## Uniform illumination and rigorous electromagnetic simulations applied to CMOS image sensors

Optics Express, Vol. 15, Issue 9, pp. 5494-5503 (2007)

http://dx.doi.org/10.1364/OE.15.005494

Acrobat PDF (700 KB)

### Abstract

This paper describes a new methodology we have developed for the optical simulation of CMOS image sensors. Finite Difference Time Domain (FDTD) software is used to simulate light propagation and diffraction effects throughout the stack of dielectrics layers. With the use of an incoherent summation of plane wave sources and Bloch Periodic Boundary Conditions, this new methodology allows not only the rigorous simulation of a diffuse-like source which reproduces real conditions, but also an important gain of simulation efficiency for 2D or 3D electromagnetic simulations. This paper presents a theoretical demonstration of the methodology as well as simulation results with FDTD software from Lumerical Solutions.

© 2007 Optical Society of America

## 1. Introduction

2. E. R. Fossum, “CMOS Image Sensors: Electronic Camera-On-A-Chip,” IEEE Trans. Electron. Devices **44**, 1689–1698 (1997). [CrossRef]

3. P. B. Catrysse, X. Liu, and A. El Gamal, “QE Reduction due to Pixel Vignetting in CMOS Image Sensors”, Proc. SPIE **3965**, 420–430 (2000). [CrossRef]

4. P. B. Catrysse and B. A. Wandell, “Optical efficiency of image sensor pixels,” J. Opt. Soc. Am A **19**, 1610–1620 (2002). [CrossRef]

5. J. Vaillant and F. Hirigoyen, “Optical simulation for CMOS imager microlens optimization,” Proc. SPIE **5459**, 200–210 (2004). [CrossRef]

7. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in Isotropic Media,” IEEE Trans. Antennas Propag. **14**, 302–307 (1966). [CrossRef]

9. Lumerical Solutions, Inc. http://www.lumerical.com.

## 2. The simulation methodology

### 2.1 The problem: the light shape

*E*from these single point sources that pass through the objective-lens. Figure 3 below shows a schematic representation of the light source seen by the pixel array.

_{i}### 2.2 The different solutions for simulation

#### 2.2.1 Impulse response superposition

#### 2.2.2 Plane wave’s superposition

## 3. Theoretical demonstration of the methodology

*D*and a focal length

*f*. Its transmission versus the coordinate inside the exit pupil is

*t(x,y)*which possibly includes vignetting and/or aberrations. This lens is illuminated by the superposition of incoherent tilted plane waves and the wavelength

*λ*. Each wave has an amplitude

*A*and its tilt is referred with its direction cosines

*α*=

_{0}*x*/

_{0}*f*,

*β*=

_{0}*y*/

_{0}*f*, and

*x*and

_{0}*y*are the coordinates in the image plane of the impulse response offset (see Fig. 6).

_{0}*I*

_{x0,y0}(

*x*,

_{f}*y*) due to each plane wave in the focal plan (

_{f}*x*,

_{f}*y*) (corresponding to the

_{f}*E*electric field in Fig. 3) is given [10

_{i}10. J. W. Goodman, *Introduction to Fourier Optics, 3 ^{rd} Edition* (Roberts & Company Publishers, Englewood, Co, 2005), Chap. 5. [PubMed]

*I*in the image plane given by:

_{f}*FT*denotes Fourier Transform.

*k*→:

*t(k*is simply the pupil function:

_{x},k_{y})*NA*the Numerical Aperture,

*k*,

_{x}*k*and amplitude

_{y}*A*) weighted by the function

*W(k*. The total intensity

_{x},k_{y})*I*in the plane

_{PW}*(x*is:

_{f},y_{f})*W(k*=

_{x},k_{y})*P(k*, i.e. a uniform distribution of plane waves inside a cone defined by the exit pupil diameter

_{x},k_{y})*D*and the focal length

*f*.

*i.e.*with

*P(k*and non-constant inside the cone defined by the pupil. In this case, we still could simulate the new thin lens by a sum of incoherent plane waves with different weight according to the wave vector

_{x},k_{y})≠*1**k*,

_{x}*k*.

_{y}## 4. Simulation results

*a*= 2 μm. The wavelength is 550nm and the aperture NA=0.26 (see Fig. 4). The Poynting vector along y direction,

*P*, is normalized to the source power per unit cell such that the total transmission

_{y}(x)*T*, normalized to the source power can be calculated by

*P*for the thin lens sources (on the left) and for the plane wave sources (on the right) for on-axis pixels,

_{y}*i.e.*pixels at the center of the sensor array. Figure 8 presents similar results for off-axis pixels,

*i.e.*pixels on the edge of the sensor with an angle-shift of 10°.

## 5. Conclusion

## References and links

1. | A. El Gamal and H. Eltoukhy, “CMOS Image Sensors. An introduction to the technology, design, and performance limits, presenting recent developments and future directions,” IEEE Circuits & Devices Magazine (May/June 2005). |

2. | E. R. Fossum, “CMOS Image Sensors: Electronic Camera-On-A-Chip,” IEEE Trans. Electron. Devices |

3. | P. B. Catrysse, X. Liu, and A. El Gamal, “QE Reduction due to Pixel Vignetting in CMOS Image Sensors”, Proc. SPIE |

4. | P. B. Catrysse and B. A. Wandell, “Optical efficiency of image sensor pixels,” J. Opt. Soc. Am A |

5. | J. Vaillant and F. Hirigoyen, “Optical simulation for CMOS imager microlens optimization,” Proc. SPIE |

6. | H. Rhodes, G. Agranov, C. Hong, U. Boettiger, R. Mauritzon, J. Ladd, I. Karasev, J. McKee, E. Jenkins, W. Quinlin, I. Patrick, J. Li, X. Fan, R. Panicacci, S. Smith, C. Mouli, and J. Bruce, “CMOS Imager Technology Shrinks and Image Performance,” IEEE (2004). |

7. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in Isotropic Media,” IEEE Trans. Antennas Propag. |

8. | A. Taflove and S. C. Hagness, |

9. | Lumerical Solutions, Inc. http://www.lumerical.com. |

10. | J. W. Goodman, |

**OCIS Codes**

(040.0040) Detectors : Detectors

(110.0110) Imaging systems : Imaging systems

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: November 10, 2006

Revised Manuscript: April 18, 2007

Manuscript Accepted: April 18, 2007

Published: April 20, 2007

**Citation**

Jérôme Vaillant, Axel Crocherie, Flavien Hirigoyen, Adam Cadien, and James Pond, "Uniform illumination and rigorous electromagnetic simulations applied to CMOS image sensors," Opt. Express **15**, 5494-5503 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5494

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### References

- A. El Gamal and H. Eltoukhy, "CMOS Image Sensors. An introduction to the technology, design, and performance limits, presenting recent developments and future directions," IEEE Circuits & Devices Magazine (May/June 2005).
- E. R. Fossum, "CMOS Image Sensors: Electronic Camera-On-A-Chip," IEEE Trans. Electron. Devices 44, 1689-1698 (1997). [CrossRef]
- P. B. Catrysse, X. Liu, and A. El Gamal, "QE Reduction due to Pixel Vignetting in CMOS Image Sensors," Proc. SPIE 3965, 420-430 (2000). [CrossRef]
- P. B. Catrysse and B. A. Wandell, "Optical efficiency of image sensor pixels," J. Opt. Soc. Am A 19, 1610-1620 (2002). [CrossRef]
- J. Vaillant and F. Hirigoyen, "Optical simulation for CMOS imager microlens optimization," Proc. SPIE 5459, 200-210 (2004). [CrossRef]
- H. Rhodes, G. Agranov, C. Hong, U. Boettiger, R. Mauritzon, J. Ladd, I. Karasev, J. McKee, E. Jenkins, W. Quinlin, I. Patrick, J. Li, X. Fan, R. Panicacci, S. Smith, C. Mouli, and J. Bruce, "CMOS Imager Technology Shrinks and Image Performance," IEEE (2004).
- K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in Isotropic Media," IEEE Trans. Antennas Propag. 14, 302-307 (1966). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics : the finite-difference time-domain method, 2nd Edition, H. E. Schrank, Series Editor (Artech House, Boston, Ma, 2000).
- Lumerical Solutions, Inc.http://www.lumerical.com.
- J. W. Goodman, Introduction to Fourier Optics, 3rd Edition (Roberts & Company Publishers, Englewood, Co, 2005), Chap. 5. [PubMed]

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