principal component analysis, to realize a broadband and sensitive wavemeter. We demonstrate sub-femtometre wavelength resolution at a centre wavelength of \SI780\nano\meter and a broad calibrated measurement range from 488 to \SI1064\nano\meter. This is comparable with or exceeding the performance of conventional wavemeters. Using this speckle wavemeter as part of a feedback loop, we stabilize a \SI780\nano\meter diode laser to achieve a linewidth better than \SI1\mega\hertz.
Light propagation in disordered media is often regarded as a randomization process by which the information contained within an optical field is destroyed, and the speckle pattern that results from repeated scattering and interference is commonly deemed detrimental to an optical system. However, although highly complex, multiple scattering is nonetheless a linear, deterministic and reversible process . A coherent beam propagating in a disordered medium yields a unique speckle pattern that depends on the spatial and temporal characteristics of the incident light field. In the spatial domain [2, 3], light scattering in complex (disordered) media can be harnessed for imaging at depth , novel endoscopy  and even sub-wavelength focusing . In a spatially-static system, the temporal variations of the speckle pattern can be used to identify characteristics of the incident light, in particular as a spectrometer or wavemeter [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], with state-of-the-art demonstrations of wavelength resolution at the picometre level and operating ranges of over \SI1\micro\meter covering the visible and near-infrared spectrum.
In this letter, we use an integrating sphere as the random medium for wavelength detection. This acts like a highly sensitive and complex interferometer due to internal path-length differences on the metre scale, delivering a different speckle pattern for each wavelength in a system with a small footprint. The resultant wavemeter is demonstrated over a wide spectral operating range of approximately \SI600\nano\meter in the visible and near-infrared spectrum, with wavelength resolution at the sub-femtometre level (\SI0.3\femto\meter measured at a centre wavelength of \SI780\nano\meter). This resolution is up to two orders of magnitude better than all previous spectrometers using complex media [16, 12, 23]. Through the subsequent use of appropriate electronic feedback, we realize an alternative approach for laser frequency stabilization. We are able to stabilize an external cavity diode laser (ECDL) over \SI10\second, (a timescale suitable for, e.g., laser cooling of Rb atoms) and observe linewidth narrowing of the ECDL by a factor of four. As this approach of using a speckle-based stabilization scheme is not based on atomic absorption, it has the advantage of being able to lock at an arbitrary wavelength, as with Fabry-Perot-based stabilization schemes .
All data in this letter were acquired with the setup shown in Fig. 1. The laser beam is split into two paths. The first path is used for reference measurements of the laser wavelength with a commercial wavemeter or a Fabry-Perot interferometer. The second path is coupled into a spectralon-coated integrating sphere through an angle-cleaved single-mode fibre to ensure a spatially-static input beam and eradicate back reflections which could otherwise hinder stable laser operation. The speckle pattern produced inside the sphere is measured by a camera placed at a distance from the exit port of the integrating sphere. In contrast to interferometer-based wavemeters, no accurate alignment of the individual components is necessary. For example, the camera and fibre can be offset by several degrees without perturbing the measurement results as long as stability is maintained after the calibration step.
The mean free path in the sphere plays a prominent role in increasing the resolution of the system, analogous to the spectral to spatial mapping principle of a spectrometer. Thus, both the sphere diameter and propagation distance from sphere to camera dictate the resolution and range of the wavemeter, as detailed in the ‘Methods’ section. Unless otherwise stated, here and . The acquisition rate of the wavemeter depends on the camera speed and can be above \SI200\kilo\hertz for a small detector area. The total power sent to the sphere was below \SI1\milli\watt for highest frame rates and can be lowered when a short exposure time and high frame rate are not required. Further details of the experimental setup can be found in the ‘Methods’ section.
The core of the speckle wavemeter is the detection algorithm used to associate wavelength changes with the speckle pattern variations. To achieve a broad operating range, a coarse analysis using the transmission matrix method (TMM) follows standard approaches [7, 8, 9, 10, 11, 12, 13] for wavelength determination with a nanometre resolution, as shown in Fig. 2a-b. The intensity distribution recorded by the camera is , where is the input spectrum and the transmission matrix. This is constructed in a calibration stage, from speckle patterns recorded using established and well-known laser sources across the required wavelength range with a step of \SI1\nano\meter. Subsequently, the unknown spectrum of incident light can be reconstructed from a speckle pattern, at the resolution of \SI1\nano\meter, by performing left-inversion of the transmission matrix. However, noise can corrupt the reconstruction, we therefore use the truncated singular value decomposition to find a pseudo inverse transmission matrix, as in . At this stage, a nonlinear search algorithm [8, 9, 10, 11, 12, 13] could be employed to further lower the noise in the retrieval of the spectrum and enable characterization of broadband light in a spectrometer configuration [7, 13]. This part of our algorithm is exclusively used to determine the wavelength of an unknown light field with a precision of 1 nm as shown in Fig. 2a-b and no further attempt has been made to retrieve a broad spectrum or increase the resolution. The demonstrated operational range of 488 to \SI1064\nano\meter is currently limited by the spectral response of the camera used.
To achieve a high resolution we use a different classification algorithm capable of accurately distinguishing small speckle changes caused by wavelength changes below \SI1\nano\meter. This algorithm is based on principal component analysis (PCA) and is further detailed in the ‘Methods’ section. Principal component analysis (or singular value decomposition) is an established tool to decorrelate multivariate data. In the field of optics PCA has, for example, been applied to wavelength detection , detector noise analysis , to distinguish surface scattering processes  and is commonly used in the spectral band recognition of mixed colourants  as well as in Raman analysis . For our wavelength meter we project the data set (speckle images) onto its principal components and thereby retrieve the underlying wavelength modulation . An initial calibration data set is acquired and used to correlate the speckle patterns to wavelength, and sets the measurement range of the high-resolution wavemeter. The density of the calibration set aids the accuracy of the wavemeter . Subsequently, all measurements are projected against this calibration set to extract the unknown wavelength. Importantly, the wavelength-dependent changes in the speckle pattern that are captured by PCA are subtle in comparison to the TMM. A comparison of both approaches is shown in the ‘Methods’ section. An example of the retrieved wavelength trace (blue) is shown in Fig. 2c together with the reference wavemeter reading (red). We note that, in order to measure an unknown wavelength to high precision, these two algorithms can be used sequentially. All subsequent measurements in this paper were performed close to \SI780\nano\meter, but could in principle be performed at any wavelength within the operational range of the camera.
After the calibration of the desired operating range, we test our speckle wavemeter by measuring the change in wavelength from an ECDL centred at \SI780.2437\nano\meter. We apply a controlled wavelength modulation by varying the current to the laser diode at \SI21\hertz, as shown in Fig. 3.
For the case of relatively large wavelength modulation of 60 fm, Fig. 3a shows the good agreement between the wavelength measured using the speckle wavemeter, where we set , and a reference wavemeter. To explore the resolution limit of our approach, we increase the distance to the camera to and decrease the wavelength modulation amplitude by a factor of 200. These measurements are below the wavelength resolution of our commercial wavemeter, but interpolating over five runs gives agreement within of interpolated Fabry-Perot interferometer measurements of the wavelength. In Fig. 3 the centre wavelength has been subtracted from the speckle wavemeter wavelength readings and displayed as in femtometres, where = 0 corresponds to an absolute wavelength of \SI780.2437\nano\meter. Clearly, with a high resolution commercial wavemeter a more accurate calibration would be possible and an absolute wavelength reading could be presented. Further reduction of the modulation amplitude was not possible due to the limited resolution of the modulation signal source. A more detailed analysis of the range and resolution of the speckle wavemeter and its dependence on experimental design parameters is presented in the ‘Methods’ section.
Using the speckle wavemeter as a detector within a calibrated rubidium absorption spectroscopy setup, we also resolve the wavelength changes of a stabilized Titanium-Sapphire laser (M Squared Lasers Ltd SolsTiS). The laser is stabilized by top-of-fringe locking  to the crossover between the and hyperfine transitions in the D2-line of Rb (see Fig. 4a). Importantly this locking technique uses a dither tone at a frequency of 24 kHz with an amplitude of \SI1.3\femto\meter. This modulation signal was wavelength calibrated and is utilized as a calibration standard for our wavemeter. The speckle wavemeter is able to correctly resolve this dither tone as shown in Fig. 4b-c.
A detailed comparison of our wavemeter to current commercial and research-grade wavemeters and spectrometers can be found in Supplementary Table 2.
Having bench tested our wavelength meter, we also incorporate it within the wavelength stabilization control loop of the ECDL, as depicted in Fig. 5a. Here, the speckle wavemeter is once again used as a detector: the speckle image is analyzed using the PCA algorithm to determine the emitted wavelength. A subsequent proportional-integral-derivative algorithm calculates an error signal from the measured wavelength to a predefined wavelength setpoint. The error signal regulates the ECDL diode drive current and thus the laser wavelength. Initially the wavemeter is calibrated across \SI±30\femto\meter (\SI±15\mega\hertz) from the stabilization setpoint which sets the capture range of the speckle lock. The grating of the ECDL is employed as an external controllable wavelength perturbation, which we use to demonstrate the stabilization performance of the speckle lock.
Fig. 5b shows a wavelength trace captured by the external wavemeter during the stabilization. The experiment starts at 2.5 seconds with the \SI60\femto\meter-wide calibration ramp, then the laser is locked to the central speckle image (at \SI780.24\nano\meter) from 4 to 18 seconds. From 18 seconds onward the ECDL is free running without feedback control. During stabilization we achieve a linewidth narrowing from \SI3.2\mega\hertz (rms) to \SI0.82\mega\hertz (rms) of the ECDL over a 10 seconds observation time (for both measurements), which is ample for applications such as the efficient laser cooling of atoms.
As shown in Fig. 5c, between 13 and \SI15\second we apply a sawtooth modulation to the ECDL grating which corresponds to a 50 fm variation of the unstabilized laser wavelength. The resultant error signal shows the compensation applied to the locked laser and the wavemeter only detects wavelength changes at the start and end of the ramp, when the perturbation is largest. Increasing the update rate of the control loop, which is currently limited to \SI200\hertz, would minimize the amplitude of these spikes.
This work presents a practical speckle-based wavemeter with sub-femtometre resolution, which retains a broad operating range. A central element of the operating principle of the device is the use of an integrating sphere to create speckle patterns with a high sensitivity to wavelength changes through a compact and near lossless propagation path. A two step algorithm based on the transmission matrix method and principal component analysis was applied to resolve wavelengths within the range 488 to \SI1064\nano\meter and changes in wavelength below \SI0.3\femto\meter respectively. A full specification of the present performance of the wavemeter is presented in Supplementary Table 1.
In comparison to the use of an integrating sphere, fibre-based [12, 9, 13] speckle wavemeters have realized picometre resolution to date. Progress towards attometre-level resolution of our wavemeter is currently limited by our 16-bit amplitude modulation source and the availability of an accurate wavelength reference to calibrate the variability of the principal components. In theory the selectivity of the integrating sphere coating (Spectralon) supports an operational range from 0.2 to \SI2.5\micro\meter at high efficiency (reflectivity 90%). In practice, this operational range was limited by the response of the CMOS chip of our cameras.
Our present system is limited in its long-term stability (see Supplementary Note 1 and Supplementary Figure 1), but mechanical and thermal stabilization engineering could form part of a future study. Alternatively, a regular recalibration step, as can be found in many commercial wavemeters, can be used to maintain accuracy over longer periods of time.
The speckle-based wavemeter was also used as part of a laser stabilization setup. Encouragingly, the system achieved laser stability over several seconds without any thermal or vibration management, and reduced the linewidth of a diode laser by a factor of four. It is important to note that the frequency stability of the speckle-locked laser is presently not compared to any reference laser and thus the measurement was limited to the accuracy of the wavelength meter. The demonstrated capture range of \SI60\femto\meter (corresponding to \SI30\mega\hertz at \SI780\nano\meter) is currently small compared to other approaches such as polarization spectroscopy  which can achieve GHz capture range for lasers with a comparable linewidth to those discussed here.
The locking technique described here could aid the development of portable atom-based quantum technologies  due to the small footprint and insensitivity to misalignment. The speckle-based stabilization is a convenient method to lock at arbitrary detuning from an atomic resonance, such as the tens of GHz typically required in atom interferometry experiments . Furthermore, compared to dither locking of the laser to an atomic saturated absorption feature, speckle-based stabilization has the advantage of eliminating any frequency modulation of the laser. Finally, the extension of our approach to the simultaneous detection and measurement of multiple wavelengths  could also allow a single device to stabilize all of the various lasers (cooling, repumper, Raman and trapping) required in quantum technologies based on cold atoms.
Details of the experimental setup
For the wavemeter study, several well known and established laser systems were employed to determine the broadband range we were able to achieve with our approach. For the resolution and stabilization studies, an ECDL (Sacher Lasertechnik Lion TE-520, \SI780\nano\meter and linewidth of \SI3\mega\hertz over 10 seconds) was used as a wavelength tunable source. A commercial wavemeter (HighFinesse WS7, resolution \SI5\mega\hertz) and a Fabry-Perot interferometer (Burleigh, free spectral range \SI2\giga\hertz, finesse 200, \SI10\mega\hertz linewidth resolution) were used for wavelength reference to benchmark our system. The speckle patterns were captured using a xiQ - USB3 Vision Camera for the wavelength measurements. A fast camera with a maximum frame rate above 200 kHz (Mikrotron monochrome EoSens 4CXP, 4 megapixel CMOS camera with an Active Silicon FireBird CoaXPress frame grabber) was used for the laser stabilization experiments. For all measurements, the camera exposure and optical power were adjusted to avoid image saturation with the aid of an image histogram as shown in Supplementary Figure 2. The speckle patterns were subsequently analyzed on a \SI2.4\giga\hertz dual core personal computer.
Speckle image analysis algorithm
In the integrating sphere wavemeter every speckle pattern serves as a marker for a wavelength. An initial training phase is used to correlate training speckle patterns to wavelengths. In a second step every new acquired frame is projected against the initial training set and the current wavelength is retrieved. In the transmission matrix method (TMM) algorithm, for our training set is a series of speckle images at known wavelengths with a step size of 1 nm. The image pixels associated with each wavelength form the column vectors in our calibration matrix, that converts between spatial patterns and input frequencies. Fig. 6a shows an example of a calibration matrix for wavelengths of 769 to 791 nm. The average correlation over the image pixels, shown in Fig. 6b, is close to zero for a wavelength shift of 1 nm.
An example training set that converts between spatial patterns and input frequencies at the femtometre level is shown in Fig. 6c. The spectral correlation function in Fig. 6d shows strong correlation over a range of at least 300 fm. Comparison of both calibration matrices in Fig. 6a and 6c makes it evident that at the femtometre scale the overall speckle variability has decreased significantly. Thus, to correlate the speckle pattern to wavelength variations of less than one nanometre, we have implemented the PCA approach using Matlab and in C++ for the laser stabilization experiment using the standard template library for linear algebra. The PCA is applied to speckle images which are acquired for a varying wavelength (subscript denotes the pixel number). In Fig. 7 we plot the projections of the measured speckle patterns in PC space, PC where corresponds to the principal components with index . The values on the -axis are a relative marker of the variation as the variance of the speckle pattern is multiplied by the coordinate of the corresponding eigenvector , with values between -1 and 1. The PCs for wavelength modulations of the ECDL at fm (Fig. 7a) and fm (Fig. 7b) are therefore represented relative to the sign of the eigenvector and can be inverted. (Here, the speckle image size is pixels, the camera distance cm and the integrating sphere diameter cm for the fm modulation and cm for the fm modulation.)
We observe that each PC shows an oscillatory behaviour that we further analyze by Fourier transforming (using the single-sided amplitude spectrum) the PC traces (Fig. 7c-d). We denote the Fourier transform of PC1 as . We observe that PC1 to 4 are showing analogous oscillations. Here, PC1 is capturing wavelength modulation as generated by the laser diode drive current modulation, while PC2 detects the second order variations of the speckle pattern. Similarly, higher order PCs detect higher order speckle pattern variations that are each orthogonal and uncorrelated to each other. This orthogonality manifests itself also in the single-sided amplitude spectrum of the PCs time series where we observe that PC3 and PC4 exhibit modulations corresponding to the third and fourth harmonic of the wavelength modulation frequency Hz as shown in Fig. 7c.
Measurement Range and Resolution.
|Image size (pixels)||Resolution||Range|
|2.5 cm||2 cm||80 fm||8 nm|
|5 cm||2 cm||3.2 fm||16 fm|
|5 cm||2 cm||1.2 fm||1 nm|
|5 cm||20 cm||0.3 fm||16 fm|
|5 cm||40 cm||0.3 fm||16 fm|
|10 cm||2 cm||0.3 fm||16 fm|
The PCA algorithm is a powerful tool to detect small changes in the overall speckle pattern. In this section we investigate the range and resolution of the PCA solely.
One method to quantify the resolution and range limits of our wavemeter is to determine the amplitude of the modulation peak of the single-sided amplitude spectrum of the first PC, as indicated in Fig. 7c. The height of this peak correlates directly with the wavelength modulation and should follow a linear dependency. Deviations from this at low (high) wavelength modulation indicate an incorrect retrieval of the modulation and hence the resolution limit (range). This method has been experimentally verified for several setups as a method to quantify the wavelength resolution. In the lower limit, this corresponds to sub-pixel changes in the speckle pattern, and in the upper limit, to changes occurring outside the finite image range. Fig. 8 collates as a function of wavelength modulation amplitude. This is investigated for different experimental parameters (shown in Fig. 8) for small wavelength modulations of 0.3 to 10 fm. In future work, the theoretical model of our configuration which is presented in Supplementary Note 2 and Supplementary Figures 2-3 could guide the optimal design parameters of subsequent generations of our wavemeter.
A reference line (shown in blue) is measured for a pixel image at cm from an integrating sphere with cm, which gives a resolution of 3.2 fm and range of 16 fm. First, we studied the influence of the total number of pixels. We observed that an eight times larger speckle image size (red line, pixels) enhances the outcome of the PCA and decreases the resolution limit from 3.2 fm to 1.2 fm and increases the range to 60 fm. This can be explained by considering that a bigger detection array is able to capture smaller distributed perturbations.
Another important parameter is the propagation distance of the speckle pattern after the output from the sphere. Indeed, finer speckle structures can be resolved by the camera as this distance increases. The upper spatial frequency limit contained within a speckle pattern  is given by where is the integrating sphere port diameter, is the wavelength and the distance between the port and the camera detector array. For the detector array the Nyquist frequency can be calculated from half the element spacing [35, 36] of \SI7\micro\meter to 71.4 cycles mm. To avoid aliasing, the maximum spatial frequency contained in the patterns should be equal to the spatial Nyquist frequency. With a port diameter of 1 cm the camera should be approximately 18 cm away from the sphere. Here, increased distances of 20 cm and 40 cm were investigated and resolution was found to increase for 18 cm (depicted for = 40 cm in Fig. 8a, green) to 0.3 fm.
Additionally, to enhance the spatial resolution of the system one can increase the mean free path within the sphere. When the sphere diameter is increased from 5 to 10 cm (depicted in Fig. 8a cm in black) a resolution of 0.3 fm can be achieved. Table 1 summarizes the parameters in detail and shows the resolvable minimum wavelength modulation amplitude as well as the range.
The resolution of the method depends on the changes in the speckle pattern with respect to the incoming wavelength. When the wavelength changes, the speckle decorrelates. In previous work [7, 12], the half-width at half maximum (HWHM) of the correlation function was used as an indicator for the ultimate resolution of the device. However, this method depends on the reconstruction algorithm and the ratio of the speckle variation to the detection noise . Therefore, the actual resolution is determined by the ability to discern speckle patterns from two closely spaced spectral lines. In this work we are able to achieve resolution well below the HWHM of the speckle correlation. In the presented ECDL setup we are driving a piezo to tune the wavelength. In order to eliminate hysteresis effects due to the piezo, we used a sinusoidal modulation signal and the frequency retrieval as a marker for the resolution of the system.
The data that support the findings of this study are available from the University of St Andrews Research Data Repository at doi:10.17630/9448553f-a806-464b-a41c-7e2f0c80522b.
N.K. Metzger acknowledges support from Dr. C.T.A. Brown and thanks Dr T.J. Edwards and Dr S. Jiang for fruitful discussions. The authors acknowledge funding from EPSRC through an Impact Acceleration Account (IAA) Case Study award and grants EP/J01771X/1 and EP/M000869/1.
NKM, MM and KD developed the project. NKM developed the algorithms and conducted the experiments. NKM and GDB analyzed the data. RS implemented the C++ real-time control algorithm, NKM developed the C++ feedback control loop. BM, GTM and GM supported the work with a laser loan and technical advice. NKM, GDB and KD wrote the paper with contributions from all the authors. KD supervised the study.
Competing financial interests
The intellectual property based around this work is owned by the University of St Andrews and licensed to M Squared Lasers Ltd.
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