## Abstract

A new design is presented to achieve a hybrid micro-diffractive-refractive lens with wide field of view (WFOV) of 80° integrated on backside of InGaAs / InP photodetector for free space optical interconnections. It has an apparent advantage of athermalization of optical system which working in large variation of ambient temperature ranging from -20 °C to 70 °C. The changing of focal length is only 0.504 μm in the ambient temperature range with the hybrid microlens, which opto-thermal expansion coefficient matches with thermal expansion coefficient of AuSn solder bump used in corresponding flip-chip packaging system. The hybrid lens was designed via CODE-V^{TM} professional software. The results show that the lens has good optical performance for the optical interconnection use.

© 2002 Optical Society of America

## Errata

Yongqi Fu and Ngoi Bryan, "Design of hybrid micro-diffractive-refractive optical element with wide field of view for free space optical interconnections: errata," Opt. Express**10**, 714-714 (2002)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-10-15-714

## 1. Introduction

Microlens integrated with laser diode and photodetector for free space optical interconnections has large potential in market of fiber communication. Some works have been done for designing of the microlens with diffractive structure and wide field of view to integrate with InGaAs/InP p-i-n photodetectors [1], which detect optical signals with working wavelength of 1.55 μm. However, working ambient temperature ranging from -20 °C to 70 °C is often required for most optical communication products. It is very difficult to ensure a satisfied coupling efficiency and small insertion loss due to variation of refractive index of optical elements under the ambient temperature range. For example, for a system working under ambient temperature of 50 °C, a focus shift of 0.5 μm ~1 μm will occur that causes a insertion loss of-2dB ~ -3dB. For many communication systems, requirement of the insertion loss is as high as -0.1dB. Therefore, athermalization is necessary for the optical system used for free space optical interconnections. In this paper, we will introduce the athermalization by virtue of hybrid micro-refractive-diffractive lens in corresponding packaging systems. To our knowledge, this is the first time that describes the athermalization of optical system in photonics packaging system.

## 2. Design Principle

The architecture of our free space optical interconnections is shown in Fig.1. The system consists of 8×8 single-mode verical-cavity surface emitting laser diode (VCSEL) array and corresponding photodetector array with working wavelength of 1550 nm, which is commonly used in fiber optical communication. In order to reduce alignment error, microlenses are integrated with VCSEL and photodetector on their backsides at wafer level. The lens with structure of hybrid refractive-diffractive is adopted for the consideration of athermalization of optical system integrated with the corresponding packaging system together. The aspherical surface is overlapped together with the diffractive structure at the same side of substrate. This lens combination can be realized by use of focused ion beam milling (FIBM) [2, 3] and laser direct writing. In this paper, we only discuss design of the hybrid lens for the photodetector. The lens design for the VCSEL’s is the same with it.

It is well know, for a thin refractive lens and a diffractive lens where the imaging space has refractive index *n _{i}*, assuming that the temperature gradient is not time varying and is linear with respect to the radial coordinate (as is the case for a lens that is absorbing radiation), their opto-thermal expansion coefficients are given by

where *x _{f,r}* is the opto-thermal expansion coefficient of the refractive lens (ppm/°C);

*x*the opto-thermal expansion coefficient of the diffractive lens (ppm/°C);

_{f,d}*n*the refractive index of lens material;

*n*the refractive index of image space;

_{i}*α*the thermal expansion coefficient of lens material (ppm/°C); and

*T*the system ambient temperature (°C).

However, our case is thick plano-convex hybrid lens. Considering this, we re-deduce the
expressions of *x _{f,r}* and

*x*in terms of geometry optics as follows.

_{f,d}For a thick refractive lens with radius curvatures of *r _{1}* and

*r*and thickness of

_{2}*d*, its focal length is given by

where *f* is the focal length of the thick lens and *n* the refractive index of lens material. For a plano-convex lens, *r _{2}*= ∞. Equation (3) can now be written as

Equation (4) means that the focal length is independent of lens thickness *d*. Differentiating equation (4) with respect to yields

but *α*= (1/*r*)(d*r*/d*T*), so equation (5) can be written as

The opto-thermal expansion coefficient of the thick refractive lens can be written as

It can be seen that the opto-thermal expansion coefficient of the thick refractive lens is still the same with equation (1) when the image space is air (*n _{i}*=1) for the thin refractive lens because the focal length is lens thickness free as stated before for our thick hybrid microlens with the type of plano-convex.

For a diffractive lens, it can be modeled as a lossless phase object, as shown in Fig.2. The zone spacing is defined such that the distance from the edge of each zone to the focal point is a multiple of the designed wavelength *λ _{0}*. For an object located at infinity, light is focused to the image plane at a distance

*f*behind the lens. The radius

*r*of the

_{m}*m*th zone is

where *f _{d}* is the focal length of the diffractive lens,

*λ*

_{0}the designed wavelength, and

*n*the refractive index of the lens material. Assuming

_{λ0}≪

*f*, the focal length can be expressed as a function of the zone radius:

_{d}As the temperature changes, the zone spacing expands and contracts. The zone radius *r _{m}* can, to the first order, be expressed as

Additionally, the refractive index of the lens changes with temperature by

The focal length, as a function of temperature, can now be written as

For most materials the second- and third-order terms in *ΔT* are negligible (≤10^{-11}). The opto-thermal expansion coefficient of the diffractive lens is given by

where *n* is the refractive index of the lens material. Equation (13) looks similar with the equation (2). Only the refractive index is different with former of lens material (*n*) and latter of image space (*n _{i}*).

For refractive lens, thermal behavior is wavelength dependent. For diffractive lens, the change in focal length of a diffractive lens is a function of thermal expansion coefficient *α* and index changes of lens material *n*, as shown in equations (7) and (13). Although the diffraction efficiency of a diffractive lens is affected by temperature changes in the refractive index of the lens, for most materials the effects are negligible. Therefore, athermalization does not require the integrated optical system to have a low opto-thermal expansion coefficient. The opto-thermal expansion coefficient of the optical system should be matched to the thermal expansion of the solder bump (its height determines the working distance of the optical system) material of the packaging system in the free-space optical interconnection system.

The change of refractive index with temperature for material InP can be described as follows [5]

where *n* is the refractive index of lens material, and *T* the lens temperature. Variation of refractive index, *dn*/*dT*, can be derived from equation (14), is 83.025 ppm/°C. Medium of imaging space is still InP with the same variation of refractive index, 83.025 ppm/°C.

For working wavelength of 1550 nm (designed wavelength *λ _{0}*) and substrate material of InP,

*x*and

_{f,r}*x*can be calculated from equations (7) and (13), are -35.262 ppm/°C and 32.191 ppm/°C, respectively.

_{f,d}Temperature compensation can be realized using athermalization. The hybrid microlens can be made to compensate the expansion of the package solders that hold it with respect to the image plane. Distribution of optical power for refractive lens and diffractive lens can be determined by solving the following equation [4,6].

where *x _{f}* is the opto-thermal expansion coefficient of the hybrid lens, ppm/°C,

*f*is the focal length of the hybrid microlens,

*f*is the focal length of diffractive lens, and

_{d}*f*is the focal length of refractive lens. Coefficient

_{r}*x*should match with thermal expansion coefficient (CTE) of solder bump, AuSn (CTE=16 ppm/ °C), which is prepared by surface patterning technology for common flip-chip bonding use, so that the change in image position compensates to the change in the position of focal plane. Total error is described by equation (16) as follows

_{f}where *Δf*
_{Total} is the total focus error caused by thermal expansion, *Δf*
_{Lens} is the focus shift of the hybrid lens, and *Δf*
_{Bump} is the focal plane shift caused by thermal expansion of solder bumps of flip-chip package system. If we set CTE _{bump} = - *X _{f}*, then

*Δf*

_{Lens}= -

*Δf*

_{Bump}, the total focus error,

*Δf*

_{Total}will be zero. In other word, the result of athermalization will be zero from the theoretical point of view. This is athermalization principle of our optical interconnection system.

The solder bump and hybrid lens can expand or contract with the same step. Focal lengths of the refractive lens and diffractive lens are all positve sign because the two surfaces combined together at the same side of substrate. Therefore, the relationship of, (*x _{f}*/

*f*)>(

*x*/

_{f,d}*f*), must be met for the hybrid surfaces. Based on this principle, for a fixed focal length,

_{d}*f*=0.35 mm (determined by the thickness of backside of photodetector), corresponding

*f*and

_{r}*f*are 0.500 mm and 1.889 mm, respectively. The designed microlens with parameters of

_{d}*f*,

_{0}*N.A.*and feature size are 350 μm, 0.27, and 3.5μm respectively.

## 3. Design Results and Discussions

The hybrid microlens will be fabricated on the backside of an InP substrate with the photodetector on the other side (see Fig. 1 and 2). The focal length of the hybrid microlens is the thickness of the InP substrate (0.35 mm) and the refractive index of InP is 3.17 for the working wavelength of 1550 nm. Diffraction order is +1. In the view of receiving signal, field of view of the lens should be as large as possible so as to increase the signal-noise ratio. Therefore, the hybrid lens should have wide field of view (FOV). On the other hand, crosstalk must be avoided for array system. For standard pixel space of 250 μm, we set diameter of the hybrid lens of 200 μm with FOV of 80°.

Designing was carried out by use of professional design tool CODE-V^{TM} [7], which is a comprehensive computer software for the design, analysis and optimization of optical system. We select aspherical surface as the refractive lens. It can be expressed by the following polynomial equation (17) with rotational symmetric form. Coefficients are for monomials in ascending order up to 20^{th} order, staring with the first order. The diffractive phase polynomial is shown in equation (18).

where r is radius of lens; *c* is vertex curvature; *k* is conic constant; and A,B C,…, J are coefficients of polynomial, set E=F=…=J=0 at here; C_{2}, C_{4}, C_{6}, … C_{20} are coefficients of diffractive phase polynomial. The coefficients for refractive lens and diffractive lens are listed in table 1.

Ray tracing layout of the hybrid lens for three different semi-fields (0°, 28° and 40°) is shown in Fig.3. It can be seen that all the rays whose incident angles are within the field angles will focus on the sensitive area of photodetector.

For the application of optical interconnections, we pay more attention to the focal spot size instead of image quality, spherical aberration and coma, should be compensated simultaneously according to the aberration theory. Astigmatism and distortion are not crucial factors, only for reference at here. Optimizing the phase polynomial of diffractive structure and coefficients of aspherical polynomial via CODE-V^{TM} software realizes the compensation. Fig.4, and 5 are graphical outputs of aberration curves and modulation transfer function (MTF) of the hybrid lens. All fields and both target orientations are included in a single plot. Fig.6 is point-spread functions for different semi-field angle of 0°, 28° and 40°, respectively. For imaging with a medium WFOV, the focus will be aberration limited rather than
diffraction limited. The effect of the point spread is to smooth and widen the image of sharp and narrow structures. It can be seen from the PSF results show that 90% of the energy will be encircled in a spot smaller than 18 μm in diameter in the three fields, as shown in Fig.7. With the device used in this system the detector area is 70 μm × 70 μm, which means that a high detection efficiency (>90%) is achievable in the system with these predicted spot sizes, and the Gaussion beam is best focused on photodetector (image plane). Combining with the MTF result and aberration curves, we can make a judgment that the hybrid lens has good optical performance. Because the lens is integrated with the photodetector together, misalignment errors caused by lateral, vertical, displacement in direction of longitudinal and tilting of photodetector doesn’t exist in our system.

Two types of cross talk that originate from optical and electrical sources exist in an optical-electrical (OE) interconnection systems. For a better optical system, the dominating cross talk in the system is electrical rather than optical origin [8]. The higher the optical link efficiency, the lower the actual optical cross talk.

For the polynomial expressions of (17) and (18), the optimized coefficients are up to order of 8 in the optimization. The influence upon aberration and MTF is so little that it can be neglected in our application for the coefficients with higher order (>8).

The hybrid lens is not only to achieve stability of focus position with temperature (insensitive to temperature), but also possible to athermalize spherical aberration of all orders, so that the on-axis image remains fully sharp and well defined as temperature changes. Because of optical power distributions, ϕ_{r}, and ϕ_{d}, for the consideration of athermalization, the refractive lens with longer focal length causes that the diffractive lens undertakes a heavy burden to control variation of spherical aberration with temperature change. Thus, we pay more attention to optimization of phase polynomial of the diffractive structure. The effects of temperature can be modeled by appropriately scaling the phase coefficients as follows: ΔC_{2}= -2αC_{2}ΔT, ΔC_{4}= -2αC_{4}ΔT, ΔC_{6}= -2αC_{6}ΔT, …, ΔC_{20}= -2αC_{20}ΔT.

We assume the source light is single-mode VCSEL with operating wavelength of 1550 nm. It can be regarded as quasi-monochromatic light. In this case, it is quite difficult to fully correct spherical aberration because spherochromatism is not large enough to be used to correct the spherical aberration. This causes that the spherical aberration is still large at the relative field of 1.0, as shown in Fig.4, and MTF curve at this field is not as ideal as that of fields at 0.0 and 0.63, as shown in Fig.5. If the light source is multi-mode VCSEL, the problem can be better solved.

## 4. Summary

A hybrid micro-refractive-diffractive lens with WFOV of 80° is presented in this paper for application of free space optical interconnections. The unique characteristic of athermalization makes it suitable for the optical system combined with packaging system used in the environment having large variation ambient temperature. The designing results show that this type of lens can be realized both in design and microfabrication. Its optical performance can meet the requirements of optical interconnection in fiber communication system.

## Acknowledgments

This work was supported in part by the Funding for Strategic Research Program on Ultra-precision Engineering from the NSTB (National Science α Technology Board, Singapore), Research Funding (ARC 9/96) from Nanyang Technological University and Innovation in Manufacturing Systems and Technology (IMST) program from Singapore-Massachusetts Institute of Technology (MIT) Alliance.

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