## Abstract

This paper proposes a method for a scheme of spectral phase encoding with resolution of up to 10 MHz and addressability of 40GHz, for a typical wide-band optical data carrying signal, by means of optical instruments and prior knowledge about the signal. The setup includes an array of Fabri-Perot Interferometers (FPI) which bypass the grating limitations, and a phase-only spatial light modulator (SLM) to properly encode the diffracted light. Their arrangement along with the method of Optical Code Division Multiple Access (OCDMA) was simulated, and promised fine results for various encoding schemes.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

The continuing growth in broadband network demand has led to the need for new, innovative techniques such as optical spectral processing, which is employed in many optical applications. Some of the common methods are pulse shaping [1], dispersion compensation elements using dispersion compensation fibers (DCF) [2–4], spectral power equalization using Micro-electromechanical mirrors (MEMs) [5–7], and a wide range of security solutions for optical communication systems using OCDMA [8–16]. Other methods include the use of photonic spectral processors (PSP) which utilize optical instruments such as waveguide grating routers (WGRs) [2–4,11,17–19], DMDs and MEMs [5–7,18], and SLM [20–22].

Currently, a novel approach has been suggested and tested which attempts to overcome the resolution limitation by spectrally dividing the bandwidth before it reaches the dispersive element [22]. This setup uses two FPIs in parallel with a slight change in their cavity length. This technique has been tested and has shown a fine optical resolution of 577 MHz with a spectral separation of 6.5 GHz. The spectral phase encoding used in this work shows the mathematical analysis and simulation of a generalized system made out of an array of FPI acting simultaneously to achieve better phase encoding and resolution, along with OCDMA technique.

The suggested system focuses on a set of 50 vertical resonators that are been tested by means of simulation. Typical optic signal of 40 GHz bandwidth was chosen and divided into 100 MHz individual bins. Several projection schemes of the bins over an SLM surface were tested and showed a successful contribution of different phase value to each one, thus implementing the phase encoding. The bins are then reconstructed to yield the desired encoded version of the data-carrying signal. To overcome the SLM physical limitations, a use of OCDMA method along with the resonator setup was also tested and accomplished finer encoding resolution of 10 MHz by enabling partial overlapping of the bins one another over the SLM surface.

## 2. System description

The suggested system is based on the traditional 4F spatial system described in [21] where the filter is implemented by a SLM. Vertical expansion of the temporal division of the input signal into *n* parallel sub-systems, as shown in Fig. 1, exploits both the *x* and *y* axes for different types of encoding. An array of equally spaced FPIs vertically spread with respect to the $[{xy} ]$ plane at $z = 0$, is configurated to output specific pass-bands so that their sum will cover all of the input signal's band-width. Thus, the input signal is temporally diffracted by the FPIs, undergoes spatial diffraction due to the grating, and is collimated by the lens for maximal use of the SLM surface. The motivation for this process is to apply different phases for each part of the signal to maximize the encoding resolution. The inverse process recombines the signal back, which yields the optical phase-only encoded signal. All of the FPI's outputs are treated as gaussian beams traveling in the positive *z* direction. The light beams are equally spaced along the *y* axis and expand on their relative $[{xy} ]$ planes in the *x* and *y* directions.

## 3. Mathematical development

#### 3.1 Signal properties

The suggested photonic signal processor (PSP) deals with a spatially collimated and temporal Fourier transform limited-optical-data carrying signal ${U_0}(t )$ with temporal bandwidth denoted as $BW$. Typical bandwidth for data carrying signal can range from ${\sim} 10\textrm{GHz}$ to about ${\sim} 200\textrm{GHz}$, where the greater the bandwidth, the more challenging it is to properly encode. Since the encoding process is closely bounded up with the characteristics of the Fabri-Perot array, we first define the main parameters in use.

The encoding resolution is derived from the passband of the FPI output, known as the Full Width Half Max (FHWM). We aim to slice the data signal into narrow-band bins of information, each with a temporal width of FWHM. Narrowing the FWHM will result with more bins and higher resolution. This paper focuses on a typical C-band optical data-carrying signal with a center wavelength of ${\lambda _C} = 1550\,\textrm{nm}$ and $BW = 40\,\textrm{GHz}$, and examines possible phase encoding resolutions of and $FWHM = 10\,\textrm{MHz}$.

#### 3.2 Fabri-Perot interferometers equations

Taking the general form of a FPI's transfer function as described in Appendix A, along with the relation derived in [21] provides a constraint for the temporal distance between two adjacent bins by substituting: $\delta \omega = 2\pi \cdot FSR$ where: $\omega = {\omega _C}$ is the carrier frequency, *B* is the grating dimensions and *d* is its period as shown in Fig. 2:

This condition enables us to set a suitable FSR that is sustainable for all of the FPIs in the array and meets typical grating characteristics. The number of cavities in the system is equal to the finesse number $\Im = \frac{{FSR}}{{FWHM}}.$

The notation throughout this paper for the FPI index is: $FP{I_j}$, where: $0 \le j < \Im \textrm{ , }j \in {\mathbb N}$. From the nature of FPI cavity, the adjustment of the $FSR$ can be set by changing cavity length, *l*, or the refractive index, *n* (assuming $\theta = 0^\circ $). The reflection coefficient for all of the cavities depends solely on the finesse number.

#### 3.3 Allocation of the bandwidth into bins

For this analysis we denote the sub-carriers’ center frequency by ${\nu _i}$ as follows: ${\nu _i}\textrm{ , }0 \le i \le N - 1$ where: $N = \lfloor{{{BW} / {FWHM}}} \rfloor $, as depicted in Fig. 3:

Each sub-carrier can easily be calculated using:

Since the Finesse number determines how many FPIs should be taken, we can easily relate the sub-carrier's index to the FPI index as follows: $i = j + k \cdot \Im \textrm{ ; }0 \le k < {{BW} / {FSR}}\textrm{ , }k \in {\mathbb Z}$.

#### 3.4 Misalignment criteria

We can link the center frequency of a specific bin with the FSR and replica index ${m_i}$ shown in Appendix A as follows:

Since *m* is the integer stating the period of the FPI's output signal, when aligning the first bin ${\nu _0}$ to the signal's predefined BE, one would get: ${m_0} = \lfloor{{{{\nu_0}} / {FSR}}} \rfloor$. The floor operation should be compensated by slightly increasing the FSR, and therefore, the practical FSR will always be somewhat bigger than the theoretical one. In our case:

Taking advantage of the flexibility of *l* (assuming a fixed refractive index) we can set an appropriate FSR. The Misalignment Criteria (MC) evaluates the error between the target and theory FSR in the sense of frequency coverage and is defined as follows:

We wish to minimize the MC as much as possible in order to avoid unwanted overlaps on one hand, and uncovered spectral ranges on the other.

#### 3.5 Free space calculations

After leaving the FPI array, the light beams meet the grating and collimated lenses. As shown in Appendix A, the width of the lens will be calculated using:

where: ${\theta _1}$ and ${\theta _2}$ are the two edge diffraction angles of a given $FP{I_j}$, and*f*is the lens's focal length an shown in Fig. 4.

Note that since each FPI results in a different set of wavelengths according to its $j$ index, the design will require a different lens per FPI. However, since the wavelengths are of negligible difference from one another, we will settle with one cylindrical lens for the entire array. Evaluating the relations between the rest of the free space parameters are described in great details on Appendix A.

#### 3.6 SLM dimensions and system constraints

The light projected over the SLM surface is made up of several spatially separated beams. Fig. 5 shows the general form of a vertical placement of stains in their elliptical shape due to the convergence in the *x* direction and expansion in the *y* direction. We denote by $2{w_{02}}$ the horizontal width and by $2{w_{02y}}$ the vertical height. The horizontal width can be set by the laser spot, and the beam's length is affected by Bruster expansion and shown in Appendix A. ${d_{py,k}}\textrm{ ; }0 \le k \le \Im $ refers to the vertical distance between two adjacent stains.

Taking the extreme case for a constraint relating the SLM dimensions with the light stains are:

*M*is a magnification constant. All are addressed in Appendix A.

## 4. Simulation results

This section deals with the simulation of the PSP. A typical C-band optical signal was chosen with ${\lambda _C} = 1550\,\textrm{nm}$ and $BW = 40\,\textrm{GHz}$. The initial resolution was set to 100 MHz. To select an appropriate choice for FSR, a grating with $d = 644000\,\,{{\textrm{grooves}} / \textrm{m}}\textrm{ , }B = 10\,\textrm{mm}$ was tested which yields $FSR > 4.78\,\textrm{GHz}$, so the FSR of 5GHz was selected. The reflection coefficient of the cavities was calculated to be 0.93911 and $l = 0.3\,\textrm{m}$, $n = 1$. The spot's width is ${w_{01}} = 0.2\,\textrm{mm}$. Since $\Im = 50$, we are left with50 FPI in the array and a maximal MC of 0.9% as shown in Fig. 6:

The free space parameter ${l_1}$ was chosen to be 40 mm and $M = 100$. The calculated values are: ${z_1} = 81.073\textrm{mm , }f = 9.055\textrm{mm}\,\textrm{,}\,\,{l_2} = 9.392\textrm{mm}\,\textrm{,}\,\,{w_{02}} = 20.869\mathrm{\mu m\ ,\ }{w_{02y}} = 243.69\mathrm{\mu m}$. The SLM used in this system is of $1920 \times 1080$ pixels with ${p_x} = {p_y} = 8\mathrm{\mu m}$.

The first encoding scheme tested included divisions of pixels into $36 \times 36$ square shapes where each shape received a whole stain and was encoded to provide different phase shift. The results yield successive phase encoding as shown in Fig. 7:

Next, two more cases were tested and presented in Fig. 8: (1) division into $36 \times 96$ rectangle shapes with different phases for each one, and (2) division into $1080 \times 40$ rectangle shapes with different phases each. The former case resulted with the same phase contribution for 3 adjacent bins and therefore degraded the resolution by a factor of 3. However, the latter encoded all the bins coming from each FPI with the same phase. Nevertheless, they are finesse-number bins separate from each other, so no degradation in phase occurred.

Since there are many options for choosing an encoding scheme, one must be familiar with the possibility of unwanted or overlapping cases. The joint bins, denoted by $J{B_x}$ and $J{B_y}$, bind this information and are defined as follows:

Bear in mind that the resolution can be degraded solely across the *y* axis, therefore:

*x*axis there will be periodic repetition of phases, but the resolution won't deteriorate.

## 5. Improved resolution using OCDMA method

#### 5.1 Method overview and theory

We can further increase the performance of the suggested PSP by utilizing a prior knowledge about the signal itself. For the vast number of information bins, the surface area of each stain will result in fewer pixels, which degrade the sampling accuracy. As stated in [23], OCDMA uses Walsh codes which can completely overlap each other and can be successfully separated. The number of Walsh code needed is bounded by $\lfloor{{{BW} / {FHWM}}} \rfloor $, however, we wish to use as few codes as possible. Spreading factor, $\eta $, refers to the number of code groups that can be overlapped over one another. The bigger the spreading factor, the fewer codes needed for the implementation of OCDMA. The maximal number of different CDMA mask sub-functions is: $2k + 1\,\,\,({k \in {\mathbb N}} )$ where each mask is defined over the following frequency range:

where: $\begin{array}{l} {g_i}(\nu ){g_j}(\nu )= 0\,\,\,\,\,\,i \ne j\\ {g_i}(\nu )= 0\,\,\,\,\,\,\, - k \le i \le k\,\,\,\,\,\,\nu \not{\in }[{ - {{\textrm{FWHM}} / 2}:{{\textrm{FWHM}} / 2}} ]\end{array}.$In our case, the spreading factor is chosen as the finesse number: $\eta = \Im $. Fig. 9 shows the division of information bins into sub-functions.

The horizontal expansion of each stain using passive optical elements provides more pixels per stain. In Fig. 10, one can see a specific example for overlapping of 4 sub-functions. In this case each bin gets 4 times more pixels compared to its original projection.

Theoretically speaking, there is no limit over the horizonal expansion, so each light stain can grab all of its row-wise pixels of the SLM. However, such an expansion will require ${{BW} / {FSR}}$ number of CDMA sub-functions, which for some resolutions can be tedious to work with. A practical trade-off can be considered, involving taking the minimal number of overlapping light beams by horizontal expansion to get an image that is accurate enough, in favor of fewer CDMA sub-functions.

#### 5.2 Simulation results

The tested parameters were repeated, this time with a resolution of 10 MHz. The calculation yields a total of 4000 information bins where each stain allocates $3 \times 2$ pixels with no overlapping. Utilizing the suggested OCDMA technique enables us to expand the pixel group to $3 \times 12$ for overlapping of 4 sub-functions. Assuming 98%-pixel accuracy gives the comparison of phase contribution shown in Fig. 11.

Taking 6 pixels per phase for 4000 bins resulted in an overall phase error of 0.51%, while expanding the number of pixels by a factor of 4 reduced the error to 0.256%. Several other measurements were examined and yield the same outcome: utilizing OCDMA to increase the stain area by an overlapping factor *k* reduces the phase error by $\sqrt k $. In other words, using OCDMA with an overlapping factor *k*, compared to the regular system's implementation, improves the PSP performance by means of phase accuracy according to $\sqrt k $.

## 6. Conclusions

In this paper, we have presented a novel concept for spectral phase encoding using an array of cavities which overcome the diffracted element of a typical PSP. In addition, by utilizing OCDMA method and taking into account the digital encoding of a signal, we've managed to outperform the expected results by the square root of a chosen overlapping factor. A set of practical system parameters proved to be capable of achieving a fine resolution of less than 0.25% phase error for 10 MHz in the C-band optical signal, with a bandwidth of 40 GHz.

## Appendix A – Mathematical Development

This appendix includes all of the mathematical developments that are mentioned in section 3.

## 7.1. Fabri-Perot interferometers equations

The general form of FPI's transfer function for an input light beam entering the cavity at an incident angle $\theta $ is:

where $F = \frac{{4R}}{{{{({1 - R} )}^2}}}$ is the finesse coefficient and $\delta (\nu )= [{4\pi nl\nu /c} ]\cos \theta$. In this notation we use*R*as the reflection coefficient of the mirrors inside the cavity,

*n*as the medium's refractive index,

*l*as the cavity length and

*c*as the speed of light as shown in Fig. 12(a). For a bandwidth limited input signal $x(t )$, the output $y(t )$ holds the following relation in the frequency domain: where the frequency axis is denoted by $\delta $ assuming the cavity parameters and incident angles are fixed. The immediate outcome shows the filtering property of such cavity as shown in Fig. 12(b).

The Free Spectral Range, abbreviated as FSR, and the FWHM introduced earlier are the two main parameters which determine the resolution quality and performance of the system. The ratio $\lfloor{{{BW} / {FSR}}} \rfloor $ determines how many bins a single FPI can output. Spatial diffraction of the light is done by the grating; however, its limitation must be considered.

The passband which defines the temporal frequency range of an individual FWHM is set by the power threshold $\gamma $ as follows:

where*m*is the replica index and: $\Gamma = \arcsin \sqrt {\frac{{1 - \gamma }}{{\gamma F}}} $ is referred to as the

*effective power threshold*. Note that $\Gamma $ depends only on the reflection coefficient

*R*of the resonance (assuming a fixed power threshold) and is defined at: $0 < \Gamma \le {\pi / 2}$. A common value refers to a 3 dB attenuation of the Band-Edge (BE), thus: $\gamma = {1 / {\sqrt 2 }}$.

To set different FSR values, assuming $\theta = 0^\circ $, we change the cavity length, *l* as can be shown to affect due to the following relation:

The reflection coefficient for all of the cavities depends on the finesse number:

## 7.2. Free space calculations

The light beams meet the grating and collimated lenses as shown in Fig. 3. The output angle with respect to the grating optical axis, in which a monochromatic light beam with wavelength $\lambda $ enters the grating at a normal incident angle, is: ${\theta _1}(\lambda )= {\sin ^{ - 1}}({{\lambda / d}} )$ where we neglect the other orders of diffraction. Bear in mind that ${\theta _1}$ differs from the FPI resonator's incident angle $\theta $ that was assumed to be equal to $0^\circ $. Since each FPI outputs several wavelengths, the output traveling light beams $[{xz} ]$ plane will be made up of a set of wavelengths ${\lambda _j},{\lambda _{j + \Im }}{\lambda _{j + 2\Im }},\ldots ,{\lambda _{j + k\Im }}$ where: $\frac{1}{{{\lambda _{j + ({m + 1} )\Im }}}} - \frac{1}{{{\lambda _{j + m\Im }}}} = \frac{{FSR}}{c}\,\,\,\,({m \in {\mathbb N}} )$.

To find the constraint for the cylindrical lens size, we take the two edge wavelengths of the given $j$-th FPI, e.g. ${\lambda _j}$ and ${\lambda _{j + \left\lfloor {\frac{{BW}}{{FSR}}} \right\rfloor \Im }}$. Denoting ${\theta _j} \buildrel \varDelta \over = {\theta _1}\textrm{ , }{\theta _{j + \left\lfloor {\frac{{BW}}{{FSR}}} \right\rfloor \cdot \Im }} \buildrel \varDelta \over = {\theta _2}$ we can set the following condition for the cylindrical lens with a focal length *f* and a width ${W_L}$ as shown in Fig. 4.

Note that since each FPI results in a different set of wavelengths according to its $j$ index, the design will require a different lens per FPI. However, since the wavelengths are of negligible difference from one another, we will settle with one cylindrical lens for the entire array.

Positioning the lens with respect to the grating while considering the beam's gaussian profile is shown in Fig. 13. The beam's longitudinal axis *z* is set to zero at the output of the FPI and its waist is denoted by ${w_{01}}$ at plane 1. The light beam expands through the grating towards plane 2 of the lens. Planes 3 and 4 are the edge of the lens and the SLM plane respectively. It's important to distinguish between the expansion along the *x* axis and along the *y* axis. The cylindrical lens focuses the light around the *x* axis while for the $y$ axis, light expands according to the Bruster angle. The travel distance of the light beam from the FPI output, through the grating up to the lens's surface is denoted by ${l_1}$, and the travel distance from the lens to the SLM surface is denoted by ${l_2}$.

The FS equations yield the following relations:

where ${z_1} = \frac{{\pi w_{01}^2}}{\lambda }$.To avoid partial overlapping, we use a magnification unit as shown in Fig. 14. $\Delta x$ denotes the horizontal distance between the center of two adjacent beams coming from the same FPI, and $\Delta {x_M}$ denotes the center’s distance from leaving the unit. The magnification value is defined as:

$\Delta x$ is the lens's width: $\Delta x = {W_L}$, which results in the outcome: $\Delta {x_M} = M\Delta x = M{W_L}$.

## 7.3 SLM Dimensions and System Requirements

The notations for the SLM dimensions and pixel sizes are shown in Fig. 15 below. The outer dimensions are the SLM length, ${L_S}$, and SLM width, ${W_S}$. Each pixel has dimensions of ${p_x}$ and ${p_y}$. The SLM has ${N_x}$ pixels in each row (the *x* axis) and ${N_y}$ pixels in each column (the *y* axis). The relation between the parameters is as follows:

The light projected over the SLM surface is made up of several spatially separated beams. In Fig. 5 from section 3, we've defined $2{w_{02}}$ and $2{w_{02y}}$, whom are the horizonal width and the vertical height respectively. Taking small angle approximation for the Bruster angle $\tan {\theta _B} \approx {\theta _B}$, one can confirm the relation:

Next, we will apply constraints for the projected light beams over the SLM surface. Denoting by ${d_{py,k}}$ the vertical distance between two stains, we can derive a general constraint:

If we assume: $0 \le k \le \Im :\textrm{ }{d_{py,k}} = {d_{py}}$ and take as the extreme case $0 \le k \le \Im :\textrm{ }{d_{py,k}} = 0$ we get:

Taking Eq. (25) and substituting into it Eq. (19) and Eq. (23) yields:

From Eq. (26), we can see that the given parameters are: ${L_S},\Im ,\lambda ,{w_{01}}$, while $f,{l_1}$ can be set to fit the characteristics of the system.

For the width of the SLM we should include a different type of constraint, primarily due to the arrangement of the beams along the *x* axis and the presence of the magnification unit. From Eq. (2) we can see that each FPI has *k* equally spaced horizonal bins. When going from one FPI to the other we can see that their first bin $({k = 0} )$ has a frequency of $\nu _{i = j}^{} = c \cdot \lambda _{i = j}^{ - 1}$. Since $j$ ranges from 0 to $\Im - 1$, the locations of each set of beams cannot be placed vertically on the SLM surface, but will be tilted as shown in Fig. 16(a) with the following tilt angle:

However, since this angle can easily be set to as small as necessary by choosing the system parameters wisely, we can ignore it in this analysis and focus on the scheme shown in Fig. 16(b).

To find the horizontal constraint we first denote the edge angles as:

The maximal distance can be expressed using the current set of parameters as follows:

## Disclosures

No conflict of interests regards this paper.

## Data availability

No data was generated or analyzed in the presented research.

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