## Design of hybrid micro-diffractive-refractive optical element with wide field of view for free space optical interconnections

Optics Express, Vol. 10, Issue 13, pp. 540-549 (2002)

http://dx.doi.org/10.1364/OE.10.000540

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### 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

## 1. Introduction

1. C. Wang, Y.C. Chan, L.P. Zhao, and N. Li, “Design of diffractive optical elements array with wide field of view for integration with photodetectors,” Opt. Commun. **195**, 63–70 (2001). [CrossRef]

## 2. Design Principle

2. Yongqi Fu Kok Ann Bryan, Investigation of micro- diffractive lens with continuous relief with vertical-cavity surface emitting lasers using focused ion beam direct milling.IEEE Photon. Techn. Lett. **13**, 424–426 (2001). [CrossRef]

3. Yongqi Fu and Ngoi Kok Ann Bryan, Hybrid micro-diffractive-refractive optical element with continuous relief fabricated by focused ion beam for single-mode coupling.Appl. Opt. **40**, 5872–5876, (2001). [CrossRef]

*n*, 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

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

_{f,r}*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).

*x*and

_{f,r}*x*in terms of geometry optics as follows.

_{f,d}*r*and

_{1}*r*and thickness of

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

*f*is the focal length of the thick lens and

*n*the refractive index of lens material. For a plano-convex lens,

*r*= ∞. Equation (3) can now be written as

_{2}*d*. Differentiating equation (4) with respect to yields

*n*=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.

_{i}*λ*. For an object located at infinity, light is focused to the image plane at a distance

_{0}*f*behind the lens. The radius

*r*of the

_{m}*m*th zone is

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

_{d}*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}*r*can, to the first order, be expressed as

_{m}*ΔT*are negligible (≤10

^{-11}). The opto-thermal expansion coefficient of the diffractive lens is given by

*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}*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.

*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.

*λ*) and substrate material of InP,

_{0}*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}6. Carmina Londono, William T. Plummer, and Peter P. Clark, “Athermalization of a single-component lens with diffractive optics,” Appl. Opt. **32**, 2295~2302 (1993). [CrossRef] [PubMed]

*x*is the opto-thermal expansion coefficient of the hybrid lens, ppm/°C,

_{f}*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}*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*, then

_{f}*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.

*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

^{TM}[7

7. CODE-V^{TM} is professional software of optical design, a product of Optical Research Associates (ORA), http://www.opticalres.com/macros/macroindex.html

^{th}order, staring with the first order. The diffractive phase polynomial is shown in equation (18).

*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.

^{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.

8. Xuezhe Zheng, Philippe J. Marchand, Dawei Huang, Osman Kibar, and Nur S. E. Ozkan, “Optomechanical design and characterization of a printed-circuit-board free-space optical interconnect package,” Appl. Opt. **38**, 5631–5639 (1999). [CrossRef]

_{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.

## 4. Summary

## Acknowledgments

## References and links

1. | C. Wang, Y.C. Chan, L.P. Zhao, and N. Li, “Design of diffractive optical elements array with wide field of view for integration with photodetectors,” Opt. Commun. |

2. | Yongqi Fu Kok Ann Bryan, Investigation of micro- diffractive lens with continuous relief with vertical-cavity surface emitting lasers using focused ion beam direct milling.IEEE Photon. Techn. Lett. |

3. | Yongqi Fu and Ngoi Kok Ann Bryan, Hybrid micro-diffractive-refractive optical element with continuous relief fabricated by focused ion beam for single-mode coupling.Appl. Opt. |

4. | M.B. Stern, Hybrid (refractive/diffractive) Optics, in: H.P. Herzing (Ed.), Micro-optics: Elements, Systems and Applications, Taloyer & Francise, London, pp.259~292, 1997. |

5. | |

6. | Carmina Londono, William T. Plummer, and Peter P. Clark, “Athermalization of a single-component lens with diffractive optics,” Appl. Opt. |

7. | CODE-V |

8. | Xuezhe Zheng, Philippe J. Marchand, Dawei Huang, Osman Kibar, and Nur S. E. Ozkan, “Optomechanical design and characterization of a printed-circuit-board free-space optical interconnect package,” Appl. Opt. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(220.1000) Optical design and fabrication : Aberration compensation

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 14, 2002

Revised Manuscript: June 20, 2002

Published: July 1, 2002

**Citation**

Yongqi Fu and Ngoi Bryan, "Design of hybrid micro-diffractive-refractive optical element with wide field of view for free space optical interconnections," Opt. Express **10**, 540-549 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-13-540

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### References

- C. Wang, Y.C. Chan, L.P. Zhao and N. Li, �??Design of diffractive optical elements array with wide field of view for integration with photodetectors,�?? Opt. Commun. 195, 63-70 (2001). [CrossRef]
- Yongqi Fu, Kok Ann Bryan, "Investigation of micro- diffractive lens with continuous relief with vertical-cavity surface emitting lasers using focused ion beam direct milling," IEEE Photon. Techn. Lett. 13, 424-426 (2001). [CrossRef]
- Yongqi Fu, Ngoi Kok Ann Bryan, "Hybrid micro-diffractive-refractive optical element with continuous relief fabricated by focused ion beam for single-mode coupling," Appl. Opt. 40, 5872-5876, (2001). [CrossRef]
- M.B. Stern, Hybrid (refractive/diffractive) Optics, in: H.P. Herzing (Ed.), Micro-optics: Elements, Systems and Applications, (Taloyer & Francise, London, 1997) pp. 259-292.
- <a href="http://www.ioffe.rssi.ru/SVA/NSM/Semicond/InP/optic.html">http://www.ioffe.rssi.ru/SVA/NSM/Semicond/InP/optic.html</a>
- Carmina Londono, William T. Plummer, and Peter P. Clark, �??Athermalization of a single-component lens with diffractive optics,�?? Appl. Opt. 32, 2295-2302 (1993). [CrossRef] [PubMed]
- CODE-VTM is professional software of optical design, a product of Optical Research Associates (ORA), <a href="http://www.opticalres.com/macros/macroindex.html">http://www.opticalres.com/macros/macroindex.html</a>
- Xuezhe Zheng, Philippe J. Marchand, Dawei Huang, Osman Kibar, Nur S. E. Ozkan, �??Optomechanical design and characterization of a printed-circuit-board free-space optical interconnect package,�?? Appl. Opt. 38, 5631-5639 (1999). [CrossRef]

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