Ultra-high precision speckle spectrometer enabling radio-frequency scale resolution of atomic spectra

Abstract

Laser speckle, the granular intensity pattern arising from random optical interference, provides a high-dimensional encoding of spectral information that can be exploited for precision metrology. Speckle-based spectrometers have advanced rapidly owing to their compact footprint, mechanical robustness and alignment agnostic nature, yet their spectral resolution has remained limited to the picometre scale. In this work, we break this limit by employing an integrating sphere as a multiply scattering cavity with access to a high range of path lengths to enhance spectral sensitivity. At 780$\,$nm, the resulting device achieves a resolution of 6$\,$fm, corresponding to a resolving power of $1.3\times10^8$, representing an approximately 80-fold improvement over previous implementations. This ultra-high resolution enables clear discrimination of laser sidebands generated by an electro-optical modulator, with extracted sideband powers agreeing with expected values to within 1%. It further permits the first direct speckle-based measurement of the hyperfine structure of the $\text{D}_{2}$ transition in $^{85}\text{Rb}$, with transmission spectra differing by no more than 3.6% from independent wavemeter-referenced measurements. These results establish speckle as a new platform for ultra-high precision spectroscopy, radio-frequency spectrometry, and microwave photonics.

Figures (4)

• Figure 1: A) Experimental setup: an amplified Er:fibre laser is frequency doubled before passing through the device under test (DUT) and being sent to the integrating sphere. Laser speckle is generated by the integrating sphere and is then captured on a CMOS camera. A cropped speckle pattern of $200\times200$ pixels is shown below the camera as an example. The scale bar is 500 µm. Two DUTs were employed. B) The first DUT: a fibre-coupled electro-optic modulator which was driven by an arbitrary waveform generator. C) The second DUT: rubidium saturation absorption spectroscopy setup (see Methods \ref{['subsec:experimental_app']}). PBS: polarisation beam splitter, HWP: half-wave plate, QWP: quarter-wave plate.
• Figure 2: Integrating sphere speckle spectrometer characterisation using an electro-optic modulator (EOM) as the device under test. A) Spectrogram obtained as the frequency of the sinusoid driving the EOM was swept linearly from approximately 0 Hz to 80 MHz in one second. In this dataset, the amplitude of the driving sinusoid was held constant. The inset is a zoom into the region where the EOM driving frequency lies between $\sim0\,\text{Hz}$ and 10 MHz. The vertical dashed line at approximately 3 MHz shows the EOM driving frequency for which the laser carrier and sidebands are resolved. B) Individual reconstructed spectrum from A corresponding to when the EOM driving frequency was 3 MHz. C) Spectrogram obtained as the amplitude of the sinusoid driving the EOM was modulated linearly from 0% to 100%. At a modulation depth of approximately 1.19, the spectrometer is able to resolve eleven separate lines ($\pm5$ sidebands and one carrier). D) Numerically integrated power in the laser carrier and sidebands as a function of phase modulation depth. The power in the $\pm n^{\text{th}}$ sidebands was averaged in each spectrum and is presented as a single data point. Solid lines are fits to the experimental data using squared Bessel functions of the first kind. The fit is performed for all functions simultaneously to ensure parameters do not change between functions. E) Residuals between numerically integrated powers and squared Bessel functions in D as a function of phase modulation depth.
• Figure 3: A) Spectrogram obtained as the EOM was driven by a sinusoid which had its frequency and amplitude modulated to produce spectra resembling wave packets. This figure shows one period of the modulation with no averaging. B) Zoom into the $+1^\text{st}$ order sideband of the spectrogram obtained after averaging over twenty modulation periods. C) Spectrogram of the expected behaviour of the $+1^\text{st}$ order sideband given the applied modulation to the sinusoid which drove the EOM.
• Figure 4: Saturated absorption spectrum of the Rb-85 D$_2$ transition. A) Comparison of the D$_2$ transition as captured by the integrating sphere speckle spectrometer and a photodiode. The photodiode trace was captured at a much higher sampling rate and has been decimated, averaged, and interpolated to match the sampling rate and wavelength of the speckle measurement. Blue vertical lines show the frequencies of the transitions from the $5S_{1/2}$, $F=3$ state to the $5P_{3/2}$, $F^{\prime}=\text{2, 3, and 4}$ states steck2025. Red vertical lines lines show the frequencies of the crossover transitions. The wavelengths for both the speckle and photodiode traces were measured separately using a reference wavemeter and then calibrated using the known $F=3\,\rightarrow\,F^{\prime}=4$ transition. B) Difference between the speckle and photodiode SAS traces in A.