Pixel design

To achieve a spatial resolution comparable with existing image sensors, an angle sensitive pixel must have a pitch of only a few microns. As several periods of the diffraction grating are necessary for self-image formation, the grating itself must have a pitch of only a few wavelengths. We have used numerical modeling and simulation to analyze diffraction effects on these small length scales. Finite difference time domain (FDTD) simulations show that Talbot-like self-images still form for a micron-pitch grating and that these periodic intensity patterns exhibit a lateral shift in response to incident angle (Fig. 5.2(b)). These simulations have assumed a linear, isotropic, homogenous dielectric and infinite gratings of perfect metal conductors under monochromatic plane illumination. Although this neglects the thin barrier layers of TiN associated with backend interconnect deposition and complex oxynitride passivation cap, our previous experience is that a simplified optical model is sufficient for angle-sensitive pixel design [28].

Fortunately, modern CMOS manufacturing easily achieves sub-micron resolu- tion, and we can use metal interconnect layers to integrate high density diffraction gratings into the individual pixels of an image sensor. Furthermore, these gratings embedded in back-end of line dielectric achieve Talbot depths on the scale of the layer stack. As an example, we consider illuminating a 1µm pitch grating with green light (wavelength λ = 532nm in vacuum). If the grating is embedded in a matrix of silicon dioxide dielectric (refractive index n_{≈ 1.46, yielding an effective} wavelength of λ = 364nm), the characteristic Talbot depth is 5.50µm. Based on classical diffraction theory, strong periodic intensity patterns will occur at multi- ples of 2.75µm. This lets us integrate structures to analyze the self-images on-chip

Photodiode Photodiode z d (a) Response (au) -60 -40 -20 0 20 40 60 Incident angle, degrees

(b)

Figure 5.3: Structure of an angle-sensitive pixel. (a) For different incident angles, the analyzer grating passes or blocks light to the photodiode, generating (b) a periodic response to incident angle.

as well.

Previous work studying macro-scale Talbot effects (where grating pitch d_λ) placed a CCD array behind the grating at the Talbot depth and directly imaged the resulting intensity patterns [54]. One approach to developing an angle-sensitive pixel is to miniaturize this arrangement, using a small sub-imager array for each micro-scale grating. This approach has two primary disadvantages. First, an array of these pixels would generate a large amount of data which requires complex processing to extract information on local angle. Second, the required sub-imager array must have pixels of 1/2 the grating pitch in order to resolve the Talbot image. Although previous work has demonstrated pixel-scale sub-imager arrays in a 0.11µm CCD process [59], carrier diffusion effects typically limit their true resolution to 1µm or worse.

the first at a depth where strong periodic intensity patterns form: half-integer multiples of the Talbot depth [28]. We term this second grating an “analyzer grating” and have implemented it in metal interconnect as well. As the incident angle of light changes, the intensity patterns generated by the first grating shift relative to the analyzer grating. When the intensity patterns align with the gaps of the analyzer grating, the total light flux passed by the two gratings is high. When the intensity patterns align with the bars of the analyzer grating, little light passes through. Measuring the total light flux with a single photodiode below the analyzer grating, we recover the alignment of the self-image and therefore an angle-sensitive response.

The photodiode placed behind the two gratings measures a periodic response to incident angle (Fig. 5.3(b)). This response I can be approximated as a function of the incident angle θ and intensity Io by the relation

I = I_o(1 + m cos(βθ + α)) (5.1) where m, α, and β are parameters dependent on the geometry of the grating pair. Modulation depth m is a measure of the strength of the incident-angle dependent behavior: lower m implies less angle selectivity. The coefficient β defines an angular sensitivity, or sensitivity of the response to small changes in incident angle. α defines which angle results in a peak photodetector response and depends on the lateral offset between diffraction and analyzer gratings.

Both the modulation depth m and the angular sensitivity β control how angular information influences the output of an angle-sensitive pixel. A good optical design will maximize m, because a larger m results in a pixel which responds more strongly to incident angle. For a given vertical separation z between primary and analyzer grating, m is proportional to cos(2πz/zt), where zt is the characteristic Talbot

depths where strong self-imaging behavior occurs, namely at separation distances z = N₂zt = N d2/λdes, where N is a positive integer, d is the grating pitch, and

λdes is the design wavelength in back-end dielectric. The separation depth z and

grating pitch d additionally determine the angular sensitivity β = 2πz/nd, where n is the index of refraction of the bulk dielectric. Available inter-layer spacings in a given manufacturing process establish constraints on possible choices of z and consequently grating pitch and angular gain for angle-sensitive pixels with large values of m.

Manufacturing variability in interconnect geometry is also a critical constraint on angle-sensitive pixel design, in particular on the choice of integer N . Sources of variation include the inter-grating dielectric thickness z, grating pitch d, inter- grating alignment, and grating wire width. Each of these can potentially influence one or more of the parameters in Eq. 5.1. Errors in grating alignment will primarily result in changes to the phase offset α. Such errors are expected to be small, as large deviations would lead to back-end connectivity failures, and can be corrected using the full quadrature information available from a set of four angle-sensitive pixels, described below. Variation in line width, such as that caused by nonuniformity in dielectric etch rates, does not alter the actual pitch of the grating, but does affect the strength of the generated diffraction pattern and therefore the strength of angular response m. Based on simulation results, the percentage variation in m is approximately 70% that of the width variation. In contrast to grating offset and wire width errors, deviations in wiring pitch d, or the center-to-center spacing of the wires in the gratings, will directly influence the Talbot depth zt. Changes in this

depth will inversely affect the angular sensitivity β and degrade the modulation depth m. Similarly, variation in inter-layer dielectric thickness z also directly influence β and m.

The most critical consequence of process variation is degradation of the mod- ulation depth m of Eq. 5.1, since sufficiently small m implies a loss of angular sensitivity. A reasonable design goal is to achieve no more than a 30% degrada- tion in modulation depth from the optimum design, when the vertical separation z between the two gratings is equal to N₂zt. Because modulation m is proportional

to a cosine function of this vertical separation, the maximum absolute variation in vertical separation which achieves 0.7 of the optimum is _{|∆z| = zt}/8. The per- missible relative tolerance in vertical inter-layer spacing from the optimum which satisfies our design goals is therefore ∆z/z = 1/4N . Similar analysis results in a permissible tolerance in wiring pitch of ∆d/d = 1/8N . The chemical-mechanical polishing used to create aluminum interconnect layers is known to generate sig- nificant intermetal oxide thickness variation [121], so we have only chosen robust angle-sensitive pixel designs where N is low (1 or 2) for our image sensor.