angular frequency ω0. since g1 is an even autocorrelation function, the spectral intensity at ωa can be written as: i n = 1 t a ∫ 0 t a cos ⁡ ( n ω a τ ) g 1 ( τ ) d τ =\int \limits _^\cos(n\omega _\tau )g1(\tau )d\tau } here, the n and ta represent the acoustic period. since the frequency spectrum is symmetric about ω0, the one side modulation depth is defined as: m 1 = i 1 i 0 = then we can consider the condition under the second approximation (weak-modulation approximation). in this situation, the term ( d / l t ′ ) ε 1 / 2 ')\varepsilon ^ is much smaller than 1. by applying the feature of sinh function, which is the main component of g1 function, the original autocorrelation g1 can be further simplified as: g 1 ( τ ) = 1 − 1 6 ( d l t ′ ) 2 ε [ 1 − c o s ( ω a τ ) ] ('}})^\varepsilon } therefore, the one side modulation depth can be simplified as m 1 = 1 12 ( d l t ′ ) 2 ε =('}})^\varepsilon } . from previous equation, we can see ε = 6 ( δ n + δ d ) ( n 0 k 0 a ) 1 / 2 +\delta _)(n_k_a)^ . therefore, in conclusion, a and m1 has quadratic relationship. such quadratic modulation can be captured by a fabry-perot interferometer. or, we can calculate the ratio between observed ac signal and he observed dc signal (also named as apparent modulation depth) which can carry enough information to represent such modulation as well. in conclusion, in uot analytic model, with the help of weak-scattering approximation, weak-modulation approximation, diffusion theory, and wiener-khinchin theorem, the relationship between acoustic amplitude and modulated light can be successfully observed. time-resolved frequency-swept uot (forward model) for single-frequency uot, the axial resolution along ultrasonic axis is always limited by the elongated ultrasonic focal zone. to improve the axial resolution, ultrasonic frequency-swept uot model is designed. in this system, the object is placed in a tank full of uot scattering medium. there will also be an ultrasound absorber at the bottom of the tank to avoid rebound of ultrasound. basically, a function generator will produce a frequency signal relating to time. after passing through a power amplifier and a transformer, such frequency command will be sent to the ultrasonic transducer to generate ultrasonic beam with different frequencies. after a brief calculation, a focused ultrasonic beam will be sent to the medium and the target. meanwhile, a laser beam which is perpendicular to the ultrasonic beam will also illuminate the scattering medium. then, on the other side of the light source, the pmt, modulated by frequency signal sent through the first function generator will detect transmitted light signal within the tank and transfer the optical signal to electrical signal. the electrical signal will then pass through an amplifier, an oscilloscope and be stored in the data base. with such data base, spectral intensity vs frequency plots at multiple points can be generated (the first spectrum is generated as a reference, produced by optical signal far from the object.). each of the spectrum can then be further converted to a 1d image showing the interior of the medium in the direction perpendicular to the tank (z direction). in the end, all the 1d image will be pieced together to generate a full view inside the medium. in summary, a frequency-swept (chirped) ultrasonic wave can encode laser light traversing the acoustic axis with various frequencies. decoding the transmitted light provides resolution along the acoustic axis. this scheme is analogous to mri. development was first proposed as a method for virus detection in 2013. recent advances in uot (2020 onwards) include 1) the development of coded ultrasound transmissions for snr gain in aoi, 2) the development of homodyne time of flight aoi, 3) the use of super-resolution techniques to improve uot beyond the acoustic diffraction limit, and 4) the use of coaxial interferometry to better enable modern high-performance cameras for parallel detection of uot signals. levi et al. discovered that the use of coded sequences of acoustic pulses can turn the speckle modulation at every time instant into the sum of acoustically modulated regions. ct-aoi can keep the spatial resolution from single cycle ultrasound pulses while increasing snr by the half of the square root of the number of cycles. in this paper, levi et al. use 79 cycles to gain an experimental 4 times increase to the snr. as a follow-up to their previous work, levi et al. developed a homodyne aoi scheme enabling the detection of tagged detection with a single low-gain photodetector. this method leads to a 4 times snr increase over more traditional high-gain photodetectors, such as photomultiplier tubes. in this homodyne time-of-flight aoi system, the reemitted light is not detected directly but is rather interfered with a reference beam in a homodyne configuration. the interference leads to an optical amplification of the us-modulated light, enabling its detection with low-gain photodetectors with a bandwidth that is higher than the ao modulation frequency. this setup does not temporally integrate the signal, allowing for much more flexibility regarding speckle decorrelation. this is because the measurement signal can be divided in post-processing, allowing the analysis of time windows wherein the speckles are stable. doktofksy et al. utilized super-resolution optical fluctuations imaging (sofi) techniques to gain massive improvements to spatial resolution in uot. naturally fluctuating speckle grains present in uot images are analogous to blinking fluorophores in sofi, which enables super-resolution. normally, single-pixel detectors (e.g. photodiode) are used in uot. these detectors suffer from limited dynamic range, which causes difficulties with low modulation depth. modulation depth can be enhanced by using multiple pixels in parallel; the enhancement is equal to n ^ (1/2), where n represents the number of pixels. modern high-performance cameras have millions of pixels but have low temporal resolution (slow framerate and long exposure time). lin et al. tackled this issue by designing a system with paired illumination from two co-propagated beams with slightly different optical frequencies.