1. Introduction

Coherent optical signal processing is a powerful tool for real-time 3D imaging of objects at distances ranging from a few hundred microns to several hundred meters with corresponding resolutions [1]. Optical Coherent Tomography (OCT) [2] is a well-developed imaging technique for objects at short distances with micron-level resolution, hence is useful for various biomedical applications [3,4]. OCT has two different forms Time Domain OCT (TD-OCT) and Fourier Domain OCT (FD-OCT) [5]. The FD-OCT has been implemented by exploiting two different approaches; the first one is Spectral Domain (SD-OCT) which utilizes a broadband source with a spectrum analyzer. The second approach is the Swept Source (SS-OCT) which utilizes a tunable laser source combined with a photodetector. Among the different implementations, SS-OCT is the most promising and can provide axial resolutions of 5 µm and depth information up to a few millimeters [3]. Other variants of OCT such as Doppler OCT also exist for specialized applications where velocity measurement is also required [6].

For measurements of 3D objects at long distances ranging from a few meters to kilometers, a Light Detection and Ranging (LiDAR) [7,8] technique is employed using a modulated source and a photodetector. LiDAR has several applications, for example: surveying [9,10], forestry [11], atmospheric physics [12], and autonomous vehicles [13]. The most common scheme to implement LiDAR is by measuring the time of flight of pulsed lasers. A more recently developed technique is frequency modulated continuous wave LiDAR (FMCW LiDAR) [8] which uses a frequency chirp. The chirped signal is transmitted to the object and its replica is made to interfere with the returned signal, reflected from the object. The beat frequency is then used to determine the distance to the object. It is worth noting that the technique of SS-OCT and FMCW lidar resemble each other in terms of using a frequency sweep and measuring distances using coherent detection [14]. The difference arises from the manner of frequency sweep, wherein SS-OCT a particular frequency interferes with itself while in FMCW LiDAR different frequencies can interfere with each other due to time lag.

In this regard, although they are treated independently in the literature, it is possible to view these methods as part of a more general universal framework: a coherent interferometer that has the capability of optical modulation in different sections of the system. The prevailing techniques can all be viewed as special cases depending on the source, modulation format, and detection procedure. In one case, a laser, quadratic phase modulation, and fast photodetector will implement FMCW LiDAR, whereas a frequency sweep and a slow photodetector become SS-OCT. According to conventional understanding, improvements to these technologies based on hardware have reached the point of diminishing returns [15]. Research has consequently shifted to alternative methods, such as superior processing algorithms, and complex modulation/detection schemes, in a bid to improve the resolution and depth performance [16–20]. The primary advantage of this universal framework, which we call Universal Photonics Tomography (UPT), is that it can be used to systematically improve existing systems, as well as formulate novel functionalities and capabilities. For example, the techniques demonstrated below produce significant gains in resolution and depth performance metrics, are universally compatible with existing tomography systems, and can be implemented with only minor hardware modifications.

In this work, we demonstrate one such example, where the addition of a phase modulator to a conventional OCT system can be exploited to scan multiple times and can be used to detect objects over longer distances by changing the resolution and depth parameters of the tomography system. These parameters are a direct consequence of the Nyquist criterion with length (or time) and frequency forming Fourier pairs. They determine the limitations and effective cost of the system, and their relations are given by Eq. (1), where the axial resolution (${l_o}$), which is also the sensitivity of the system, is mainly determined by the bandwidth ($B$) of the laser sweep while the maximum distance ($L$) by the frequency resolution $\nu_{o} $

Nearby object imaging is limited by the axial resolution ${l_o}$ (determined by the optical bandwidth) and far object imaging is limited by maximum distance $L$ (determined by the frequency resolution). We mathematically develop this tomography system from first principles and show how the fundamental resolution and depth limitations can be pushed using phase modulation and a multirate filter bank interpretation. Our implementation of UPT is novel to the best of our knowledge, and no other previous work has developed or anticipated the use of phase modulation, multirate filter banks, and signal processing for improvement in axial and frequency resolution, which are both limited by the hardware in coherent detection schemes.

Multirate signal processing is a widely used technique in many areas of modern engineering such as wireless and satellite communication systems, image processing, video processing, etc. Multirate filter banks are used for such signal processing applications which require data compression, detection of harmonics, de-noising, sub-band decomposition, recognition of one and two-dimensional signals, adaptive filtering, design of wavelet bases, and wireless communication [21]. Specifically, multirate systems can split the original input signal into multiple signals or combine multiple signals into a single signal. The specific idea of developing multi-rate systems is their ability to split the original input signal into multiple signals or to combine multiple signals into a single composite signal in the frequency domain. Although the application of multi-rate filter banks is different for conventional signal processing we borrow the process of splitting and combining, translating it to optical modulation and post-detection signal processing. In the application demonstrated here, this allows the sampling to be increased through additional measurements. This enables the performance of the system to be improved simply by increasing the measurement time.

2. Formulation

Multirate filter banks are sets of filters, decimators, and interpolators used widely in conventional digital systems [22]. Usually, decimators downsample the signal after passing through analysis filters. This compressed information is stored or transmitted via a channel. On the other end of the channel, the signal is interpolated or upsampled and passed through synthesis filters to retrieve the original information. The process of downsampling means decreasing the resolution of the system which is similar to an undersampled tomography system. The tomography systems are also discrete, and analog filters can be implemented by phase modulation of the optical carrier signal and by digital processing after detection. Hence, the imaging system can be considered as a multirate filter bank with each scanning cycle representing a single channel and carrying object information in a compressed form. Here, we demonstrate a 2-channel filter bank implementation which results in a twofold improvement in both length and frequency resolution of the tomography system. Using this scheme, both near and far objects, as well as their density profiles, can be measured with improved parameters. In this way, it is more versatile than conventional approaches.

2.1 Universal photonics tomography

The setup for our implementation of UPT, without the phase modulators, is shown in Fig. 1(a). Consider a heterogeneous object with multiple optical media and their corresponding surfaces present only at an effective optical distance $il_{o}$ from the first surface. Since a reflected beam will pass through each section twice (once in the transmission direction, and once in the reflected direction), the effective optical path length of each section is defined as twice the distance multiplied by the effective index of the medium. i is an integer in [1, N-1]. N is the total number of surfaces that can be present, including the first surface. ${l_o}$ determines the axial resolution of the imaging system. A complex field E, with a certain carrier frequency ν, is incident on this sample. The reflection from the i^th surface is given by:

The field in the sample arm will be proportional to the r_total. The proportionality constant depends on: 1) coupling coefficient of the 3db fiber coupler, losses, etc. which are neglected as they are scaling terms, and 2) sample arm length, which is set to be equal to that of the reference arm and can therefore also be neglected such that:

The field in the reference arm is the original field that is transmitted to the object and is given by:

 

Fig. 1. Structure for Universal Photonics Tomography (UPT). a, Base case: Setup for our implementation of UPT without phase modulators which resembles the Swept Source OCT in single mode fiber. b, Case A: Phase modulator is added in the sample arm to the base case. A waveform generator (not shown) is used to give slow modulation which assists to improve the resolution in the length domain. c, Case B: Phase modulator is added just after the tunable laser to the base case. A signal generator (not shown) is used to give fast modulation which assists to increase the maximum unambiguous range. d, Schematic for the working of UPT and the required post-processing in the filter bank form. The horizontal dashed lines indicate photodetection. Parts b and c represent the physical system of UPT which forms the left-hand side (pre-photodetection process) of the filter bank scheme shown in part d. The right-hand side (post-detection process) is implemented digitally. The down-arrow and up-arrow blocks correspond to downsampling and upsampling respectively, both by a factor of an integer M. Upsampling is performed digitally, while downsampling is inbuilt in the UPT system as the resolution of the system is less than needed. The transfer functions (represented as Z transforms) are in the frequency domain for case A, while in the length (i.e., time) domain for case B. u_i(n) is the detected signal, H_i(z) is the analysis filter and F_i(z) is the synthesis filter in the i^th channel. n is the time vector in case A while the frequency vector in case B. H_i(z) is implemented optically using a phase/intensity modulator while F_i(z) is implemented digitally. a(n) is the high-resolution OCT information that we wish to obtain while y(n) is its reconstruction using the UPT system. e, An illustration of the reconstruction signal formed by combining two channels (M=2) in the frequency domain for both cases. The red and black dots represent the effective frequency measured by both channels. In case A, both channels contribute to extending the bandwidth in the frequency domain, thus improving time domain resolution. While in case B, both channels interleave to increase the frequency resolution, thus extending the maximum ambiguous range. The graphs are presented to give an intuition of the placement of frequency points in the reconstructed signal and do not represent a physical situation.

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Let P be the power detected by the photodetector.

Assume a new set of phases with mapping Φ_i → Φ_i for $i > 0$, -Φ_-i → Φ_i for $i < 0$, and Φ₀ = 0. Also mapping $a(i )\to \hat{a}(i )$ for $i > 0$, $a\ast ({ - i} )\to \hat{a}(i )$ for $i < 0$, and $\hat{a}(0 )= 0$.

Note that the effective distance between the 1^st and i^th surface (as defined before) is il_o for i > 0. Performing the mapping for Φ_i as above, it can be shown that Eq. (9) holds for all possible values of i:

Physically speaking, for remote measurement, the light reflected from the i^th surface will be delayed by ${l_o}$i/c prior to arriving at the photodetector. Consequently, the finite integration time of the photodetector must be considered, since the net integration time will be reduced for light from surfaces deeper in the sample. Assuming Δt to be the time for which the laser is active for a given frequency the interference power measured by the system will be:

For many applications, it is safe to assume that loN/c << Δt, as the integration time of the detector to measure one frequency point in the sweep (∼100 µs) is significantly more than the time taken by light to travel as long as a kilometer ($ < $5 µs). Furthermore, in the case that the integration time cannot be neglected, the area of the integral and ∣E∣² can be corrected by normalizing $\hat{a}(i )$, which we represent as $\bar{a}(i )$:

Next, the discrete nature of the frequency characterization must be considered. For a swept laser, we can measure discrete frequencies directly with a photodetector, whereas for a broadband source the spectral power density can be characterized by a spectrum analyzer. In either case, as we have 2N+1 terms in the summation, we must measure the interference term at 2N+1 frequencies with resolution $\nu_{o} $. For an integer k in [0, 2N]

Usually, the frequency sweep would not start from ν = 0, but we ignore an offset term in Eq. (12) as it would only contribute to a constant phase term in Eq. (11) and can be omitted as a scaling factor. To comply with the Nyquist sampling condition, which is the result of the Nyquist-Shannon sampling theorem, the frequency resolution is chosen such that $\nu_{o} $lo/c= (2N+1)^-1 and the laser measures at 2N+1 points. Then the discretized version of the interference term is given by:

Equation (13) represents the inverse Discrete Fourier Transform relationship between the measured power and reflection coefficients that forms the basis of FD-OCT. Consequently, taking the 2N+1 point DFT of P_intf gives back the depth information of the object. Note that only the positive part of $\bar{a}$ is needed to obtain $a$. The i’s for which a(i) is non-zero can be used to calculate the optical distance (being ilo), while the a(i)’s can be used to calculate the refractive index using the Fresnel equations.

2.2 Frequency dependent slow modulation

Next, consider the addition of a phase modulator to the sample arm, as shown in Fig. 1(b), and the use of a signal generator to introduce a phase modulation ϕ (t). The field in the sample arm then becomes:

Assuming that the modulation is slow compared to the time taken by the laser to measure a single frequency (the time bin), P_intf can be calculated in a manner similar to the previous section (see Supplementary Information, S-II):

Let $\tilde{H}(k )= \exp ({j\phi ({k\Delta t} )} )$ and the 2N point DFT of $\tilde{H}(k )$ be $h(n )$, where n is the integer that represents the frequency component of the Fourier Transform. For satisfying the Nyquist criterion we should have $\nu_{o} $lo/c= (2N)^-1 and the laser must sweep over 2N points. Taking the 2N DFT on both sides of Eq. (15) results in:

Equation (17) resembles a filter h. (n) applied to a(n) in a linear system with convolution in length (i.e., time) domain. A transfer function can then be defined in the frequency domain, and this provides the opportunity to apply digital signal processing to the depth information. This connection to signal processing is the most important insight of the manuscript.

2.3 Frequency independent fast modulation

In this case, we place the phase modulator just after the laser, as shown in Fig. 1(c). We use fast modulation which repeats after every sweep frequency, i.e., it is periodic with Δt. It can then be shown that the interference term in Eq. (13) becomes

Here the P_intf has been replaced by u(n) and variable k is replaced n. Here $\tilde{a}(n )= \mathrm{{\cal F}}[{\bar{a}(i )} ]$, and $h(n )= \mathrm{{\cal F}}[{\bar{H}(i )} ]$ determines the filter coefficients (see Supplementary Information, S-III). Note that here the convolution is in the frequency domain, as opposed to the previous case. Hence, the transfer function can be implemented in the length domain.

2.4 Multirate filter bank interpretation

The main theoretical results of the manuscript are Eq. (17) and Eq. (19). They represent a linear system in which multirate signal processing can be used to increase the resolution of the system as shown in Fig. 1(d). These equations can be represented in Z domain by taking the Z-transform such that:

The capital letters U, A and H in the above equation are Z-transform of their corresponding small letter functions $u(n )$, $a(n )\textrm{ [or }\tilde{a}(n )\textrm{]}$ and $h(n )$, and z is the complex variable in Z domain. Hence phase modulation can be interpreted as a transfer function [ $H(z )$] on the resolution limited signal [$A(z )$]. Consequently, Eq. (20) corresponds to a single channel on the left hand side of Fig. 1(d). Likewise, additional scans can be used to obtain information for all the channels. Next, the channels on the right-hand side are implemented on a digital computer. The depth information can be retrieved numerically by implementing the synthesis filters [$F(z )$] and then combining the various channels in the appropriate domain.

We again stress that although Eq. (20) looks the same for both the slow and fast modulation scenarios, it is important to know that their domain is opposite (specifically frequency and length respectively). By performing multiple scans, axial resolution can be improved in the slow case while the maximum depth is increased in the fast case. Consequently, the slow case is of interest when it is desired to overcome the limitation of the bandwidth of the source laser, while the fast case is of interest when it is desired to overcome the frequency resolution limit of the system. Conceptually, these two cases are equivalent to the presence of a downsampled block in the system. Analysis filters are implemented using phase modulators while the synthesis filters and upsampling blocks are implemented on a digital computer. As the number of channels possible can only be integers, the resolution/maximum unambiguous range can only be improved by an integral multiple. In the result section, we experimentally demonstrate a 2-channel filter bank for both cases, which improves the resolution/maximum unambiguous range by a factor of two. An illustration of the reconstructed signal formed by a 2-channel filter bank is presented in Fig. 1(e). Additionally, we numerically validate the 4-channel filter bank (see Supplementary Information, S-V).

For slow modulation, we use a linear phase modulation, which is effectively a z^-1 transfer function in the Z domain (see Supplementary Information, S-IV). This results in a so-called lazy filter bank. For fast modulation, sinusoids are the only cost-effective option. The transfer function then corresponds to a Bessel function (first kind, zeroth-order) of a sinusoid (see Supplementary Information, S-IV). The synthesis filters [${F_m}$], which need to be implemented on a digital computer, can be calculated from the perfect reconstruction conditions of filter banks [22], as given by Eq. (21) and Eq. (22). K_D is an integer and corresponds to the delay due to signal processing.

3. Experimental results

The experimental results demonstrate the working principle of the device, which is developed in the section above, under the universal framework. We then experimentally demonstrate how various modulation schemes provide the opportunity for novel detection and post-processing strategies.

3.1 Universal photonics tomography

To demonstrate the UPT, we use two microscope slides as objects [Fig. 2(a)], one placed directly in front of the other, hence a total of four different interface surfaces separating two different media (namely air and glass). The microscope slides are about 1mm thick, and the two slides are placed about 12 cm apart. The refractive index of glass is assumed to be n_glass ≈ 1.5, and the refractive index of air is taken to be n_air ≈ 1.0. Figure 2(b) shows the detected interferogram after using an offset equal to its mean. The bandwidth is 5 nm at a wavelength of 1.55 µm and the resolution $\nu_{o} $ is 0.3 pm. As shown in Fig. 2(c), the Fourier transform clearly distinguishes the four surfaces and gives accurate distances of all surfaces. The normalization of FT magnitude is performed using its maximum value.

 

Fig. 2. Experimental demonstration of UPT without modulators (Base Case). a, The objects used are a couple of microscope slides placed one behind the other. b, The measured interference pattern on the photodetector as a function of the frequency sweep. c, Fourier transform (FT) of the interference pattern. The four larger peaks predict the distances of the surfaces present. The smaller peaks (barely visible) are due to the autocorrelation of the sample arm signal in the interferogram and can be removed by balanced photodetection.

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3.2 Increasing axial resolution

To demonstrate how to increase the axial resolution we use a microscope slide and a mirror behind it, as shown in Fig. 3(a). We create a situation where the bandwidth of the laser is not high enough to clearly distinguish the two surfaces of the slide. The laser sweeps a bandwidth of 1 nm with 0.2 pm resolution. This results in an axial resolution (${l_o}$) of 2.4 mm, while the normalized distance between the slide surfaces is 3 mm. This measurement is referred to as the unmodulated signal (curve Ch 0 in Fig. 3(b), green curve, and the green curve in Fig. 3(d)). It corresponds to conventional SS-OCT, but the surfaces are not resolvable due to insufficient resolution resulting from the limited bandwidth of the tunable laser source. Next, we use a waveform generator to provide a linear phase modulation to the sample arm, referred to as Ch 0 in Fig. 3(c). The Ch 1 and Ch 0 interferograms have the same resolution (${l_o}$). The minor difference between them is due to a constant phase offset that causes the points to be sampled at slightly different depth positions (Fig. 3(b) blue curve and Fig. 3(d) blue curve). Next we combine the two signals, treating them as two different channels of a multirate filter bank [Fig. 3(e)]. This improves ${l_o}$ from 2.4 mm to 1.2 mm. The surfaces can be distinguished much more easily now, and their positions are known twice more accurately than before. The axial resolution of the synthesized signal with a 1 nm bandwidth optical source is equal to that of a single channel system with a 2 nm source, a 100% improvement! Essentially, in this UPT implementation, the frequency domain signal of the second channel is forced to behave like an extended signal of the first channel using modulation. Hence, the two channels can be concatenated to give a longer signal in the frequency domain [Fig. 1(e)]. This direct concatenation is possible because we were able to implement a lazy filter bank that only adds a delay to the second channel. It is even possible to implement a more complex filter at the analysis side, however, then a corresponding synthesis filter will be needed before final concatenation. Additionally, as the system corresponds to a lazy filter bank, the synthesis signals (before combination) in the two channels will look exactly like the modulated signal except that both will be upsampled and the first channel will be delayed by one step of resolution in length [Fig. 3(d)]. Further, note that multiple channels can be used to improve the axial resolution even more. Essentially, the resolution can be increased by a factor equal to the number of channels, with the only trade off being the extra time needed to measure the additional channels. This is a highly significant result, as it provides the best path to ultrahigh resolution devices by a large margin.

 

Fig. 3. Demonstration of UPT for increasing axial resolution (Case A). a, The objects are a microscope slide and a mirror with the former in front of the latter. b, The measured interference pattern of the unmodulated channel 0 (green curve) and modulated channel 1 (blue curve) as a function of frequency. c, Schematic of the linear modulation given to Channel 1 (Ch 1). d, Fourier transform (FT) of Ch 0 (green curve) and Ch 1 (blue curve). e, Synthesized distance estimation of the objects by combining both the channels as part of a lazy filter bank. The resulting curve has a twice better length resolution compared to the ones detected in the individual channels.

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3.3 Increasing maximum depth

For a simple demonstration of how to increase the maximum unambiguous depth, we again use the microscope slide with a mirror behind it [Fig. 3(a)]. We define a balanced point which is the zero position in the length domain and physically represents the point where the delay of the reference signal is equal to that of the signal from the object. The microscope slide is used as a reference, which is at 2.51 m from the balanced point, while the mirror, which is at 3.41 m from the balanced point, is the target object. Here we consider the situation when the resolution of laser sweep is limited to 0.4 pm, which corresponds to a maximum unambiguous depth ($L$) equal to 3 m, and the position of the target (mirror) is beyond it. Hence the frequency resolution is not enough to measure any object beyond 3m. We first measure this object with 50 MHz sinusoidal phase modulation as shown in the green curve of Fig. 4(b) and Fig. 4(c). The peak for the mirror appears at 2.62 m which is an aliasing artifact that arises due to undersampled measurement. To predict the true position of the target we perform a second measurement where the transfer function of the phase modulation has a zero at the unaliased position of the target but not at the aliased position. If the target peak disappears then it indicates that the target is indeed at a much further distance, otherwise, the original peak gives the correct position. Thus, we use adaptive phase modulation and signal processing to determine the position of a single target which is often the requirement of a conventional LiDAR system. This is a valuable method as it is often difficult to determine the accurate transfer function of the optical modulation due to factors such as nonlinearity, variable V$_{\pi}$, RF impedance mismatch, et cetera. In contrast, this method only requires the knowledge of zero crossings of the transfer function. In our case, an 80 MHz sinusoidal phase modulation gives a transfer function that has a zero at 3.41 m and we show that this makes the 2.62 m peak disappear (Fig. 4(b) blue curve). Therefore, we can conclude that position of the target is actually at 3.41 m. We also demonstrate in Fig. 4(d) that the peak would not have disappeared if the true position of the mirror were actually at 2.62 m, by physically placing a mirror at this position. Also, the 50 MHz and 80 MHz measurements can be treated as two different channels in a multirate filter bank and combined, as shown in Fig. 4(e), to give a graph that has twice the maximum unambiguous range of individual channels. Note that the SNR is same is similar in both the channels as well as the synthesized result. Essentially, for an increase in maximum depth, a higher resolution in frequency is needed. The second channel in this case is forced to fit in between the existing frequency points of the first channel [Fig. 1(e)]. Hence, by combining both, the effective frequency resolution is increased. This method will perform better for more complex objects but also require an accurate structure of the analysis of the transfer function produced by phase modulation (see Supplementary Information, S-IV). Thus, we have shown that distances up to 6 m can be measured by using a laser sweep resolution which corresponds to only a maximum depth of 3 m in the unmodulated case. As mentioned above, multiple channels (scans) can be used to increase the limit even more. Also, for simple targets, adaptive measurements can be performed which will require a smaller number of channels, but which can still measure much farther positions from the target. This is a highly significant result for the same reasons.

 

Fig. 4. Demonstration of UPT for increasing maximum unambiguous depth (Case B). a, Schematic depicting the voltage applied to phase modulation as laser frequency is tuned. b, Measured power when phase modulator is given 50 MHz sinusoidal signal at channel 0 (green curve) and 80 MHz sinusoidal signal at channel 1 (blue curve). c, Fourier transform of the measured power in the two channels (The normalization is performed using the maximum value of the channel, is to simplify the FT scale and holds little signal processing significance). These both are combined to predict the true position of the mirror, which is beyond the Nyquist limit d, Fourier transform of the measured power when a mirror is actually placed in the aliased position of the original mirror. Comparing the 80 MHz RF modulation between the blue and brown curves it can be seen that the length-dependent transfer function has a different effect on the two peaks that are at different physical positions. Note that this occurs even though they show up at the same place in the Fourier domain due to aliasing, and can be used to identify spurious peaks. e, Synthesized signal by combining the two channels and passing them through the synthesis filters. The peak at 3.41 m predicts the true position of the mirror.

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4. Discussion

We have theoretically developed and demonstrated UPT which is a universal framework for measuring the depth and position of objects at various distances by adjusting the laser sweep frequency and bandwidth, and which includes conventional OCT and LIDAR techniques as special cases. Experimentally we validated SS-OCT as a special case under this framework, and we used the UPT framework to design and demonstrate an alternative approach that improves the resolution and depth performance through the use of slow and fast modulation of the optical carrier. This is a significant technological development, as it provides a method to improve the performance of existing tomography systems through the addition of simple, inexpensive hardware components.

According to conventional understanding, improving the performance of tomography systems would necessitate simultaneously increasing the tunable bandwidth of the laser, as well as the tuning resolution, while also increasing the output power so the spectral power density is maintained. This is an uneconomical prospect at best. The bandwidth and frequency resolution of a tunable laser is limited by size, power, material properties, etc. [24,25]. Improving them directly will make the system bulky, cost-ineffective, and difficult to implement. In contrast, the alternative approach demonstrated here only requires a simple phase modulator and waveform/signal generator which are far more economical and much easier to integrate into the system. By making multiple scans, ultrahigh resolutions can be achieved both in the frequency and length domain. The only drawback of this method is the extra time taken to perform multiple scans. The design is agnostic to the type of phase modulators used, which can be mechanical, acousto-optic, electro-optic, etc. In our experiments, we used Lithium Niobate phase modulators [26] which have promising specifications of low V_π and high RF bandwidths.

This multichannel detection scheme works on the principle of multirate filter banks, and the number of channels can be increased to more than two and can be used for more complex objects, similar to how a multichannel filter bank works [22]. Given enough channels with appropriate modulation, they can be theoretically combined by multirate signal processing to get a reconstructed signal with arbitrarily high resolution. In the multirate filter bank formulation, the resolution improvement has no theoretical limit. However, physically speaking, for long distances, the detected power might drop below the noise levels of the photodetectors. Another practical challenge that exists is the imprecision in the frequency sweep. If all the frequency values reported by the laser do not have a constant frequency difference, the Fourier transform will be noisy when measuring near or beyond the Nyquist limit. We observe this in case B where the noise floor is due to the improperly spaced frequency values. The power on the photodetector comprises the DC term (reference autocorrelation), the sample autocorrelation, and the interference term (cross-correlation). To efficiently extract the interference term with high SNR, it is important to filter out the remaining two terms. One way is to attenuate the signal in the sample arm and subtract the mean of the total interference power. This method can still produce small peaks in the Fourier transform due to the presence of autocorrelation term, which can also be observed in our base case [ Fig. 2]. A better way to remove the other two terms is using balanced photodetection, where subtracting the two interference powers cancels out the two unnecessary terms.

To implement synthesis filters, it is essential that Δ(z) as described in Eq. (2)2 is invertible. This is not the case when sinusoidal phase modulation is given only to one channel with no modulation on the other. Hence, for case B, both channels should have sinusoidal modulation. Other modulation shapes can also be used if the speed of the waveform generator permits. More generally, under the UPT framework other novel configurations are also possible. For example, using intensity modulators instead of phase modulators to implement more complex filters, or developing a system similar to SD-OCT and using optical modulation to virtually improve the bandwidth of the source and frequency resolution of the spectrum analyzer.

It is also important to emphasize the universality of the UPT framework. Notably, the formulation and implementation demonstrated here is compatible with existing devices, including those that use parametric estimation, spectral estimation, photon-counting, frequency comb, supercontinuum laser, etc. [16–20,27,28]. All the various systems can have improved axial and frequency resolution by using different channels and multirate signal processing.

Finally, from an engineering standpoint, the most significant results are the improvements in axial resolution and maximum depth measurement without increasing the signal bandwidth and frequency resolution of a tunable laser. This is because of the factors that form a hard limit on the source bandwidth in conventional systems. Specifically, these include source limitations, transparency windows of the optical components, and power tolerance. Similarly, the frequency resolution is limited by factors depending on the tunable laser. For example, external cavity lasers require large cavities for a small free spectral range. These can only be improved through costly modifications to the hardware and the use of increasingly sophisticated detection schemes. Operation under the UPT framework bypasses all these hardware challenges without the use of exotic and expensive equipment.