Improvement of misalignment tolerance in free-space optical interconnects

ABSTRACT


INTRODUCTION
With the large growing demand for data processing and transmission in the current information age, the required data rate per channel increases and more channel bandwidth is required.Free space optical interconnects (FSOIs) present a promising technology that can be used as wideband channels to support the very high data rate transmission in many computing systems [1]- [9].However, misalignment of different optical components as a result of manufacturing or installation reasons is considered as relevant issue that affects the design and performance of FSOIs systems [10]- [14].Therefore, studying the impact of misalignment on FSOIs systems and introduce solutions to reduce its effect is essential in these optical systems [15]- [17].In general, the misalignment of different components in FSOIs increases the optical crosstalk that already exists in these systems.Therefore, methods to deal with crosstalk to increase the system tolerance to misalignment are required.Most of solutions discussed in the literature were created and developed based on presenting optimized models for the crosstalk of the misaligned optical system.For example, in [18]- [20] modifications on the mode expansion model were introduced and optimized in the presence higher transverse operation modes of the laser beam.In [21] the use of Hermite-Gaussian beams and the theory of Collins diffraction were introduced to model the optical diffractions.In [22], the generalized diffraction integral formula assuming a Laguerre-Gaussian (LG) model for the input beam was used to model crosstalk.An approximation formula for the output optical field was obtained in terms of a Int J Elec & Comp Eng ISSN: 2088-8708  Improvement of misalignment tolerance in free-space optical interconnects (Nedal Al-Ababneh) 427 sum of complex-valued Gaussian models.In [23] a model that uses a micro lens with a Gaussian transmittance profile is used to reduce crosstalk.In this method it was shown that crosstalk can be reduced as a result of reducing the optical power that is coupled to the neighboring channels.Reducing the coupled light might also improve the optical systems' misalignment tolerance, which is the main goal of this paper.
In this paper, we show that the use of a micro lens with Gaussian profile increases the misalignment tolerance in lens based FSOIs.To the extent of authors' knowledge, the use of tapered profile for micro lens transmittance to increase the system's tolerance to misalignment in FSOIs models has not been studied yet.Section 2 demonstrates the research methods investigated and proposed in this study.This includes analyses of a general misaligned optical model and derivations of light irradiance distributions assuming three cases for the micro lenses: an open aperture, a uniform circular aperture, and a Gaussian profile aperture, respectively.In addition, crosstalk calculations are provided for all of the considered cases.In section 3, numerical results and discussion are provided.Finally, the paper is concluded in section 4.

METHOD 2.1. A general misaligned optical system
Figure 1 shows a general diagram for a two-dimensional optical model, without misalignment as described in Figure 1(a) and with misalignment as described in Figure 1(b).P1 and P2 represent the input and output planes for an optical system without misalignment, on the other hand, P1m and P2m represent the misaligned planes. is the distance between the input and the output planes along the z-axis.The output field  2 ( 2 ,  2 ) at the output plane can be obtained by exploiting Collins diffraction integrals as (1): where  1 ( 1 ,  1 ), , and  are the input field, wave number, and wavelength, respectively.The elements of the transfer matrix of aligned optical system are a, b, c, and d.The misalignment parameters , , , and ℎ are given as (2) to (5): The parameters   and   represent the misalignments in the x and y directions, respectively.  and   represent the angular misalignments.  ,   ,   and   represent the misalignment elements that can be written as (6): Assuming  = cos () and  = sin (), the optical field in (1) can be written in cylindrical coordinates for slightly misaligned system as [24]: where  1 ,  1 and  2 ,  2 are the cylindrical coordinates in the misaligned planes, respectively.In addition,  2 ′ and  2 ″2 are defined as and

Open aperture misaligned FSOIs
For this section the FSOIs system is shown in Figure 2.An array of vertical cavity surface emitting lasers (VCSELs) is located at the front focal length of an array of micro lenses with focal length of   for each.The output plane contains an array of photodetectors. is the distance between detector and micro lenses arrays.  and   are the transverse misalignment of the micro lens.The output field at the detector's plane can be found using (7).Neglecting the angular misalignment (  =   = 0) the transfer matrix elements are: Assuming a LG beam model, the field at the input surface of the microlens can be given by ( 10): where    is the Laguerre polynomial [24], [25]. 1 is the beam radius and is given by [26].
where  0 is the LG beam's waist radius.Using ( 10) and ( 7) along with ( 8) and ( 9), the optical field at the array of detectors can be formulated as (11): In the context of slightly misaligned open aperture lens based FSOIs system, (11) represents the optical field irradiance calculated at the array of detectors for the LG beams propagating.Importantly, this equation can be used to estimate the crosstalk in a misaligned open aperture optical system.This is because the equation accounts for the transverse misalignment parameters.

FSOIs with uniform circular aperture
In this section, we assume the presence of a circular aperture with uniform hard edge transmission profile at the plane of the micro lenses array.Let ( 1 ) be the uniform hard edge circular aperture function with radius, then In this case, the optical field at the array of detectors is: The integral in (13) can be found using the following expansion for the aperture function [24] ( 1 ) = ∑  1 exp (− where  1 and  2 are complex coefficients.By substituting ( 14) into ( 13) we obtain Equation ( 15) represents the optical output field distribution in a misaligned FSOIs systems of a uniform circular aperture profile.Both the aperture size and the transverse misalignment of the microlens are considered in the equation.In subsection 2.5, we use the equation to estimate the crosstalk taking into account the effect of the uniform finite aperture of the microlens.

FSOIs with Gaussian circular aperture
Herein, we assume the utilization of microlens with circular aperture and Gaussian transmittance.Let ( 1 ) be the circular Gaussian aperture function with  1 radius.
where ( 1 ) = exp ( 2 ) and   2 is the Gaussian aperture's width.Again, by expanding ( 1 ), ( 1 ) can be written as (17): Substituting ( 17) into (7) and performing the integral, we obtain: In (18) represents a closed form relation for the output fields in the misaligned FSOIs systems of circular Gaussian aperture.It includes the aperture size and the misalignment of the lens at the same time.Moreover, the dependence of the output filed on the misalignment parameters is obvious in the above equation.In the following subsection, we use the equation to estimate the crosstalk taking into account the effect of circular Gaussian aperture of the microlenses.

Crosstalk calculation
To study the effect of using tapered aperture on the misalignment tolerance we need to evaluate the crosstalk.For the sake of estimating the crosstalk, we refer back to Figure 2. Let the channel with the blue detector as the active channel.In this paper, two sources of crosstalk are taken into account.The first source is the stray light which is the mount of light from the active VCSEL that reaches to other detectors through neighboring micro lenses.Based on this, the stray crosstalk can be found using the following equations for the different apertures where the subscripts , , and  are used to denote for no aperture, uniform hard aperture, and Gaussian aperture, respectively.In ( 19), (20), and ( 21)  11 and  12 are the crosstalk noises at the eight neighbor detectors.Ω is the detector' area.Diffraction induced crosstalk represents the second source of crosstalk which arises due to light diffraction from other VCSELs through other micro lenses to the active detector.In this case crosstalk can be obtained from the following (22): The subscript i should be replaced by u, a, or g to find the crosstalk for no aperture, uniform aperture, or Gaussian aperture case, respectively.Useful signal power can be defined as the power received by the active detector through its active channel and is given as (23): Using (19) to (23), the signal-to-crosstalk (SCR) ratio can be calculated as (24).

RESULTS AND DISCUSSION
To show the impact of using lenses with tapered aperture on the crosstalk of a misaligned system we refer to Figure 2 and assume the following parameters: wavelength (λ) of 0.850 µm, LG beam's waist radius (  ) for the VCSEL of 3 µm, micro lenses focal length (  ) of 720 µm, and micro lenses diameter of 250 µm.The interconnection distance and the radius of the hard aperture are considered to be 2.5 mm and 125 µm, respectively.In addition, the width of the Gaussian aperture is assumed to be 125 µm.In Figure 3, we showed the SCR as a function of the detector size for the aligned FSOIs model for both uniform and Gaussian apertures.Using uniform aperture, the maximum SCR is 25 and it is 93 using the Gaussian aperture.Moreover, the optimum detector size for the Gaussian aperture (14 µm) is less than that of the uniform hard-edge aperture (23 µm).The presence of such optimum is due to the fact that the diffraction induced crosstalk is mainly determined by the detector size and the stray crosstalk is determined by the microlens size.For a given microlens size, the stray crosstalk is fixed and is larger than the diffraction crosstalk at small detector size.Given this, the decrease of the SCR with small detector size is because of the reduction in the signal power.For large detector size, the diffraction crosstalk is larger and increases with the detector size.
Figure 4 demonstrates the SCR as a function of the detector radius with a misalignment of 10 µm for the micro lens in the x-direction.It is clear from the figure that the SCR decreases significantly with misalignment.The optimum level of SCR is decreased to 18.5 using Gaussian aperture and to 7.5 using uniform aperture.In addition, the optimum size of the detector decreases with increasing the misalignment to tolerate the decrease in the SCR.Optimum detector radii when using uniform and Gaussian apertures are 18 µm, and 12 µm, respectively.Figure 5 shows the SCR versus transverse misalignment for a fixed detector size.The figure clearly demonstrates that the SCR decreases when the misalignment increases.If we set the minimum acceptable SCR to 10 for proper transmission, the optical system can tolerate misalignment up to 14 µm for the case of Gaussian aperture and up to 6 µm for the uniform hard aperture case.The increase in the system tolerance can be explained by the aid of using Figures 6 and 7. Figure 6 shows the normalized crosstalk versus transverse misalignment.As we can see the crosstalk when using the Gaussian aperture is less than that of the uniform hard aperture for all misalignment values.Even though the normalized signal power is less when using the Gaussian aperture as shown in Figure 7 but the reduction of the crosstalk results in more SCR as expected.

Figure 1 .
Figure 1.Schematic diagram of optical model (a) without misalignment and (b) with misalignment

Figure 3 .Figure 4 .
Figure 3. SCR as a function of detector size for aligned FSOIs systems using uniform and Gaussian apertures