Design and Simulation of Ultra low loss Spiral Delay line for Integrated Optical Coherence Tomography

We report a design of 55 cm long spiral delay line at 850 nm wavelength. Propagation characteristics were simulated and analysed with a fully vectorial EigenMode Expansion (EME) method. Bending losses were calculated and analysed for the optimization of minimum bending radius for reported structure. We have also simulated and analysed excess loss for a bandwidth of 200 nm at 850 nm operating wavelength. An excess loss of less than 0.2 dB is reported for the structure over the entire bandwidth range. Reported structure is easy to fabricate, ultra-low loss and also broadband over 200 nm with a footprint of only 6 × 6 sq. mm. So, this design will improvise the reference arm section and will enhance the depth scanning of integrated optics based optical coherence tomography systems.


Introduction
Photonic integrated components have recently gained wide interest in medical and industrial applications (Singh et al. 2018;Srivastava et al. May 2020). One such biomedical application is optical coherence tomography (OCT). OCT is a non-contact imaging technique which works on the principle of low coherence interferometry (Fercher et al. 2003). The generic OCT system consists of a broadband monochromatic light source which is directed on to a 2 × 2 fibre optic coupler in order to split the incident power evenly into the sample and reference arm. Light from the reference arm is sent upon a reference delay and redirected back whereas light from the sample arm is sent to a scanning mechanism which is structured to focus the beam on the sample. The recombined light from the sample and reference arms gives rise to interference pattern only when constructive interference takes place. The scanning depth of the sample is controlled by the delay in the reference arm for deep scanning (Izatt and Choma 2008).
Optical delay lines (ODL) are important components which are used to provide a timed delay in a system. When these ODLs are implemented using integrated chips they show various advantages such as reduced cost, size, weight and power consumption (Shahoei and Jianping. 2014). They have various applications such as optical beam forming networks for antenna arrays, imaging processing units, radio on fibre links and devices, OCT systems and sensing units. There are different techniques to obtain optical delay in photonic integrated circuits such as integrated grating (Sharma, et al. 2020) delay lines(IGDL) wherein the spatial structure or refractive index parameter undergoes a periodic change mainly by modulating its structural parameters such as waveguide width and height or by modulating its effective refractive index (Zhou et al. 2018), photonic crystal waveguides (PhCWs) based delay lines in which the photonic crystals consist of periodic optical micro and nano structures resulting in manipulation of light propagation while maintaining its efficiency (Watcharakitchakorn and Silapunt 2018), micro ring resonators(MRR) having the advantage of compact size and scalability consists of cascaded optical ring resonators for obtaining tunable delay lines mainly in side coupled integrated space sequence of resonators (SCISSSOR) or coupled resonator optical waveguides (CROW) fashion and optical waveguide spiral delay lines which are simple length dependent delay structures that can be implemented using various material technologies such as Indium Phosphide (InP), silicon nitride ( Si 3 N 4 ) and silicon on insulator(SOI) (Stopinski et al. 2013).
While PhCWs offer wide bandwidth but incur high propagation losses, MRRs offer small footprint but usually have narrow bandwidth and bandwidth delay product is rather small. IGDLs offer compact resonant delay lines but fabrication is rather challenging and insertion loss is also high. Spiral delay lines are the most promising candidate as they offer optimal area utilization with low loss, large bandwidth and does not require any external control. Moreover, these delay lines can be easily integrated with other components.
Various components in the OCT system can be integrated on a chip so as to miniscule the existing fibre optic based OCT system. Numerous research groups have designed various components on an integrated optics platform for OCT systems. Yurtsever et al. reported a Mach-Zehnder interferometer (MZI) based integrated circuit consisting of numerous y-splitters and 190 mm reference arm using Si 3 N 4 /Si O 2 waveguides at 1320 nm. Upon its realization, the system was used to scan different layers of finger tip (Yurtsever et al. 2014). Cascaded MZI with tunable couplers-based delay line with delay tuning range of 124 ps was reported at 1550 nm central wavelength (Waqas et al. 2018). Recently cascaded MRRs on Si 3 N 4 platform was proposed. These structures showed a maximum delay tuning range of 394.6 ps @ 2 GHz bandwidth using continuous tuning method (Lin et al. 2019).
In this paper, a spiral reference ODL for depth scanning in an integrated optics-based OCT system is reported. This reference delay line is structured for 850 nm central wavelength using Si 3 N 4 and SiO 2 waveguides. The proposed structure is compact, low loss, broadband and long enough to provide deep scanning for OCT systems. This structure was modelled for fully vectorial (FV) mode type and was found to be broadband over 200 nm bandwidth with an excess loss of 0.17 dB. The eigenmode expansion method is used for calculations which are fully vectorial and fully bidirectional which overcomes the drawbacks of semi vectorial analysis and provide accurate results. Comparative analyses for FV mode in terms of different spiral length have also been calculated for TE like and TM like polarization. The reported compact reference delay line upon realization along with integrated directional coupler (Sharma et al. 2019a, b) will completely replace the bulky fiber-based interferometer in a Fourier domain OCT system.

Waveguide Design and Modelling
The reported spiral structured optical delay line is designed for integrated optics OCT system at 850 nm central wavelength to compensate for the time delay required in the reference arm. The waveguides used in our structure are based on Si 3 N 4 and Si O 2 technology called Triplex (Blumenthal et al. 2018). We preferred channel buried waveguide geometry to achieve low propagation losses and better mode confinement. Smaller bending radius is required to miniaturize the device, so we encapsulated Si 3 N 4 between SiO 2 layers so to have a medium refractive index contrast between Si 3 N 4 (1.98) and SiO 2 (1.45). Schematic of the waveguide structure is shown in Fig. 1.
Reference delay lines for OCT prefer single mode operation as multimode might alter its axial resolution and sensitivity. To optimize the width and thickness of the waveguide core for single mode operation, we have done modelling of the waveguide using a fully vectorial Finite Difference Mode (FDM) solver of the Fimmwave simulation tool (http:// www. photo nd. com/). Effective indices were calculated at the lower spectral end i.e. 750 nm of the broadband structure. The calculated effective indices for TE and TM mode as a function of waveguide core width (W) and thickness (H) are shown in Fig. 2.
It has been observed from Fig. 2a that for waveguide core width below 1.2 µm only fundamental TE and TM mode will propagate. For various width variations, waveguide core height was kept at a constant of 0.07 µm. Also, from Fig. 2b it can be viewed that fundamental TE and TM modes will propagate for waveguide core thickness below 75 nm. Similarly, for waveguide thickness variations, waveguide core width was kept at a constant of 1 µm. So, after analysis of results, the width and thickness for single mode operation are optimised as 1 µm and 70 nm respectively.
Once we have achieved maximum values of waveguide core width and thickness, transmission below these values is an essential factor which needs to be contemplated. For this we have considered values of W ranging from 0.7 um to 1.4 µm and H ranging from 0.05 to 0.09 µm. A plot of waveguide W and H vs power efficiency at output is shown in Fig. 3. It can be interpreted from the graph that the best suitable values for maximum transmission below cut-off will be for W greater than 0.9 µm and H greater than 0.065 µm. So, after analysis of results, the W and H for single mode operation with transmission efficiency are optimised as 1 and 70 nm respectively.

Spiral Reference Section Design
The reported spiral delay line has been constructed using an interleaved Archimedean spiral with continuous change of bending radius. General descriptive schematic for spiral reference delay line at 850 nm is shown in Fig. 4. The reported structure covers an area of approximately 6 × 6 sq. mm.
The structure consists of 2 sets of half circles of which one set is used to bring light into the centre and other outwards towards the output. These inwards and outwards semi circles are connected to a S bend shaped structure whose radius is kept adiabatically large enough so as to have loss-less transmission and change of mode location between CW and CCW spiral waveguides. Design parameters and propagation characteristics were modelled and  optimized using Eigen Mode Expansion (EME) method of Fimmprop. EME is a fully vectorial and fully bidirectional technique that relies on a scattering matrix approach wherein all reflections are taken into account. The method is correct for an indefinite number of modes. When a large number of modes are considered, this allows for the propagation of modes with wide angles and arbitrary precision (Chrostowski and Hochberg 2015). When travelling in a homogeneous medium, each mode is transmitted separately by multiplying by the complex propagation constant. Scattering parameters are utilised to link to the next component of the device. Propagation loss (PL) is calculated using a scattering matrix approach at individual joints which are connected together to form the complete structure.
Excess loss (EL) and Insertion loss (IL) have been arbitrarily used here to signify the loss in the spiral structure corresponding to the ratio of total input power to the power received at the output, expressed in dB. Fimmprop is integrated with Fimmwave and allows great flexibility for constructing complex structures. Circular bends with a bend section along the circular path were used for designing the structure. The structure began with a set of bending arcs having 180° waveguide arcs each for CW rotation. Each successive bending arc was set with a decreasing radius so as to form an inward spiral. A total of 30 sets of semi-circular bending arcs were used to form this inward spiral. Next an S-bend section was designed using two bending arcs of the same radius having + 180° and − 180° phase shift each. The bending section exiting the S-bend was then connected through a set of bending arcs having − 180° phase shift with increasing radius for all successive arcs.
The minimum bending radius is a very significant parameter as the overall length of the spiral is dependent on it. The minimum bending radius corresponds to the smallest radius in the spiral structure. For the optimization of minimum bending radius, we have calculated the transmission and excess loss over a range of 0.5 to 3.5 mm. Results obtained for variation in transmission and excess loss vs bending radius is plotted in Fig. 5.
It is evident from Fig. 5 that the transmission power variation is high below 1.29 mm. The transmission power shows little variation above 1.47 mm and so on. We have

Mathematical Calculations
The length of consecutive semi circles is then related to the length of minimum bending arc by following relation.
where A is the minimum bending arc length and dA is 10 µm and A2 is the bending arc length of a consecutive semi-circle from S bend.
The number of semi-circular arcs is mainly responsible for the overall length of the spiral. For this structure, the number of half circles are taken to be 30 i.e. 30 half circle are deployed both in CW and CCW direction and the relation of each arc length can be determined by following relation Once the length of each half arc is being determined, the overall length (L) of spiral structure can then be defined by following relation as

Spiral structure results
For a minimum bending radius of 1.47 mm, an overall delay line of 55.11 cm is designed. The relation between length of propagation across spiral structured delay lines vs power (1) A2 = (A × 2) + dA (2) A n = A n−1 + 10; for n = 3 to 30 Min. bending radius vs power at output obtained and loss can be seen in Fig. 6. The overall propagation loss across the length of propagation was 0.00047 dB.
The time delay can then can be calculated by following relation where t d is the time delay of the spiral delay line, n g is the group effective index and C is the speed of light. For a delay length (L) of 55.11 cm and effective group index ( n g ) of 1.62 (TE mode) and 1.51 (TM mode), a delay time of 2.35 and 2.19 ns is reported for this structure at 850 nm. Now in order to analyse the structure performance, certain parameters such as its bandwidth, normalized power at output and excess loss are to be determined. The reported structure has been scanned over 200 nm i.e. from 750 to 950 nm at 850 nm central wavelength and is found to be broadband over this bandwidth range. Also, at 850 nm central wavelength, a normalized power of 98.64% and an excess loss of 0.06 dB is obtained. A maximum excess loss of 0.18 dB is observed across 200 nm bandwidth. The relation between bandwidth, normalized power and excess loss at 850 nm centre wavelength can be seen through Fig. 7.

Performance analysis for FV modes at different structural length of spiral
The calculations for the reported structure as stated above were performed for fully vectorial (FV) mode type which consists of both TE and TM modes. Now in order to study the individual mode performance for reported structure it is important to study the structural (4) t d = n g × L C Fig. 6 Length of propagation at 850 nm wavelength behaviour at TE and TM modes individually. Certain calculations were performed at different bending radii for FV mode type and analysis for the same is reported in Table 1. The variation in the minimum bending radius corresponds to the different length of spiral structures resulting in time delay for both TE like and TM like polarization for FV method. Here a variation of time delay for TE and TM mode types are provided for different length structures which are application specific and can be used for various OCT applications.

Tolerance analysis
The design tolerance has been calculated for the proposed structure so as to show the feasibility of our structure to fabrication variation if any. Since the structure has been simulated for very precise values of waveguide core width and height, a design tolerance analysis would justify the feasibility of the proposed structure when fabricated. The design tolerance was calculated at ± 2% deviation for waveguide core width and  thickness. The simulations for design tolerance were performed for FV mode type in terms of EL and loss across length of spiral parameters. The EL calculations were performed for a 200 nm wavelength range which can be seen through Fig. 8. It is observed Tolerance analysis for loss across length of spiral at 850 nm from the plot that the maximum EL of 0.27 dB is accounted for, which is little higher than the original EL of 0.17 dB. Also, at 850 nm, an almost similar EL of 0.06 dB was observed for all deviation in width and thickness. Next, loss across the length of the spiral was calculated at ± 2% deviation for waveguide core width and thickness. The plot for loss across the length of spiral in terms of its design tolerance can be seen through Fig. 8. It can be seen that a constant loss was observed for the entire length of propagation at all deviations in waveguide width and thickness at 850 nm central wavelength.
It can thus be analysed from Figs. 8 and 9 that the reported structure is fabrication tolerant and can be used for IO based OCT systems.
The reported structure was also compared to some of the existing structures available in literature. A comparison analysis can be seen through Table 2. For Si 3 N 4 material-based delay line structures, a time delay of 395 ps and 17.2 ns was observed with a loss of 3.6 and 0.5 dB respectively. It can be observed from the table that the lowest PL was calculated for the reported spiral delay line with a compact footprint of only 6 × 6 mm sq. It lists the comparison of this reported spiral delay line with various state of art integrated optical delay lines over the years. Comparisons are made in terms of loss obtained, delay time acquired, material and wavelength used, length of delay structure and area occupied and bandwidth. To the author's knowledge, this is the first ever attempt to design an integrated spiral delay line at 850 nm wavelength using Si 3 N 4 / Si O 2 material. The reference structures shown in Table 2 have been fabricated and correspond to their experimental losses. Since, the proposed structure is based on theoretical and simulation outcomes, a design tolerance results comparison will be more feasible to compare with referenced structures.

Conclusion
In this work, design and simulation of compact spiral delay line for application as parameterized building block in integrated optics-based OCT has been presented. Simulation results show a potential of Si 3 N 4 / Si O 2 material to be used for further integration of other components in the OCT system. The reported delay line showed very promising results over 200 nm bandwidth at 850 nm central wavelength. Over the length of propagation of 55.11 cm, an excess loss of 0.06 dB and propagation loss of 0.027 dB was accounted for an area of 6 × 6 sq. mm. Also, the design was analysed for FV mode type and their performance comparison for different spiral length has been shown. The structure was also analysed in terms of its tolerance parameter and was found to be design tolerant. When further optimized with directional coupler, the combined system shall replace the beam splitters in existing fibre optics-based OCT systems.