FAINMAN YESHAIAHU (US)
GAUR PRABHAV (US)
ALSHAMRANI NAIF (US)
ALMUTAIRI DHAIFALLAH (US)
WO2008048263A1 | 2008-04-24 |
US20090219193A1 | 2009-09-03 | |||
US20120188555A1 | 2012-07-26 | |||
US20140125952A1 | 2014-05-08 |
CLAIMS: 1. A method for determining axial resolution or depth in an optical coherence tomography system configured for imaging an object having one or more surfaces, the method comprising: 5 scanning the one or more surfaces by projecting light from a tunable narrowband laser source into an interferometer to generate an interferogram while applying phase modulation to the projected light; and applying multirate signal processing to the interferogram to determine positional information for the one or more surfaces of the object. 10 2. The method of claim 1, wherein applying phase modulation comprises inserting a signal generator into a sample arm of the interferometer to apply phase modulation that is slow compared to the time taken to measure a single frequency. 3. The method of claim 1, wherein applying phase modulation comprises inserting a signal generator immediately downstream of the laser source to apply fast 15 modulation to increase a maximum unambiguous range, wherein the fast modulation repeats after every sweep frequency. 4. The method of claim 1, further comprising repeating scanning and applying for multiple iterations. 5. The method of claim 1, wherein applying multirate signal processing 20 comprises defining multiple channels within the interferogram and combining the multiple channels in a frequency domain to increase time domain resolution. 6. The method of claim 1, wherein applying multirate signal processing comprises defining multiple channels within the interferogram and interleaving the multiple channels to increase frequency resolution. 25 7. A method for measuring one or more of axial resolution and depth of an object using an optical imaging system, the method comprising: applying phase modulation while scanning the object by projecting light from a tunable narrowband laser source into an interferometer to generate an interferogram, wherein the phase modulation changes resolution and depth parameters within the 30 imaging system; and applying multirate signal processing to the interferogram to determine positional information for the object. 8. The method of claim 7, wherein applying phase modulation comprises inserting a signal generator into a sample arm of the interferometer to apply phase 5 modulation that is slow compared to the time taken to measure a single frequency. 9. The method of claim 8, wherein the optical imaging system is a swept source optical coherence tomography (SS-OCT) system. 10. The method of claim 7, wherein applying phase modulation comprises inserting a signal generator immediately downstream of the laser source to apply fast 10 modulation to increase a maximum unambiguous range, wherein the fast modulation repeats after every sweep frequency. 11. The method of claim 10, wherein the optical imaging system is a Light Detection and Ranging (LiDAR) system. 12. The method of claim 7, further comprising repeating scanning and applying 15 for multiple iterations. 13. The method of claim 7, wherein applying multirate signal processing comprises defining multiple channels within the interferogram and combining the multiple channels in a frequency domain to increase time domain resolution. 14. The method of claim 13, wherein the optical imaging system is a swept 20 source optical coherence tomography (SS-OCT) system. 15. The method of claim 7, wherein applying multirate signal processing comprises defining multiple channels within the interferogram and interleaving the multiple channels to increase frequency resolution. 16. The method of claim 15, wherein the optical imaging system is a Light 25 Detection and Ranging (LiDAR) system. 17. An assembly for determining axial resolution or depth in an optical imaging system configured for imaging an object having one or more surfaces, the assembly comprising: a tunable narrowband laser source; an interferometer configured to generate an interferogram at a detector using light from the laser source; a phase modulator inserted within an arm of the interferometer; and a multirate filter bank configured for processing the interferogram to determine 5 positional information for the object. 18. The assembly of claim 17, wherein the phase modulator is inserted into a sample arm of the interferometer, and wherein the phase modulator is configured to apply slow modulation to the light to improve axial resolution in a length domain. 19. The assembly of claim 18, wherein the optical imaging system is a swept 10 source optical coherence tomography (SS-OCT) system. 20. The assembly of claim 17, wherein the phase modulator is inserted downstream of the laser source, and wherein the phase modulator is configured to apply fast modulation to the light to increase a maximum unambiguous range of detection. 21. The assembly of claim 20, wherein the optical imaging system is a Light 15 Detection and Ranging (LiDAR) system. 22. The assembly of claim 17 wherein the multirate filter bank is configured to define multiple channels within the interferogram and combine the multiple channels in a frequency domain to increase time domain resolution. 23. The assembly of claim 22, wherein the optical imaging system is a swept 20 source optical coherence tomography (SS-OCT) system. 24. The assembly of claim 17, wherein the multirate filter bank is configured to define multiple channels within the interferogram and interleave the multiple channels to increase frequency resolution. 25. The method of claim 24, wherein the optical imaging system is a Light 25 Detection and Ranging (LiDAR) system. |
Finally, combine the integrals to obtain an integral transform of the sample reflectivity over the entire domain. 5 ) The Fourier transform relationship then becomes apparent. 10 In this case, the conjugate variables are the time it takes for light to be reflected from different depths of the sample, and the effective frequency which is tuned via the delay line (or more precisely, each point of the effective frequency is mapped to tuning the delay line at a constant rate). Note that this is the simplest conceptual case of the effective frequency. The method can be simplified by using a chirp in the delay. 15 The total reflected field as a function of time (and ultimately depth) can be recovered by an inverse transform. Nyquist Sampling Constraints The time integral in the continuous case derived above goes over all time, which 5 is unphysical. In reality, the measurement limits will be truncated and the measurement points will be digital and discrete rather than continuous. This discreteness will impose Nyquist limits on the “bandwidth” (measurement depth) and measurement resolution. These may be determined in a manner directly comparable to the Nyquist theorem of signal processing. As the derivation is well known to those of skill in the art, only the 10 main results are presented here. The effective frequency interval Δf and sampling rate Δt are related as follows: 4) Similarly, the maximum effective frequency translates into a sampling rate of: 15 These constraints can be significant depending on the type of wave used in the device. Since the wave velocity appears in the expressions, there will be a major difference between a fast wave, e.g., light, versus a slow wave such as sound. To provide a sample illustration, a thermo-optic phase shifter can be used. This 5 device can be mapped to the effective frequency of the proposed OCT device and will help provide a sense of what performance can be expected and what applications are feasible. The mathematical conventions used in the following discussion vary somewhat from those listed previously. Specifically, t = time, and T = temperature. 10 A basic thermo-optic phase shifter is a length of waveguide overlain with a heater, which is used to change the refractive index. The first order phase response of such a device is as follows: 6) 15 We can substitute this into the Nyquist conditions for the device to obtain analytical expressions for the resolution: Similarly, the per point measurement time and number of measurement points are: A particularly attractive feature of these results is that the resolution can be improved be increasing the length of the thermo-optic phase shifter. 5 There are a number of general intuitions that can be developed from these expressions: • The faster the time derivative of the phase change, the better the resolution. • The longer the phase shifter, the better the resolution. • Total measurement time is determined by the maximum depth you want to look 10 into the sample. • The maximum sampling rate is determined by the maximum depth you want to look into the sample. • Slower waves will have longer measurement times and higher number of sampling points. 15 Tables 2 and 3 below provide sample results, respectively, for a silicon waveguide phase shifter (SOI) (operating thermally) and a high speed fiber optic phase shifter (based on lithium niobate (LiNO 3 ). Silicon-on-Insulator (SOI) Waveguide t 20 Fiber/LiNO 3 Waveguide 5 TABLE 3 It is worth noting that in conventional signal processing, a known time signal is used and transformed to work with the frequency components. In this method, we do the reverse, namely start with a signal of modulated frequency and transform it to work with the time components. This has important differences that defy DSP intuition -- rather 10 than being concerned with minimum sampling rate as in conventional DSP, the present focuses on maximum sampling rate. This is easy to explain physically. For the interferogram to include points deep in the sample, we must measure long enough for light to travel to that point and back. The total measurement time is this round trip travel time multiplied by the number of points. This is the minimum measurement time. It is 15 interesting because the alternative of measuring the time response would require extremely fast detectors and short integration times for short distances. Thus, the same advantages of the Fourier transform method are similar to that of spectroscopy. Biomedical, geomorphic, remote sensing, et cetera. This invention can be used in all the same applications as conventional tomography. Additionally, the narrow spectral 20 sensitivity can be useful for several new applications such as creating tomographic profiles of molecules if the operating wavelength is tuned to one of their absorption spectra peaks. The new method should also be able to be made more resistant to noise, which should extend the maximum range of the measurement. Example 1: Implementation of UPT 25 FIG.3A illustrates a base case for an exemplary implementation of UPT without phase modulators, which resembles the Swept Source OCT in single mode fiber. The normalized interference term measured at the photodetector is given by: 9) where ^ത^( ^^) consists of reflection and transmission coefficients in the ^^-th surface present at a particular position with its magnitude determined from Fresnel equations and the transmitted optical power to the object, ^^ is the total number of surfaces present, and each value of ^^ represents a frequency in laser sweep. The negative arguments of the 5 summation represent the conjugate part of the interference. ^ത^( ^^) can be obtained by taking the Discrete Fourier Transform (DFT) of ^^ ^^ ^^ ^^ ^^ . The position, ^^, of the non-zero elements of ^ത^( ^^) give the optical distance of the surface, while their magnitude can be used to determine the optical index of the layer which in turn can be used to extract the true physical distance. 10 Next, we add a phase modulator to the sample arm and use a signal generator to introduce a phase modulation ϕ( ^^). FIG. 3B shows a first variation on the base case in which a phase modulator is added in the sample arm. A waveform generator (not shown) is used to give slow modulation which assists in improving the resolution in length domain. Assuming that the modulation is slow compared to the time taken (time 15 bin) by the laser to measure a single frequency, the DFT (transformation from ^^ to ^^) of the interference term ( ^^ ^^ ^^ ^^ ^^) is given by, where | ^^| ଶ ^^( ^^) = ^ത^( ^^) for ^^ > 0 and | ^^| ଶ is the transmitted optical power to the object. 20 ℎ( ^^) = ^[exp( ^^ϕ( ^^Δ ^^))], where ^[.] is the DFT function and Δ ^^ (time bin) is the time taken to measure the power at a single frequency. Eq. (20) can be truncated to ^^ > 0 r egime and then normalized by | ^^ |ଶ to give ^^( ^^). ) Eq. (21) resembles a filter ℎ( ^^) applied to ^^( ^^) in a linear system with convolution in 25 length (i.e., time) domain. A transfer function can then be defined in frequency domain, and this provides the opportunity to apply digital signal processing on the depth information. FIG. 3C illustrates a second variation in which a phase modulator is added to the base case just after the tunable laser. A signal generator (not shown) is used to apply fast modulation which assists to increase the maximum unambiguous range. The fast modulation repeats after every sweep frequency, i.e., it is periodic with At. It can then be shown that the interference term is given by, where H(i) is the autocorrelation function of the phase modulation. Eq. (22) can be written in the convolution form.
Here the Pintf has been replaced by u(n ) and variable k is replaced n. d(n) = f [a(t)], and h(n ) = f[H(i)] determines the filter coefficients. Note that here the convolution is in frequency domain, as opposed to previous case. Hence, the transfer function can be implemented in length domain.
Eq. 21 and Eq. 23 represent a linear system in which multirate signal processing can be used to increase the resolution of the system as shown in FIG. 3D, which provides a schematic for the working of UPT and the required post-processing in the filter bank form. The horizontal dashed lines indicate photodetection. The physical system of UPT corresponds to elements on the left-hand side (pre-photodetection process) of the filter bank scheme of FIG. 3D. The right-hand side (post-detection process), i.e., the “Signal Processing” elements are 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 frequency domain for the first case (FIG. 3B) while in length (i.e., time) domain for the second case (FIG. 3€). In FIG. 3D, thai) represents the detected signal, H,(z) is the analysis filter, and Fi{z) is the synthesis filter in the i th channel, n is the time vector in the first case, and the frequency vector in the second case. Hi(z) is implemented optically using a phase/intensity modulator while Fi(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. Note that while Eq. (21) and Eq. (23) appear to be the same, the former is convolution in length domain while the latter in frequency domain. Hence, the interpretation of transfer function will be domain inverted compared to the previous case. Taking the Z -transform results in 5 ^ ^( ^^) = ^^( ^^) ^^( ^^) (24) By performing multiple scans, axial resolution is improved in the first case (FIG. 3B) while maximum depth is increased in the second case (FIG.3C). The first case may arise when bandwidth of laser is limited while second case may arise when frequency 10 resolution is limited. 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 digitally. In Example 2 below, we demonstrate a 2-channel filter bank for both the cases. For slow modulation we use a linear phase modulation, which is effectively a ^^ -1 15 transfer function in Z domain. As the maximum bandwidth of the laser is usually limited, it may cause the resolution in length domain (axial resolution) to be less than desired, resulting in under sampling. Let the laser have a bandwidth that is ^^ times smaller than required so that the axial resolution is down sampled by a factor of M from the desired ^^ ^ . This can be depicted by a block diagram as shown in FIG.3D. The block 20 diagram resembles a single channel of M channel filter bank. If we make the measurement M times with M different synthesis filters ( ^^ ^ ), the ideally sampled signal can be reconstructed using analysis filters ( ^^ ^ ). On the other hand, Eq. (24) corresponds to a transfer function block with the Z-transform in length domain. The same multirate filter bank analysis can be used to deal with under sampling problem. In this case, 25 downsampling is in frequency domain as the resolution of sweeping laser is limited. Hence, the filter bank can be used to reconstruct the signal with increased frequency resolution and detect object at greater depth without aliasing. For demonstration purposes, we discuss the situation when M=2. The perfect reconstruction (PR) of ^^( ^^) is said to be achieved when ^^( ^^) = ^^( ^^ − ^^), i.e., ^^( ^^) is 30 perfect replica of ^^( ^^) and is with a shift of ^^ points. This removes both aliasing and distortion from the reconstruction. For two channel filter bank, the PR condition is given by ^ ^^^( ^^) 2 ^^ି^ ^^^(− ^^) (15) ^ ^ ( ^^)൨ = ^− ^^ (− ^ ൨ ^ ∆( ^^) ^ ^) where ∆( ^^) is given by 6) The simplest implementation of this is the lazy filter bank, in which the first channel is detected without any modulation while the second channel shifts the input by one time step. ^ ^^ ( ^^ ) = 1 ; ^^^ ( ^^ ) = ^^ ି^ ∆( ^^) = −2 ^^ ି^ ; ^^ = 1 (37) ^ ^^( ^^) = ^^ ି^ ; ^^^( ^^) = 1 5 For the first case, గ^ ^ ^. Let total time of scan be ^^ = ^^Δ ^^. As only half of required number of points are scanned the phase modulation should be 8 ) This corresponds to a linear phase modulation from 0 to ^^ phase shift in time ^^ and thus the voltage provided by signal generator vary from 0 to ^^ గ in this time. For large ^^ (of the order of 10,000), the assumption that ^^ ( ^^ ) varies slowly in Δ ^^ holds. 10 For the second case, we assumed that the frequency of the modulation is comparable to ^^/ ^^ ^ , which can be of the order of 10s of megahertz. It is difficult as well as cost ineffective to produce arbitrary waveforms at such high frequency. The easiest modulation is sinusoidal, produced using an RF signal generator. e xp ( ^^ ^^( ^^) ) = exp ( ^^ ^^ sin ( 2 ^^ ^^^ ^^ )) (5) 15 ^^ ^ is the sinusoidal phase modulation frequency and ^^ is its amplitude. This gives 0) Where ^^ ^ is the Bessel function of first kind and zeroth order. The corresponding filter coefficients and transfer function can be calculated from Eq. (30). To carry out the filter 20 bank analysis, it is important that Δ( ^^) is invertible. For this purpose, the amplitude ( ^^) and modulation frequency ( ^^ ^ ) can be engineered so as to make the analysis filter stable. Alternatively, other types of waveforms can be used, but that would require high speed analog waveform generators. For fast modulation, sinusoids are the only cost-effective option. The synthesis filters can be calculated from the perfect reconstruction conditions of filter banks, as 5 given by Eq. 31 and Eq. 32, where ^^ is an integer and corresponds to the delay due to signal processing. 1) ∆ ( ^^ ) = ^^^ ( ^^ ) ^^^ ( − ^^ ) − ^^^(− ^^) ^^^( ^^) (32) FIG. 3E illustrates samples of reconstruction signals formed by combining two channels (M=2) in the frequency domain for the first and second cases. The dots represent the effective frequency measured by both channels. In the first case (top), the results of both 10 channels (the two channels are separated by the dashed line) contribute to extending the bandwidth in the frequency domain, thus improving time domain resolution. While in the second case (lower), the two 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 15 represent a physical situation. Example 2: UPT Results The following results demonstrate the working principal of the inventive UPT approach under the universal framework. We then experimentally demonstrate how various modulation schemes provides the opportunity for novel detection and post- 20 processing strategies. The laser used for performing all the experiments was the 81608A Tunable Laser Source from Keysight Technologies (Santa Rosa, CA, US) which can give frequency resolution up to 0.1 pm and has a narrow linewidth (<10 kHz). The photodetector is the 81635A Dual Optical Power Sensor, also from Keysight. The phase modulator employed25 in both the cases is the Thorlabs Lithium Niobate 40 GHz phase modulators (LN27S- FC). The linear waveform is produced using Keysight B2960 series power supply while the sinusoidal signal is generated using Keysight MXG series 6 GHz Analog Signal Generator. The entire setup (excluding objects) is built upon SMF-28 single mode fiber. To demonstrate how to increase the axial resolution using the inventive UPT, we 30 used two microscope slides as objects (FIG. 4A), 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 ^^ ^^ ^^ ^^ ^^ ^^ ≈ 1.5 , and the refractive index of air is taken to be ^^ ^^ ^^ ^^ ≈ 1.0. This creates a situation where the 5 bandwidth of 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 ( ^^ ^^) of 2.4 mm, while the normalized distance between the slide surfaces is 3 mm. Base case: To establish the base case without modulators, the measured 10 interference pattern on the photodetector as a function of frequency sweep is shown in FIG.4B after using an offset equal to its mean. The bandwidth is 5 nm at a wavelength of 1.55 µm and resolution ( ^^ ^^) is 0.3 pm. FIG.4C provides the Fourier transform (FT) of the interference pattern. The four larger peaks clearly distinguish the four surfaces and predict the distances between them. The smaller peaks (barely visible) are due to 15 autocorrelation of the sample arm signal in the interferogram and can be removed by balanced photodetection. 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. 5A. We create a situation where the bandwidth of laser is not high enough to clearly distinguish the two 20 surfaces of the slide. The laser sweeps a bandwidth of 1 nm with 0.2 pm resolution. This results in an axial resolution ( ^^ ^^ ) of 2.4 mm, while the normalized distance between the slide surfaces is 3 mm. This measurement referred to as unmodulated signal, (curve Ch 0 in FIGs. 5B and 5D) corresponds to conventional SS-OCT, but the surfaces are barely resolvable due to limited bandwidth of the tunable laser source. Next, we use a 25 waveform generator to provide a linear phase modulation to the sample arm, as shown in FIG.5C. The interferogram that is obtained can distinguish the surfaces better or worse depending on the position of the surfaces, but the resolution ( ^^ ^^) remains the same (FIG. 5D, lower curve). The two signals are combined and treated as two different channels of a multirate filter bank (FIG.5E). This improves ^^ ^^ from 2.4 mm to 1.2 mm. The surfaces 30 can be distinguished more easily now, and their positions are known twice as accurately as before. Thus, 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. Further, note that multiple channels can be used to improve the axial resolution even more. This is a highly significant result, as it provides the best path to ultrahigh resolution devices by a large margin. Increasing Maximum Depth: For a simple demonstration on how to increase the maximum unambiguous depth, we again use the microscope slide with a mirror behind it 5 (see FIG. 5A). We define a balanced point which is the zero position in the length domain and physically represents the point where delay of reference signal is equal to that of 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. FIG.6A provides a schematic depicting the voltage applied to 10 phase modulation as laser frequency is tuned. Here we consider the situation when the resolution of laser sweep is limited to 0.4 pm, which corresponds to maximum unambiguous depth ( ^^) equal to 3 m, and the position of the target (mirror) is beyond it. We first measure this object with 50 MHz sinusoidal phase modulation as shown in FIG. 6B, upper curve. The peak for mirror appears at 2.62 m which is an aliasing artifact that 15 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 much further distance, otherwise the original peak gives the correct position. Thus, we use adaptive phase 20 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 nonlinearity, variable ^^ ^^ , RF impedance mismatch, etc., but this method only requires the knowledge of zero crossings of the transfer function. In our case, an 80 MHz 25 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. 6B, lower curve). Therefore, we can conclude that position of the target is actually at 3.41 m. We also demonstrate in FIG.6C 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 30 80 MHz measurements can be treated as two different channels in a multirate filter bank and combined, as shown in FIG. 6D, to give a graph that has twice the maximum unambiguous range than individual channels. This method will perform better for more complex objects but also requires an accurate structure of the analysis of the transfer function produced by phase modulation. This demonstrates that distances up to 6 m can be measured by using laser sweep resolution which corresponds to only 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 5 can be performed which will require lesser number of channels and can still measure more distant positions of the target. This is a highly significant result for the same reasons. The above discussion demonstrates UPT as a universal method to measure depth and position of objects at various distances by adjusting the laser sweep frequency and 10 bandwidth. The inventive UPT framework provides an alternative approach to improve the resolution and/or depth performance through the use of slow and/or fast modulation of the optical carrier. This approach requires only a simple phase modulator and waveform/signal generator which are more economical and easier to integrate in the system. By making multiple scans, ultrahigh resolutions can be achieved both in 15 frequency and length domain. The only drawback in this method is the extra time required 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 which have promising specifications of low ^^ ^^ and high RF bandwidths. 20 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. Given enough channels with appropriate modulation, they can be theoretically combined by multirate signal processing to get a reconstructed signal with arbitrarily high resolution. 25 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 constant frequency difference, the Fourier transform will be noisy when making 30 a measurement near or beyond the Nyquist limit. We observe this in the second case where the noise floor is due to the improperly spaced frequency values. The power on the photodetector comprises of 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 presence of autocorrelation term, which can also be observed in the base case 5 (FIGs. 4A-4C). A better way to remove the other two terms would be to use balanced photodetection, where subtracting the two interference powers cancels out the two unnecessary terms. To implement synthesis filters, it is essential that Δ( ^^) as described in Eq. (32) is invertible. This is not the case when sinusoidal phase modulation is given to only one 10 channel with no modulation on the other. Thus, for second case, both channels should have sinusoidal modulation. Other modulation shapes can also be used if the speed of waveform generator permits. 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 the system similar to SD-OCT and using 15 optical modulation to virtually improve the bandwidth of the source and frequency resolution of the spectrum analyzer. 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 tunable laser. This is because many factors form a 20 hard limit on the source bandwidth in conventional systems. Specifically, these include source limitations, transparency windows of the optical components, and power tolerance. Similarly, frequency resolution is limited by factors depending on the tunable laser, for example, external cavity lasers require large cavities for small free spectral range. Operation under the UPT framework bypasses all these hardware challenges 25 without the need for exotic and costly equipment. Method for Optimizing Adiabatic Tapers A review of the well-known results of basic coupled mode theory is the starting point for this disclosure. Here, the waveguides are assumed to be composed of nonmagnetic and dielectric materials, which simplifies the mathematics while still 30 illustrating the fundamental concepts underlying the method. The most general form of the derivation that relaxes these conditions is found using the approach of Koegelnik’s well-known coupled wave theory (See, e.g., H. Koegelnik, "Theory of Dielectric Waveguides," in Integrated Optics Topics in Applied Physics, vol. VII, T. Tamir, Ed., Berlin, Heidelberg, Springer, 1975, pp. 66-79, which is incorporated herein by reference.) In this formalism, the electric and magnetic fields are expressed as a combination of eigenmodes that are determined by Maxwell’s equations combined with the boundary 5 conditions imposed by the waveguide geometry and composition. The effect of a perturbation to the waveguide permittivity is to transfer energy from one mode to another. For single frequency fields: 3) 10 Add the two equations to obtain the basis for the coupled mode relationship. 4) Integrate over the volume of space and Apply Gauss’ theorem to the left side: 5) 15 Consider the limit in which the transverse integral is taken infinitely far away in the plane perpendicular to propagation, and that the integral in the direction of propagation is infinitesimally small. Taking z as the propagation direction, since physical fields vanish at infinity the integral reduces to: 6) 20 Next, apply the fundamental theorem of calculus to write how the fields vary with one another in the propagation direction: S Next, expand the fields in terms of the mode expansion of the unperturbed system. Note that the perturbed fields will have variable amplitude coefficients since they are not in their natural basis. 8) 5 The permittivity perturbation causes coupling between a waveguide modes. To see this, using the above mode convention, substitute the following fields into the field coupling equation. For the unprimed field we take one incident waveguide mode, and for the primed field take the unknown projection of that field in the perturbed system. Similarly, use Ohm’s law to express the free currents in terms of the fields and 10 conductivity. S Note that may be a technical issue concerning the behavior of the z-components of the fields arising from the orthogonality condition that requires the perturbations to be 15 small for this expression to be accurate. This can be trivially satisfied as adiabatic tapers are inherently gradual. In other contexts, however, large perturbations can be handled by a slight modification to the portion of the coupling coefficient arising from the longitudinal fields. Next, use the orthogonality condition to simplify the left hand side:
0) Isolate the amplitude and simplify: 1) 5 Finally, express in terms of coupling coefficients: 2) By inspection, the form of the most general coupling equation is suitable for application of the optimization method described below. The fields may be decomposed in terms of the mode amplitudes A l : E = ^ A l E l ( x, y ) exp ^ ^ i ( ω t − β l z + ϕ l ) ^ ^ l (43) The mode orthonormalization is chosen so that the modal fields carry a power P: 4) The differential equations that govern the mode amplitudes are: 5 5) The strength of the coupling is governed by coefficient κ, which is a function of the dielectric perturbation and the extent to which it overlaps with the interacting modes. The sign difference between the equation depends on whether the modes are co- propagating or counter-propagating. 10 Essentially, the inventive method devises an adiabatic taper by formally minimizing the mode amplitudes along the taper. Since all possible mode interactions are governed by this equation, this will result in a taper that is optimized in the most general sense. In a tapered waveguide, the strength of the coupling coefficient will become 15 variable along the taper, as well as the mode properties. Notably, the variation of the propagation constant is equal to the self-coupling coefficient (e.g. Δβl = κll ). Following Eq. (45), for a z-dependent taper the mode amplitudes may be expressed as the integrals of the form: 6) 20 Minimizing the amplitudes results in a condition of the form: ^ ^ ^ ^ ^ ^ d − lm z ^ κ ( ) A ( z ) exp i ^ β ( z ) − β ( z ) ^ z + i ( ϕ − ϕ dz ^ ^ ^ dz m { ^ l m ^ l m ) } ^ ^ = dA l ( z ) = ^ ^ 0 ^ − ^ κ lm ( z ) exp { i ^ ^ β l ( z ) − β m ( z ) ^ ^ z + i ( ϕ l − ϕ m ) } dA m ( z ) ^ ^ (47) ^ ^ ^ d β z d β ^ ^ ^ ^ ( ) ( z ) z l − m ^ ^ ^ − κ z A z i ^ ^ dz dz ^ ^ ^ exp i ^ β z β z ^ z i ϕ ϕ dz ^ ^ l ( ) ( ) ^ ^ ^ ^ ^ ^ { ( ) − ( ) + ( − ) } ^ ^ m m ^ ^ l m ^ l m ^ ^ ^ ^+ ^ ^ β l ( z ) − β m ( z ) ^ ^ ^ ^ ^ ^ ^ This second term vanishes identically from the condition that dA = 0. Otherwise the remaining terms must vanish to meet the adiabatic condition. Applying this condition to the coupling coefficient will enable us to determine the optimal permittivity taper. The 5 solution of this equation is straightforward, however in the most general case it will need to be performed numerically (because the propagation constants generally can’t be solved analytically). The variation of the coupling coefficient throughout a taper occurs not only through dielectric perturbation, but also through the evolution of the normal mode field 10 profiles (represented below as the correction terms ΔEl and ΔHm). For a z-dependent taper the coupling coefficient may be expressed as: dy If we consider only a small step in the taper, then all the correction terms and the 15 perturbation will be small. This results in a first order expression for the coupling coefficient as: dy Adiabatic transitions will be gradual, so the first order approximation will be 20 valid for the structures under consideration. This expression is perfectly amenable to the type of optimization we wish to perform. This solves the problem for the case of a continuously variable perturbation, however it is straightforward to extend to all the other important cases (such as waveguide bending or width tapering) using the technique of conformal mapping. The technique of conformal mapping reformulates a waveguide that is curved within the x-z plane into an equivalent straight waveguide in the conformal u-v plane (along lines of u = constant). The new mapping is compatible with standard numerical mode solvers, although it comes at a price of complicating the refractive index profile in 5 the plane transverse to propagation. The problem of converting a graded index change to an equivalent taper profile may be arrived at in a manner similar to the derivation of M. Heiblum and J. H. Harris in "Analysis of Curved Optical Waveguides by Conformal Transformation," IEEE Journal of Quantum Electronics, Vols. QE-11, pp. 75-83, 1975, incorporated herein by reference, but performed in reverse. 10 Finally, the case of waveguide bending is the easiest to handle, since an equivalent conformal expression of the permittivity of a curved waveguide can be found in literature. The broader implications of this result are also highly significant. While a primary focus of this approach is on optical applications, the method itself relies on 15 general coupled mode theory, which can be applied to essentially any wave phenomenon. Since the derivation involves the well-known process of minimization, it can be extended in the same ways, such as through the incorporation of constraints using Lagrange multipliers. Proper tapering is critical for all integrated photonics. Similarly, given the 20 generality of the underlying proof, the approach described above is applicable to any phenomenon that can be described using coupled mode theory. This includes wave phenomenon in general, not only physical but abstract (such as traffic flow waves). Low Reflection Optical Couplers Conventional grating designs often rely on partially-etched gratings or binary 25 blazed gratings in order to enhance the coupling efficiency while attempting to reduce the reflected light. However, the impedance mismatch in such designs remains an issue. In some embodiments of the inventive approach, a metamaterial structure (tapers) is used to facilitate the adiabatic transition of the refractive index (n), resulting in an impedance matching grating. The inventive approach is effective in suppressing the reflections that 30 frequently occur due to the impedance mismatch when coupling light into/out of a photonic chip through integrated optical couplers (I/O’s). The inventive design is a combination of both binary gratings and metamaterial structures (tapers) that are optimized as described above. The working principle of the inventive scheme employs features of diffraction gratings, where uniform gratings are used to couple light into and/or out of a photonic 5 chip. The key improvement involves the optimization of metamaterial structure (tapers) that were introduced/fabricated at the end of the binary gratings and at the interface of the waveguide as shown in FIG.7. The optimized tapers enhance the adiabatic transition of the refractive index (n), suppressing the reflections that frequently occur at the interface(s), as a result of the 10 abrupt change in the refractive index. Furthermore, this transition benefits from the optimization of parameters as described in the adiabatic taper discussion above. The inventive approach is based on the mathematical relationship: 0) 15 where n is the index of refraction, I is the initial value of the individual metamaterial structure (tapers), and F is the final value of the individual metamaterial structure (tapers). FIG. 8 provides plots of simulated spectral S parameters of optical I/O couplers 20 with (lower) and without (upper) the inventive impedance matching structure. When incorporating photonic chips into optical transmission assemblies, I/O couplers are essential elements in the majority of designs. The inventive couplers can be implemented in any photonic chip for any application (telecommunication, biomedical, geomorphic, remote sensing, etc.). 25
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