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Lecture Notes 10 Image Sensor Optics • Imaging Optics • Pixel Optics • Microlens

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Lecture Notes 10 Image Sensor Optics • Imaging Optics • Pixel Optics • Microlens Lecture Notes 10 Image Sensor Optics • Imaging optics ◦ Space-invariant model ◦ Space-varying model • Pixel optics ◦ Transmission ◦ Vignetting • Microlens EE 392B: Image Sensor Optics 10-1 Image Sensor Optics Microlens Imaging Optics Pixel Optics Image sensor optics consist of (a) imaging optics and (b) pixel optics EE 392B: Image Sensor Optics 10-2 Imaging Optics: Linear Space-Invariant Model Y Y Entrance Exit Pupil Pupil X X Z Object Generalized Model Image L (W/m2 sr) (f/#, T, R) I (W/m2) A generalized model for rotationally symmetric imaging optics J.W. Goodman, Introduction to Fourier Optics, 2nd Ed, Ch 6, pp. 126-171 (1996) EE 392B: Image Sensor Optics 10-3 Imaging Optics: Camera Equation T (λ ) R(x, y;λ ) ( λ ) ≅ π x y λ Iideal x, y; Lscene , ; 1+ 4(1+ m)2 ( f /#)2 m m 2 Iideal : ideal, unaberrated, geometric image irradiance distribution (W m ) 2 Lscene: Lambertian object radiance distribution (W sr m ) = − m zi zo : magnification T : transmittance R : relative illumination factor (e.g., cos4 fall-off) f /# = f D: f-number Imaging optics map 2D object radiance distribution (W/sr m2) into 2D image irradiance distribution (W/m2) M.V. Klein and T.E. Furtak, Optics, , 2nd Ed, Ch 4, pp. 193-262 (1986) EE 392B: Image Sensor Optics 10-4 Resolution Limited by Diffraction Plane Converging Plane Converging wave wave wave wave Perfect Diffraction geometric limited point point Infinite Aperture Optics Finite Aperture Optics Diffraction, which is caused at aperture edges, is the fundamental reason why a plane wave can not be focussed into a perfect geometric point EE 392B: Image Sensor Optics 10-5 Diffraction: Space Domain Description Image formation process with incoherent illumination can be described in the space domain by a convolution operation ( λ) = ( λ)∗ ( λ) Iimage x, y; PSF x, y; Iideal x, y; Iimage: blurred, aberrated, distorted image irradiance distribution Iideal : ideal, unaberrated, geometric image irradiance distribution PSF: point spread function, i.e., response of optical system in image plane to a point excitation in object plane ∗: convolution operator EE 392B: Image Sensor Optics 10-6 Diffraction: Frequency Domain Description Alternatively, image formation in shift-invariant systems can be viewed as a linear filtering process in the frequency domain ( λ) = ( λ) ⋅ ( λ) F {Iimage x, y; } F {PSF x, y; } F {Iideal x, y; } = λ ⋅ ( λ) OTF( fx , f y ; ) F {Iideal x, y; } ( λ) = −1 λ ⋅ ( λ) Iimage x, y; F {OTF( f x , f y ; ) F {Iideal x, y; }} The optical transfer function (OTF) is normalized to preserve radiometry. Its magnitude is the modulation transfer function (MTF) F {PSF(x, y;λ)} λ = λ = MTF( fx , f y ; ) OTF( fx , f y ; ) F {PSF(x, y;λ)} = = fx 0, f y 0 EE 392B: Image Sensor Optics 10-7 Example: Diffraction-limited Lens 1 f/# = 2.0, λ = 486 nm 0.9 0.8 0.7 0.6 0.5 MTF 0.4 0.3 0.2 0.1 0 0 200 400 600 800 1000 1200 Spatial frequency (lp/mm) 2 − −1 1 MTF(fr) = π(' cos ' sin ') with ' = cos (frλf=#) and fr;cutoff = λf=# EE 392B: Image Sensor Optics 10-8 System MTF If the system is linear and space-invariant, the system MTF (optics + sensor) in the frequency domain can then be easily computed λ = λ ⋅ ⋅ λ MTFsystem ( f x , f y ; ) MTFoptics ( fx , f y ; ) MTFgeometric ( fx , f y ) MTFdiffusion ( fx , f y ; ) To obtain the image spectrum we apply the MTF as a linear filter λ = λ ⋅ λ Iimage ( fx , f y ; ) MTFsystem ( fx , f y ; ) Iobject ( fx , f y ; ) EE 392B: Image Sensor Optics 10-9 Modeling Real Imaging Lenses Vignetting Distortion Illumination Effects Wavefront Point Spread Modulation Transfer Aberrations Function (PSF) Function (MTF) P. Y. Maeda, P. B. Catrysse, and B. A. Wandell, Proc. SPIE Int. Soc. Opt. Eng. 5678, pp. 48-58 (2005) EE 392B: Image Sensor Optics 10-10 Imaging Optics: Linear Space-Varying Model Y Y W Entrance Exit k Pupil Pupil X X Z Object Generalized Model Image L (W/m2 sr) (f/#, T, R) I (W/m2) Image plane is segmented into isoplanatic regions: Ω1, Ω2, : : :, Ωn P. Y. Maeda, P. B. Catrysse, and B. A. Wandell, Proc. SPIE Int. Soc. Opt. Eng. 5678, pp. 48-58 (2005) EE 392B: Image Sensor Optics 10-11 Linear Space-Varying Image Formation Let Ωk, k = 1; : : : ; n be aplanatic image segments, then I (x, y;λ) = ∑PSFΩ (x, y;λ)∗ I Ω (x, y;λ) image k ideal, k k or −1 I (x, y;λ) = F ∑OTFΩ ( f , f ;λ)⋅ F {I Ω ( x, y;λ)} image k x y ideal, k k P. Y. Maeda, P. B. Catrysse, and B. A. Wandell, Proc. SPIE Int. Soc. Opt. Eng. 5678, pp. 48-58 (2005) EE 392B: Image Sensor Optics 10-12 Example: Double Gauss f/2.0 Lens P. Y. Maeda, P. B. Catrysse, and B. A. Wandell, Proc. SPIE Int. Soc. Opt. Eng. 5678, p. 48-58 (2005) EE 392B: Image Sensor Optics 10-13 Double Gauss f/2.0 Lens: MTF P. Y. Maeda, P. B. Catrysse, and B. A. Wandell, Proc. SPIE Int. Soc. Opt. Eng. 5678, p. 48-58 (2005) EE 392B: Image Sensor Optics 10-14 Pixel Optics Incident Photons Reflected Transmitted Photons Photons iciency f Rejected or Accepted Scattered Photons Optical Ef Photons Uncollected Collected Electrons iciency Electrons f Quantum Ef P. B. Catrysse and B. A. Wandell, J. Opt. Soc. Am. A 19, pp. 1610-1620 (2002) H. Rhodes et al., IEEE Workshop on Microelectronics and Electron Devices, pp. 7-18 (2004) EE 392B: Image Sensor Optics 10-15 Incident Photons -3 x 10 10 9 c) 8 x-se u 7 e (l ur s 6 po x 4000 E 5 c i tr e 4 3 hotom 2000 P 2 1000 1 2 3 4 5 6 7 8 Pixel size (mm) P. B. Catrysse and B. A. Wandell, Proc. SPIE Int. Soc. Opt. Eng. 5678, pp. 1-13 (2005) EE 392B: Image Sensor Optics 10-16 Pixel Transmittance F+(z’) F-(z’) Layer 0 I01 Layer 1 L1 Layer j-1 Transmission Ij-1,j Layer j Lj Layer m Lm Im,m+1 Layer m+1 F+(z”) F-(z) F. Abeles, Ann. de Phys. 5, pp. 596-640 (1950) P. B. Catrysse and B. A. Wandell, J. Opt. Soc. Am. A 19, pp. 1610-1620 (2002) EE 392B: Image Sensor Optics 10-17 Example: Transmittance 0.18µm CMOS 1 0.8 0.6 0.4 air Transmittance 0.7 mm Si3N4 0.2 8.3 mm SiO2 Si 0 400 500 600 700 800 Wavelength (nm) Pixel transmittance is λ-dependent (even for dispersion-free materials) P. B. Catrysse and B. A. Wandell, J. Opt. Soc. Am. A 19, pp. 1610-1620 (2002) EE 392B: Image Sensor Optics 10-18 Example: Transmittance 0.18µm CMOS 0.8 0.6 Si3N4 0.4 nsmission T a r SiO2 T 0.2 Si 0 0 20 40 60 80 100 Angle (degrees) Pixel transmittance (λ-averaged) is approximately independent of angle P. B. Catrysse and B. A. Wandell, J. Opt. Soc. Am. A 19, pp. 1610-1620 (2002) EE 392B: Image Sensor Optics 10-19 Pixel Vignetting No Vignetting Pixel Vignetting Pixel Cross-talk P. B. Catrysse and B. A. Wandell, J. Opt. Soc. Am. A 19, pp. 1610-1620 (2002) EE 392B: Image Sensor Optics 10-20 Pixel Vignetting: Effect of Pixel Height 0.7 0 degrees 11 degrees 0.6 23 degrees 0.5 0.4 0.3 Pixel response 0.2 0.1 0 1 2 3 4 5 6 7 Number of metal layers Reduction in optical efficiency as a function of the number of metal layers in a 0.18µm standard CMOS process (f=1.8 imaging lens) P. B. Catrysse and B. A. Wandell, J. Opt. Soc. Am. A 19, pp. 1610-1620 (2002) EE 392B: Image Sensor Optics 10-21 Pixel Vignetting: Effect of Technology Scaling 0.8 0 degrees 11 degrees 0.7 23 degrees se 0.6 pon res l e x 0.5 i P 0.4 0.3 0.4 0.35 0.3 0.25 0.2 0.15 Feature size (mm) Reduction in optical efficiency for a standard APS pixel with a 30% fill-factor using 2 metal layers as a function of the feature size of CMOS technology P. B. Catrysse and B. A. Wandell, J. Opt. Soc. Am. A 19, pp. 1610-1620 (2002) EE 392B: Image Sensor Optics 10-22 Optical Efficiency • Definition: ◦ Optical efficiency (OE) is the ratio of the photons incident of the substrate and the photons incident on the pixel surface • Sources of photon loss: ◦ Back-reflections in dielectric stack (air-SiO2, SiO2-Si) ◦ Photons absorbed in dielectric stack (SiON) ) Pixel transmittance T(λ,θ) ◦ Photons scattered away from pixel (pixel cross-talk) ◦ Photons rejected by metal ) Pixel vignetting V(x,y,θ) • Description: ◦ OE(x,y,λ,θ) = T(λ,θ) V(x,y,θ) EE 392B: Image Sensor Optics 10-23 Microlens • Focus light onto photo-sensitive region { increases effective fill factor from 25-40% to 60-80% (and sensitivity by ≥ 2X) • Less effective if photosensitive area is irregularly shaped A. Theuwissen, \Solid State Imaging with Charge-Coupled Devices," Kluwer (1995) S.G. Wuu et al., \High performance 0.25 µm CMOS color imager technology with non-silicide source/drain pixel," IEDM Tech. Dig., pp. 705-708 (2000) EE 392B: Image Sensor Optics 10-24 Microlens Fabrication • Lens material requirements: ◦ Highly transparent in the visible light region ◦ Index of refraction > 1.59 ◦ Can be applied below 500C ◦ No degradation or aging ◦ Semiconductor processing com- patible ◦ Can be patterned with feature size commensurate with the pixel size • Lens materials are typically i-line or DUV resists • Base materials are acrylic-based resists, polyimide resists, epoxy resists, poly- organosiloxane, polyorganosilicate EE 392B: Image Sensor Optics 10-25 Microlens and the Main Lens • Microlens is optimized for a specific main lens system • Rays incident on the microlens form a cone with NA = sin ' • NA varies as a function of the size and position of the exit pupil • Principle ray at the periphery of the sensor has an angle δ, chief ray angle (CRA), between the ray and the optical axis (δ depends on the position of the pixel on the sensor) A.
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