Derivation of Crowell and Labuda for Special Cases using SymPy

Detector Comparison: CMOS, CCD, and MCT

This table groups imaging detectors by their physical architecture, light-entry geometry, and the resulting internal layer logic.

Detector Category	Architecture	Geometry (Light Path)	Internal Layer Order	Core Assumption / Key Feature
Standard CMOS	FSI (Front-Illuminated)	Light hits circuitry side first.	Depletion → Field-Free	Shadowing: Metal wiring reduces fill factor; requires microlenses to focus light.
Scientific CMOS (sCMOS)	BSI (Back-Illuminated)	Light hits thinned silicon substrate.	Field-Free → Depletion	Diffusion: Uses PPD (Pinned Photodiode) and DTI (Deep Trench Isolation) to prevent crosstalk.
Standard CCD	FSI (Front-Illuminated)	Light hits polysilicon gate electrodes.	Depletion → Field-Free	Charge Transfer: High CTE is required to move charge across the chip without “smearing.”
Scientific CCD	BSI (Back-Illuminated)	Light hits thinned back-surface.	Field-Free → Depletion	Thinning: Wafer is thinned (<20 µm) so electrons don’t wander before hitting the depletion zone.
Deep Depletion CCD	Thick-Substrate	High-bias voltage applied.	Fully Depleted (No Field-Free)	High Energy: Eliminates the field-free region entirely to capture NIR or X-ray photons accurately.
MCT (Infrared)	Hybrid / BSI	Light hits MCT absorber layer.	Field-Free → Depletion	Hybridization: Detector material is bump-bonded to a Silicon ROIC (Readout Integrated Circuit).

Key Definitions:

• Field-Free (Neutral) Region: A region with no internal electric field where electrons move via slow, random diffusion.

• Depletion Region: A region with a strong electric field that “sweeps” electrons toward the storage well instantly.

• PPD (Pinned Photodiode): A structure that “pins” the surface potential to reduce dark current and eliminate “image lag.”

• DTI (Deep Trench Isolation): Physical “trenches” etched between pixels to prevent electrons in the field-free region from drifting into adjacent pixels.

Notes on C&L Applicability

• BSI detectors (sCMOS, Scientific CCD, MCT) match the C&L geometry: light enters the field-free region first, then reaches depletion. These are the primary targets for the C&L model.

• FSI detectors (Standard CMOS, Standard CCD) have reversed geometry: light enters the depletion region first. The C&L equation does not directly apply — a reversed-geometry derivation from first principles is needed.

• Deep Depletion CCD has no field-free region, so diffusion MTF is unity (no degradation).

• For Scientific CCD, the dead back surface (\(S \to \infty\)) is a key concern — this maps directly to Case 1.

Detector → C&L Parameter Mapping and Derivation Cases

Detector Type	Light Direction	\(S\)	\(R\)	\(L_b\)	Case Number
Scientific CCD (BSI, thinned)	Field-Free → Depletion	\(S \to \infty\)	\(R = 0\)	finite	Case 1
sCMOS (BSI, passivated)	Field-Free → Depletion	\(S \approx 0\)	\(R = 0\)	finite	Case 2
MCT Infrared (BSI, thick)	Field-Free → Depletion	finite \(S\)	\(R \approx 0\)	\(L_b \to \infty\)	Case 3
Deep Depletion CCD	Fully Depleted	—	—	—	No field-free → MTF = 1
FSI (CMOS/CCD)	Depletion → Field-Free	finite \(S\)	\(R = 0\)	finite	Reversed geometry

Note on light direction: The original C&L model derives the diffusion equation for light entering the field-free region first (BSI geometry). For FSI detectors, where light enters the depletion region first, the generation profile in the field-free region changes and the boundary conditions are different. Both directions must be derived from the diffusion equation.

Crowell and Labuda Diffusion MTF Model

The most general analytical diffusion MTF model, from the original Crowell & Labuda (1969) paper “The Silicon Diode Array Camera Tube” (BSTJ Vol. 48, No. 5, Eq. 5).

Geometry (Fig. 3 in the paper): Light enters at \(y = 0\) (illuminated surface), passes through the undepleted field-free region (\(0 \le y \le L_a\)), and reaches the depletion region edge at \(y = L_a\). The total n-type region extends to \(y = L_b\), so the depletion width is \(L_b - L_a\).

Boundary conditions (Eq. 2):

• At \(y = 0\) (illuminated surface): \(S p = D \frac{\partial p}{\partial y}\) (surface recombination)

• At \(y = L_a\) (depletion edge): \(p = 0\) (carriers swept away by E-field)

Generation (Eq. 3): \(G(x, y) = \frac{N_0}{2} \alpha (1 - R)(1 + \cos kx) e^{-\alpha y}\)

where \(R\) is the back-surface reflectivity and \(k = 2\pi f\) is the spatial frequency.

The collection efficiency at spatial frequency \(k\) is given by Eq. (5):

\[\eta_k = \frac{\alpha L (1 - R)}{\alpha^2 L^2 - 1} \left[ \frac{2(\alpha L + SL/D) - (\beta_+ - \beta_-) \exp(-\alpha L_a)}{\beta_+ + \beta_-} - (\alpha L)^{-1} \exp(-\alpha L_a) \right] - (1 - R) \exp(-\alpha L_b)\]

where:

• \(\beta_{\pm} = (1 \pm SL/D) \exp(\pm L_a / L)\)

• \(1/L^2(k) = 1/L_o^2 + k^2\) (so \(L(k) = L_o / \sqrt{1 + (2\pi f L_o)^2}\), our \(L_f\))

• \(L_o = \sqrt{D \tau}\) is the diffusion length

• \(L_a\) = undepleted (field-free) region thickness

• \(L_b\) = total n-type region thickness (\(L_b - L_a\) = depletion width)

• \(S\) = surface recombination velocity at the illuminated surface (cm/s)

• \(D\) = minority carrier diffusion coefficient (cm2/s)

• \(R\) = back-surface reflectivity

The diffusion MTF is: \(R_D(k) = \eta_k / \eta_0\)

Note on the last term: The \(-(1-R)\exp(-\alpha L_b)\) term accounts for photons that traverse the entire substrate without being absorbed. For thick substrates (\(\alpha L_b \gg 1\)), this term is negligible.

C&L Equation in SymPy

import sympy as sp

# Define symbols
alpha = sp.Symbol("alpha", positive=True)  # absorption coefficient [1/length]
k = sp.Symbol("k", nonnegative=True)  # spatial frequency [rad/length]
L_o = sp.Symbol("L_o", positive=True)  # zero-frequency diffusion length
L_a = sp.Symbol("L_a", positive=True)  # field-free (undepleted) region thickness
L_b = sp.Symbol("L_b", positive=True)  # total substrate thickness (L_b > L_a)
S = sp.Symbol("S", nonnegative=True)  # surface recombination velocity
D = sp.Symbol("D", positive=True)  # minority carrier diffusion coefficient
R = sp.Symbol("R", nonnegative=True)  # back-surface reflectivity (0 <= R <= 1)

# Frequency-dependent diffusion length  L(k) = L_o / sqrt(1 + k^2 * L_o^2)
L = L_o / sp.sqrt(1 + k**2 * L_o**2)

# Dimensionless groups
aL = alpha * L  # alpha * L(k)
sLD = S * L / D  # S*L/D  (dimensionless surface recombination term)

# beta_+/-  (Eq. 5 of Crowell & Labuda 1969)
beta_p = (1 + sLD) * sp.exp(L_a / L)
beta_m = (1 - sLD) * sp.exp(-L_a / L)

# Bracket numerator and denominator
bracket_num = 2 * (aL + sLD) - (beta_p - beta_m) * sp.exp(-alpha * L_a)
bracket_den = beta_p + beta_m

# Full C&L collection efficiency eta_k  (Eq. 5)
eta_k = alpha * L * (1 - R) / (alpha**2 * L**2 - 1) * (
    bracket_num / bracket_den - sp.exp(-alpha * L_a) / (alpha * L)
) - (1 - R) * sp.exp(-alpha * L_b)

print("eta_k defined. Symbols:", eta_k.free_symbols)

eta_k defined. Symbols: {alpha, D, L_o, S, k, R, L_b, L_a}

Check \(\beta_+\), \(\beta_-\) and the bracket before looking at \(\eta_k\) in full.

sp.init_printing(use_unicode=True)

print("β₊ =")
display(beta_p)
print("β₋ =")
display(beta_m)
print("β₊ + β₋ =")
display(sp.simplify(beta_p + beta_m))
print("β₊ - β₋ =")
display(sp.simplify(beta_p - beta_m))

Note that \(\beta_+ + \beta_-\) and \(\beta_+ - \beta_-\) can be rewritten using hyperbolic functions:

\[\beta_+ + \beta_- = 2\cosh\!\left(\frac{L_a}{L}\right) + 2\frac{SL}{D}\sinh\!\left(\frac{L_a}{L}\right)\]

\[\beta_+ - \beta_- = 2\sinh\!\left(\frac{L_a}{L}\right) + 2\frac{SL}{D}\cosh\!\left(\frac{L_a}{L}\right)\]

This is a cleaner form that we can verify below.

# Rewrite beta sums using hyperbolic functions and verify they are equivalent
xi = L_a / L  # shorthand for L_a / L(k)

beta_sum_hyp = 2 * sp.cosh(xi) + 2 * sLD * sp.sinh(xi)
beta_diff_hyp = 2 * sp.sinh(xi) + 2 * sLD * sp.cosh(xi)

# Verify identities (should be zero)
check_sum = sp.simplify(
    (beta_p + beta_m).rewrite(sp.exp) - beta_sum_hyp.rewrite(sp.exp)
)
check_diff = sp.simplify(
    (beta_p - beta_m).rewrite(sp.exp) - beta_diff_hyp.rewrite(sp.exp)
)
print("β₊ + β₋  check (should be 0):", check_sum)
print("β₊ - β₋  check (should be 0):", check_diff)

β₊ + β₋  check (should be 0): 0
β₊ - β₋  check (should be 0): 0

Full \(\eta_k\) in hyperbolic form

Substituting the hyperbolic identities, the bracket ratio becomes:

\[\frac{\beta_+ - \beta_-}{\beta_+ + \beta_-} = \frac{\sinh(L_a/L) + (SL/D)\cosh(L_a/L)}{\cosh(L_a/L) + (SL/D)\sinh(L_a/L)}\]

and the full collection efficiency is:

\[\eta_k = \frac{\alpha L (1-R)}{\alpha^2 L^2 - 1} \left[ \frac{2(\alpha L + SL/D) - \left[2\sinh\!\frac{L_a}{L} + 2\frac{SL}{D}\cosh\!\frac{L_a}{L}\right]e^{-\alpha L_a}}{2\cosh\!\frac{L_a}{L} + 2\frac{SL}{D}\sinh\!\frac{L_a}{L}} - \frac{e^{-\alpha L_a}}{\alpha L} \right] - (1-R)\,e^{-\alpha L_b}\]

# Rewrite eta_k using hyperbolic form and verify equivalence
eta_k_hyp = alpha * L * (1 - R) / (alpha**2 * L**2 - 1) * (
    (2 * (aL + sLD) - beta_diff_hyp * sp.exp(-alpha * L_a)) / beta_sum_hyp
    - sp.exp(-alpha * L_a) / (alpha * L)
) - (1 - R) * sp.exp(-alpha * L_b)

# Verify equivalence with the original exponential form
check_eta = sp.simplify(eta_k.rewrite(sp.exp) - eta_k_hyp.rewrite(sp.exp))
print("eta_k equivalence check (should be 0):", check_eta)

eta_k equivalence check (should be 0): 0

Reversed Geometry: FSI (Depletion → Field-Free)

For front-side illuminated (FSI) detectors (Standard CMOS, Standard CCD), light enters the depletion region first, then reaches the field-free region. We derive the collection efficiency from the diffusion equation.

Geometry

• Light enters at \(z = 0\) (illuminated surface / depletion region edge)

• Depletion region: \(0 \le z \le L_D\) (carriers collected instantly, no diffusion)

• Field-free region: \(L_D \le z \le L_D + L_a\) (carriers diffuse)

• Far surface at \(z = L_D + L_a\) with surface recombination velocity \(S\)

We define a local coordinate \(y\) within the field-free region: \(y = z - L_D\), so \(0 \le y \le L_a\).

Differences from the original C&L (BSI) model

	C&L / BSI (Field-Free → Depletion)	Reversed / FSI (Depletion → Field-Free)
BC at :math:`y = 0`	Surface recombination: \(Sp = D \frac{\partial p}{\partial y}\)	Depletion edge: \(p = 0\)
BC at :math:`y = L_a`	Depletion edge: \(p = 0\)	Surface recombination: \(-D \frac{\partial p}{\partial y} = Sp\)
Generation	\(G \propto \alpha\, e^{-\alpha y}\) (full intensity)	\(G \propto \alpha\, e^{-\alpha L_D}\, e^{-\alpha y}\) (attenuated by depletion)
Depletion collection	Included via back-surface term	Separate term: \(\eta_{\text{dep}} = 1 - e^{-\alpha L_D}\)
Back-surface reflection	\(R\) can be nonzero	\(R = 0\) (photons exit through back)
Collection current	Diffusion current at \(y = L_a\) (depletion edge)	Diffusion current at \(y = 0\) (depletion edge)

Note the sign convention change for the surface recombination BC: at \(y = L_a\) the outward normal is \(+y\), so the flux into the surface is \(-D \frac{\partial p}{\partial y}\).

Simplification: Dead far surface (\(S \to \infty\))

For standard FSI detectors, the far surface (back of the substrate) typically has very high recombination (\(S \to \infty\)), giving the simpler BC:

• At \(y = 0\): \(p = 0\) (depletion edge)

• At \(y = L_a\): \(p = 0\) (dead far surface)

Diffusion equation

\[\frac{d^2 p_k}{dy^2} - \frac{p_k}{L^2} = -\frac{\alpha}{D}\, e^{-\alpha L_D}\, e^{-\alpha y}\]

The general solution is:

\[p_k(y) = A \sinh\!\frac{y}{L} + B \cosh\!\frac{y}{L} + P\, e^{-\alpha y}\]

where \(P = \frac{\alpha L^2}{D(\alpha^2 L^2 - 1)} e^{-\alpha L_D}\).

BC1 (\(p(0) = 0\)): \(B = -P\)

BC2 (\(p(L_a) = 0\)): \(A \sinh\xi + B \cosh\xi + P e^{-\alpha L_a} = 0\), giving:

\[A = P\,\frac{\cosh\xi - e^{-\alpha L_a}}{\sinh\xi}\]

where \(\xi = L_a/L\).

Collection efficiency

The diffusion current at the depletion edge (\(y = 0\)) gives the field-free contribution:

\[\eta_{\text{ff},k} = -D \frac{dp_k}{dy}\bigg|_{y=0} = -D\left(\frac{A}{L} - \alpha P\right)\]

The total collection efficiency is:

\[\eta_k^{\text{rev}} = \underbrace{(1 - e^{-\alpha L_D})}_{\text{depletion}} + \eta_{\text{ff},k}\]

# --- Reversed geometry (FSI), S -> inf: p(0) = 0, p(L_a) = 0 ---

y = sp.Symbol("y", nonneg=True)
L_D = sp.Symbol("L_D", positive=True)

xi = L_a / L

# Particular solution coefficient
P_coeff = alpha * L**2 / (D * (alpha**2 * L**2 - 1)) * sp.exp(-alpha * L_D)

# BC1: p(0) = 0  =>  B = -P
# BC2: p(L_a) = 0  =>  A = P * (cosh(xi) - exp(-alpha*L_a)) / sinh(xi)
A_coeff = P_coeff * (sp.cosh(xi) - sp.exp(-alpha * L_a)) / sp.sinh(xi)

# Collection: J = -D * (A/L - alpha*P)  at y=0
J_diff_0_rev = -D * (A_coeff / L - alpha * P_coeff)
J_diff_0_rev = sp.simplify(J_diff_0_rev)

print("Field-free collection eta_ff (reversed, S->inf):")
display(J_diff_0_rev)

# Verify the closed-form from the markdown
eta_ff_expected = (
    alpha
    * L
    * sp.exp(-alpha * L_D)
    / (alpha**2 * L**2 - 1)
    * (alpha * L - (sp.cosh(xi) - sp.exp(-alpha * L_a)) / sp.sinh(xi))
)

check = sp.simplify((J_diff_0_rev - eta_ff_expected).rewrite(sp.exp))
print("\nVerification vs closed form (should be 0):", check)

Field-free collection eta_ff (reversed, S->inf):

$\displaystyle - \frac{L_{o} \alpha \left(L_{o} \alpha e^{\alpha \left(L_{D} + L_{a}\right)} + \frac{\sqrt{L_{o}^{2} k^{2} + 1} e^{L_{D} \alpha}}{\sinh{\left(\frac{L_{a} \sqrt{L_{o}^{2} k^{2} + 1}}{L_{o}} \right)}} - \frac{\sqrt{L_{o}^{2} k^{2} + 1} e^{\alpha \left(L_{D} + L_{a}\right)}}{\tanh{\left(\frac{L_{a} \sqrt{L_{o}^{2} k^{2} + 1}}{L_{o}} \right)}}\right) e^{- \alpha \left(2 L_{D} + L_{a}\right)}}{- L_{o}^{2} \alpha^{2} + L_{o}^{2} k^{2} + 1}$

Verification vs closed form (should be 0): 0

# Total collection efficiency and MTF (reversed geometry, S->inf)
eta_dep = 1 - sp.exp(-alpha * L_D)
eta_total_k_rev = sp.simplify(eta_dep + J_diff_0_rev)

print("Total collection efficiency eta_k (reversed, S->inf):")
display(eta_total_k_rev)

# Diffusion MTF: MTF(k) = eta_total_k / eta_total_0
eta_total_0_rev = eta_total_k_rev.subs(k, 0)
check_mtf0 = sp.simplify((eta_total_k_rev / eta_total_0_rev).subs(k, 0).rewrite(sp.exp))
print("MTF(k=0) check (should be 1):", check_mtf0)

Total collection efficiency eta_k (reversed, S->inf):

$\displaystyle - \frac{\left(L_{o}^{2} \alpha^{2} e^{\alpha \left(3 L_{D} + L_{a}\right)} + L_{o}^{2} k^{2} e^{\alpha \left(2 L_{D} + L_{a}\right)} - L_{o}^{2} k^{2} e^{\alpha \left(3 L_{D} + L_{a}\right)} + \frac{L_{o} \alpha \sqrt{L_{o}^{2} k^{2} + 1} e^{2 L_{D} \alpha}}{\sinh{\left(\frac{L_{a} \sqrt{L_{o}^{2} k^{2} + 1}}{L_{o}} \right)}} - \frac{L_{o} \alpha \sqrt{L_{o}^{2} k^{2} + 1} e^{\alpha \left(2 L_{D} + L_{a}\right)}}{\tanh{\left(\frac{L_{a} \sqrt{L_{o}^{2} k^{2} + 1}}{L_{o}} \right)}} + e^{\alpha \left(2 L_{D} + L_{a}\right)} - e^{\alpha \left(3 L_{D} + L_{a}\right)}\right) e^{- \alpha \left(3 L_{D} + L_{a}\right)}}{- L_{o}^{2} \alpha^{2} + L_{o}^{2} k^{2} + 1}$

MTF(k=0) check (should be 1): 1

Simplifications

Case 1: Dead Illuminated Surface + No Back-Surface Reflection

This is the Stanford / El Gamal (EE392B) special case of Crowell & Labuda.

Physical meaning of the two assumptions:

Assumption	Parameter change	Physical meaning
Dead illuminated surface	\(S \to \infty\)	Every minority carrier reaching \(y=0\) recombines instantly. The surface is a perfect sink, so the minority carrier density vanishes there: \(p(0)=0\).
No back-surface reflection	\(R = 0\)	Photons that reach the back of the substrate (\(y = L_b\)) are fully absorbed or transmitted — none are reflected back into the field-free region.

Effect on :math:`beta_pm` as :math:`S to infty`: The \(SL/D \to \infty\) term dominates \(\pm 1\) inside \(\beta_\pm\), so using the hyperbolic forms:

\[\beta_+ + \beta_- = 2\cosh\!\frac{L_a}{L} + 2\frac{SL}{D}\sinh\!\frac{L_a}{L} \;\xrightarrow{S\to\infty}\; 2\frac{SL}{D}\sinh\!\frac{L_a}{L}\]

\[\beta_+ - \beta_- = 2\sinh\!\frac{L_a}{L} + 2\frac{SL}{D}\cosh\!\frac{L_a}{L} \;\xrightarrow{S\to\infty}\; 2\frac{SL}{D}\cosh\!\frac{L_a}{L}\]

The \(SL/D\) factors cancel in the ratio, leaving \(\coth(L_a/L)\).

In the numerator, \(2(\alpha L + SL/D) \xrightarrow{S\to\infty} 2\,SL/D\), which when divided by \(2\,(SL/D)\sinh(L_a/L)\) gives \(1/\sinh(L_a/L)\).

The simplified collection efficiency (verified by SymPy below) is:

\[\boxed{\eta_k^{\rm SE} = \frac{\alpha L}{\alpha^2 L^2 - 1} \left[ \frac{1}{\sinh(L_a/L)} - \coth\!\frac{L_a}{L}\cdot e^{-\alpha L_a} - \frac{e^{-\alpha L_a}}{\alpha L} \right] - e^{-\alpha L_b}}\]

or,

\[\boxed{\eta_k^{\rm SE} = \frac{\alpha L}{\alpha^2 L^2 - 1} \left[ \frac{1 - \cosh(L_a/L) \cdot e^{-\alpha L_a}}{\sinh(L_a/L)} - \frac{e^{-\alpha L_a}}{\alpha L} \right] - e^{-\alpha L_b}}\]

where \(L = L_o/\sqrt{1 + k^2 L_o^2}\) as before.

# Apply S -> inf and R = 0 to eta_k_hyp (hyperbolic form is cleaner for this limit)
eta_k_SE = sp.limit(eta_k_hyp.subs(R, 0), S, sp.oo)
eta_k_SE = sp.simplify(eta_k_SE)
print("eta_k (S->inf, R=0):")
display(eta_k_SE)

# Verify against the merged boxed formula
eta_k_SE_expected = alpha * L / (alpha**2 * L**2 - 1) * (
    (1 - sp.cosh(xi) * sp.exp(-alpha * L_a)) / sp.sinh(xi)
    - sp.exp(-alpha * L_a) / (alpha * L)
) - sp.exp(-alpha * L_b)
check = sp.simplify(eta_k_SE.rewrite(sp.exp) - eta_k_SE_expected.rewrite(sp.exp))
print("\nVerification vs boxed formula (should be 0):", check)

eta_k (S->inf, R=0):

$\displaystyle \frac{\left(L_{o}^{2} \alpha^{2} e^{L_{a} \alpha} - L_{o}^{2} k^{2} e^{L_{a} \alpha} + L_{o}^{2} k^{2} e^{L_{b} \alpha} + \frac{L_{o} \alpha \sqrt{L_{o}^{2} k^{2} + 1} e^{L_{b} \alpha}}{\tanh{\left(\frac{L_{a} \sqrt{L_{o}^{2} k^{2} + 1}}{L_{o}} \right)}} - \frac{L_{o} \alpha \sqrt{L_{o}^{2} k^{2} + 1} e^{\alpha \left(L_{a} + L_{b}\right)}}{\sinh{\left(\frac{L_{a} \sqrt{L_{o}^{2} k^{2} + 1}}{L_{o}} \right)}} - e^{L_{a} \alpha} + e^{L_{b} \alpha}\right) e^{- \alpha \left(L_{a} + L_{b}\right)}}{- L_{o}^{2} \alpha^{2} + L_{o}^{2} k^{2} + 1}$

Verification vs boxed formula (should be 0): 0

Case 2: Passivated Illuminated Surface + No Back-Surface Reflection

This is the BSI sCMOS special case of Crowell & Labuda.

Physical meaning of the two assumptions:

Assumption	Parameter change	Physical meaning
Passivated illuminated surface	\(S = 0\)	The illuminated surface is perfectly passivated — no carriers recombine there. The boundary condition becomes \(\frac{\partial p}{\partial y}\big|_{y=0} = 0\) (zero flux, reflecting boundary).
No back-surface reflection	\(R = 0\)	Photons that reach the back of the substrate (\(y = L_b\)) are fully absorbed or transmitted — none are reflected back into the field-free region.

Effect on :math:`beta_pm` as :math:`S = 0`: With \(SL/D = 0\), the hyperbolic forms simplify to:

\[\beta_+ + \beta_- = 2\cosh\!\frac{L_a}{L} + 2 \cdot 0 \cdot \sinh\!\frac{L_a}{L} = 2\cosh\!\frac{L_a}{L}\]

\[\beta_+ - \beta_- = 2\sinh\!\frac{L_a}{L} + 2 \cdot 0 \cdot \cosh\!\frac{L_a}{L} = 2\sinh\!\frac{L_a}{L}\]

In the numerator, \(2(\alpha L + SL/D) = 2\alpha L\).

Substituting into \(\eta_k\):

\[\eta_k^{\text{S0}} = \frac{\alpha L}{\alpha^2 L^2 - 1} \left[ \frac{2\alpha L - 2\sinh(L_a/L) \cdot e^{-\alpha L_a}}{2\cosh(L_a/L)} - \frac{e^{-\alpha L_a}}{\alpha L} \right] - e^{-\alpha L_b}\]

which simplifies to:

\[\boxed{\eta_k^{\text{S0}} = \frac{\alpha L}{\alpha^2 L^2 - 1} \left[ \alpha L\,\operatorname{sech}(L_a/L) - \tanh(L_a/L) \cdot e^{-\alpha L_a} - \frac{e^{-\alpha L_a}}{\alpha L} \right] - e^{-\alpha L_b}}\]

where \(L = L_o/\sqrt{1 + k^2 L_o^2}\) as before.

Comparison with Case 1: The difference is in the first term inside the bracket — Case 1 has \(1/\sinh\xi\) (from \(S \to \infty\)), while Case 2 has \(\alpha L\,\operatorname{sech}\xi\) (from \(S = 0\)). Case 1 also has \(\coth(L_a/L)\) multiplying \(e^{-\alpha L_a}\), while Case 2 has \(\tanh(L_a/L)\).

# Case 2: S = 0, R = 0 (passivated BSI sCMOS)
eta_k_S0 = sp.simplify(eta_k_hyp.subs([(S, 0), (R, 0)]))
print("eta_k (S=0, R=0):")
display(eta_k_S0)

# Verify against the closed-form in the text above
eta_k_S0_expected = alpha * L / (alpha**2 * L**2 - 1) * (
    alpha * L / sp.cosh(xi)
    - sp.tanh(xi) * sp.exp(-alpha * L_a)
    - sp.exp(-alpha * L_a) / (alpha * L)
) - sp.exp(-alpha * L_b)
check = sp.simplify(eta_k_S0.rewrite(sp.exp) - eta_k_S0_expected.rewrite(sp.exp))
print("\nVerification vs boxed formula (should be 0):", check)

eta_k (S=0, R=0):

$\displaystyle \frac{\left(L_{o}^{2} \alpha^{2} e^{L_{a} \alpha} - \frac{L_{o}^{2} \alpha^{2} e^{\alpha \left(L_{a} + L_{b}\right)}}{\cosh{\left(\frac{L_{a} \sqrt{L_{o}^{2} k^{2} + 1}}{L_{o}} \right)}} - L_{o}^{2} k^{2} e^{L_{a} \alpha} + L_{o}^{2} k^{2} e^{L_{b} \alpha} + L_{o} \alpha \sqrt{L_{o}^{2} k^{2} + 1} e^{L_{b} \alpha} \tanh{\left(\frac{L_{a} \sqrt{L_{o}^{2} k^{2} + 1}}{L_{o}} \right)} - e^{L_{a} \alpha} + e^{L_{b} \alpha}\right) e^{- \alpha \left(L_{a} + L_{b}\right)}}{- L_{o}^{2} \alpha^{2} + L_{o}^{2} k^{2} + 1}$

Verification vs boxed formula (should be 0): 0

Case 3: Thick Substrate + No Back-Surface Reflection (General \(S\))

This is the BSI thick-substrate case of Crowell & Labuda, applicable to MCT (HgCdTe) infrared detectors where the absorber substrate is much thicker than the absorption depth (\(\alpha L_b \gg 1\)).

Assumptions:

Assumption	Parameter change	Physical meaning
Thick substrate	\(L_b \to \infty\)	Carriers generated deep in the substrate recombine before reaching the back surface. The back-surface boundary has no effect.
No back-surface reflection	\(R = 0\)	Photons reaching the back surface are not reflected back.
General surface recombination	\(S\) finite	The illuminated surface may be passivated (\(S \approx 0\)), dead (\(S \to \infty\)), or anywhere in between.

Effect of :math:`L_b to infty`: The only change is that the \(-e^{-\alpha L_b}\) tail term vanishes. The rest of \(\eta_k\) remains unchanged with general \(S\).

Simplification: Substituting the hyperbolic forms \(\beta_+ + \beta_- = 2\cosh\xi + 2\frac{SL}{D}\sinh\xi\) and \(\beta_+ - \beta_- = 2\sinh\xi + 2\frac{SL}{D}\cosh\xi\), the factors of 2 cancel and we can split the fraction:

\[\boxed{\eta_k^{\text{thick}} = \frac{\alpha L}{\alpha^2 L^2 - 1} \left[ \frac{\alpha L + SL/D}{\cosh\xi + (SL/D)\sinh\xi} - \frac{\sinh\xi + (SL/D)\cosh\xi}{\cosh\xi + (SL/D)\sinh\xi} \cdot e^{-\alpha L_a} - \frac{e^{-\alpha L_a}}{\alpha L} \right]}\]

where \(\xi = L_a/L\).

Sub-case 3a: Dead illuminated surface (\(S \to \infty\))

As \(S \to \infty\), \(SL/D\) dominates and the ratios simplify:

\[\frac{\alpha L + SL/D}{\cosh\xi + (SL/D)\sinh\xi} \;\to\; \frac{1}{\sinh\xi} = \operatorname{csch}\xi, \qquad \frac{\sinh\xi + (SL/D)\cosh\xi}{\cosh\xi + (SL/D)\sinh\xi} \;\to\; \frac{\cosh\xi}{\sinh\xi} = \coth\xi\]

This is Case 1 without the tail term — applicable to thick-substrate MCT with dead illuminated surface.

Sub-case 3b: Passivated illuminated surface (\(S = 0\))

With \(SL/D = 0\):

\[\frac{\alpha L + 0}{\cosh\xi + 0} = \alpha L\,\operatorname{sech}\xi, \qquad \frac{\sinh\xi + 0}{\cosh\xi + 0} = \tanh\xi\]

This is Case 2 without the tail term — applicable to thick-substrate MCT with passivated illuminated surface.

# Case 3: R = 0, L_b -> inf, general S (thick substrate BSI, MCT)
eta_k_thick = sp.limit(eta_k_hyp.subs(R, 0), L_b, sp.oo)
eta_k_thick = sp.simplify(eta_k_thick)
print("eta_k (R=0, L_b->inf, general S):")
display(eta_k_thick)

# Verify against the simplified (split fraction) boxed formula
eta_k_thick_expected = (
    alpha
    * L
    / (alpha**2 * L**2 - 1)
    * (
        (aL + sLD) / (sp.cosh(xi) + sLD * sp.sinh(xi))
        - (sp.sinh(xi) + sLD * sp.cosh(xi))
        / (sp.cosh(xi) + sLD * sp.sinh(xi))
        * sp.exp(-alpha * L_a)
        - sp.exp(-alpha * L_a) / (alpha * L)
    )
)
check = sp.simplify(eta_k_thick.rewrite(sp.exp) - eta_k_thick_expected.rewrite(sp.exp))
print("\nVerification vs boxed formula (should be 0):", check)

# Sub-case 3a: S -> inf (dead surface, thick substrate) = Case 1 without tail
eta_3a = sp.limit(eta_k_thick_expected, S, sp.oo)
eta_3a = sp.simplify(eta_3a)
eta_3a_expected = (
    alpha
    * L
    / (alpha**2 * L**2 - 1)
    * (
        1 / sp.sinh(xi)
        - sp.cosh(xi) / sp.sinh(xi) * sp.exp(-alpha * L_a)
        - sp.exp(-alpha * L_a) / (alpha * L)
    )
)
check_3a = sp.simplify(eta_3a.rewrite(sp.exp) - eta_3a_expected.rewrite(sp.exp))
print("\nSub-case 3a (S->inf) vs Case 1 without tail (should be 0):", check_3a)

# Sub-case 3b: S = 0 (passivated, thick substrate) = Case 2 without tail
eta_3b = sp.simplify(eta_k_thick_expected.subs(S, 0))
eta_3b_expected = (
    alpha
    * L
    / (alpha**2 * L**2 - 1)
    * (
        alpha * L / sp.cosh(xi)
        - sp.tanh(xi) * sp.exp(-alpha * L_a)
        - sp.exp(-alpha * L_a) / (alpha * L)
    )
)
check_3b = sp.simplify(eta_3b.rewrite(sp.exp) - eta_3b_expected.rewrite(sp.exp))
print("Sub-case 3b (S=0) vs Case 2 without tail (should be 0):", check_3b)

eta_k (R=0, L_b->inf, general S):

$\displaystyle \frac{\left(- 2 D L_{o}^{4} \alpha^{2} k^{2} e^{\frac{2 L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + D L_{o}^{4} k^{4} e^{\frac{L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + D L_{o}^{4} k^{4} e^{\frac{L_{a} \left(L_{o} \alpha + 3 \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} - D L_{o}^{3} \alpha k^{2} \sqrt{L_{o}^{2} k^{2} + 1} e^{\frac{L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + D L_{o}^{3} \alpha k^{2} \sqrt{L_{o}^{2} k^{2} + 1} e^{\frac{L_{a} \left(L_{o} \alpha + 3 \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} - 2 D L_{o}^{2} \alpha^{2} e^{\frac{2 L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + 2 D L_{o}^{2} k^{2} e^{\frac{L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + 2 D L_{o}^{2} k^{2} e^{\frac{L_{a} \left(L_{o} \alpha + 3 \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} - D L_{o} \alpha \sqrt{L_{o}^{2} k^{2} + 1} e^{\frac{L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + D L_{o} \alpha \sqrt{L_{o}^{2} k^{2} + 1} e^{\frac{L_{a} \left(L_{o} \alpha + 3 \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + D e^{\frac{L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + D e^{\frac{L_{a} \left(L_{o} \alpha + 3 \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} - 2 L_{o}^{4} S \alpha k^{2} e^{\frac{2 L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + L_{o}^{4} S \alpha k^{2} e^{\frac{L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + L_{o}^{4} S \alpha k^{2} e^{\frac{L_{a} \left(L_{o} \alpha + 3 \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} - L_{o}^{3} S k^{2} \sqrt{L_{o}^{2} k^{2} + 1} e^{\frac{L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + L_{o}^{3} S k^{2} \sqrt{L_{o}^{2} k^{2} + 1} e^{\frac{L_{a} \left(L_{o} \alpha + 3 \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} - 2 L_{o}^{2} S \alpha e^{\frac{2 L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + L_{o}^{2} S \alpha e^{\frac{L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + L_{o}^{2} S \alpha e^{\frac{L_{a} \left(L_{o} \alpha + 3 \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} - L_{o} S \sqrt{L_{o}^{2} k^{2} + 1} e^{\frac{L_{a} \left(L_{o} \alpha + \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}} + L_{o} S \sqrt{L_{o}^{2} k^{2} + 1} e^{\frac{L_{a} \left(L_{o} \alpha + 3 \sqrt{L_{o}^{2} k^{2} + 1}\right)}{L_{o}}}\right) e^{- L_{a} \left(2 \alpha + \frac{\sqrt{L_{o}^{2} k^{2} + 1}}{L_{o}}\right)}}{\sqrt{L_{o}^{2} k^{2} + 1} \left(- L_{o}^{2} \alpha^{2} + L_{o}^{2} k^{2} + 1\right) \left(D \sqrt{L_{o}^{2} k^{2} + 1} e^{\frac{2 L_{a} \sqrt{L_{o}^{2} k^{2} + 1}}{L_{o}}} + D \sqrt{L_{o}^{2} k^{2} + 1} - L_{o} S \left(1 - e^{\frac{2 L_{a} \sqrt{L_{o}^{2} k^{2} + 1}}{L_{o}}}\right)\right)}$

Verification vs boxed formula (should be 0): 0

Sub-case 3a (S->inf) vs Case 1 without tail (should be 0): 0
Sub-case 3b (S=0) vs Case 2 without tail (should be 0): 0

Addendum: Published Models as Special Cases

The reversed geometry (FSI) collection efficiency derived above is:

\[\eta_k^{\text{rev}} = \underbrace{(1 - e^{-\alpha L_D})}_{\text{depletion}} + \underbrace{\eta_{\text{ff},k}}_{\text{field-free diffusion}}\]

with \(S \to \infty\) (dead far surface) and \(R = 0\). We now show that three well-known published models are recoverable from this single expression.

Overview of Models

The most general diffusion MTF model is the Crowell & Labuda (1969) equation (Eq. 5), which includes surface recombination velocity \(S\) at the illuminated surface and back-surface reflectivity \(R\). All other models implemented here are special cases or independent re-derivations:

• Crowell & Labuda (1969) — General form (Eq. 5): Full model with \(S\), \(R\), finite field-free region \(L_a\), and depletion region \((L_b - L_a)\). Parameters: \(\alpha\), \(S\), \(R\), \(L_o\), \(L_a\), \(L_b\).

• Stanford/El Gamal (EE392B) — Dead-surface finite model: Independent re-derivation of C&L assuming \(S \to \infty\) (dead surface) and \(R = 0\) (no back-reflection), with light entering through the depletion region first. Parameters: \(\alpha\), \(L_D\), \(L_{Diff}\), \(L\).

• Seib (1974) — Semi-infinite special case of C&L: \(S \to \infty\), \(R = 0\), and \(L \to \infty\). Also cited via Fiete and Kopeika. Suitable for thick-substrate CCD and standard CMOS.

• Shepherd & Day (1982) — equivalent to Stanford/El Gamal with \(L_D = 0\) (no depletion region). Derived independently for MCT infrared detectors.

Stanford / El Gamal (EE392B)

The Stanford/El Gamal (EE392B, p. 9-14) formula for the FSI geometry with \(S \to \infty\) and \(R = 0\) is:

\[D(f) = \frac{1 + \alpha L_f - e^{-\alpha L_D}}{1 + \alpha L_f} - \frac{L_f \alpha\, e^{-\alpha L_D}\left(e^{-\alpha L} - e^{-L/L_f}\right)}{(1 - (\alpha L_f)^2)\,\sinh(L/L_f)}\]

where \(D(f)\) is the raw collection efficiency, \(L_f = L\) is our frequency-dependent diffusion length, and \(L = L_a\) is the field-free thickness. The diffusion MTF is \(D(f)/D(0)\).

This is identical to our reversed-geometry \(\eta_k^{\text{rev}}\) — the same model in different notation. No simplification is needed; they are the same derivation (verified symbolically below).

Fiete-Sieb (Semi-Infinite Substrate)

Taking \(L_a \to \infty\) (substrate much thicker than diffusion length), \(\sinh\xi \to \infty\) and the field-free boundary term vanishes. The raw collection efficiency becomes:

\[\eta_k^{\text{Sieb}} = 1 - \frac{e^{-\alpha L_D}}{1 + \alpha L}\]

The normalised diffusion MTF is then:

\[\text{MTF}_{\text{diff}}(k) = \frac{\eta_k^{\text{Sieb}}}{\eta_0^{\text{Sieb}}} = \frac{1 - \dfrac{e^{-\alpha L_D}}{1 + \alpha L}}{1 - \dfrac{e^{-\alpha L_D}}{1 + \alpha L_o}}\]

which is exactly the Fiete-Sieb formula used for bulk silicon CCDs and standard CMOS (verified symbolically below).

Shepherd & Day Dead Surface

Setting \(L_D = 0\) (no depletion region — the entire absorber is field-free), the depletion term vanishes and \(\eta_k^{\text{rev}}\) reduces to the field-free diffusion contribution alone, with \(L_a = d\) (total absorber thickness). This is the Shepherd & Day dead-surface model for MCT infrared detectors (verified numerically below).

# === Stanford/El Gamal: symbolic check ===
stanford = (1 + alpha * L - sp.exp(-alpha * L_D)) / (
    1 + alpha * L
) - L * alpha * sp.exp(-alpha * L_D) * (sp.exp(-alpha * L_a) - sp.exp(-xi)) / (
    (1 - (alpha * L) ** 2) * sp.sinh(xi)
)
check_stanford = sp.simplify((eta_total_k_rev - stanford).rewrite(sp.exp))
print("Stanford/El Gamal check (should be 0):", check_stanford)

# === Fiete-Sieb: symbolic check (L_a -> inf) ===
eta_sieb = sp.limit(eta_total_k_rev, L_a, sp.oo)
eta_sieb = sp.simplify(eta_sieb)

eta_sieb_published = 1 - sp.exp(-alpha * L_D) / (1 + alpha * L)

check_sieb = sp.simplify((eta_sieb - eta_sieb_published).rewrite(sp.exp))
print("Fiete-Sieb (L_a -> inf) check (should be 0):", check_sieb)

# === Shepherd & Day dead surface: numerical check (L_D = 0) ===
import numpy as np


def reversed_geometry_eta(f_arr, alpha_v, L_o_v, L_a_v, L_D_v):
    """Reversed geometry (FSI, S->inf) collection efficiency at frequency f."""
    k_arr = 2 * np.pi * f_arr
    L_v = L_o_v / np.sqrt(1 + k_arr**2 * L_o_v**2)
    xi_v = L_a_v / L_v

    eta_dep = 1 - np.exp(-alpha_v * L_D_v)
    P = alpha_v * L_v**2 / (alpha_v**2 * L_v**2 - 1) * np.exp(-alpha_v * L_D_v)
    A = P * (np.cosh(xi_v) - np.exp(-alpha_v * L_a_v)) / np.sinh(xi_v)
    eta_ff = -(A / L_v - alpha_v * P)  # -D * (...) / D cancels
    return eta_dep + eta_ff


def shepherd_day_dead(f_arr, alpha_v, L_o_v, d_v):
    """Shepherd & Day dead surface raw collection efficiency."""
    2 * np.pi * f_arr
    gamma = np.sqrt(1 / L_o_v**2 + (2 * np.pi * f_arr) ** 2)

    num = gamma * np.sinh(d_v / L_o_v) * (
        np.cosh(gamma * d_v) - np.exp(-alpha_v * d_v)
    ) - alpha_v * np.sinh(d_v / L_o_v) * np.sinh(gamma * d_v)
    den = (1 / L_o_v) * np.sinh(gamma * d_v) * (
        np.cosh(d_v / L_o_v) - np.exp(-alpha_v * d_v)
    ) - alpha_v * np.sinh(gamma * d_v) * np.sinh(d_v / L_o_v)
    raw = (alpha_v**2 - 1 / L_o_v**2) / (alpha_v**2 - gamma**2) * num / den
    return raw


test_cases = [
    ("MWIR 4µm", 1.0, 15.0, 7.0),
    ("LWIR 9µm", 0.15, 25.0, 10.0),
]
f_test = np.linspace(0.01, 33.0, 200)  # cy/mm

print("\nShepherd & Day dead surface vs reversed C&L (L_D=0):")
for label, alpha_v, L_Diff, d in test_cases:
    eta_rev = reversed_geometry_eta(f_test / 1000, alpha_v, L_Diff, d, 0.0)
    mtf_rev = eta_rev / eta_rev[0]

    eta_sd = shepherd_day_dead(f_test / 1000, alpha_v, L_Diff, d)
    mtf_sd = eta_sd / eta_sd[0]

    max_diff = np.max(np.abs(mtf_rev - mtf_sd))
    print(f"  {label}: max |MTF_rev - MTF_SD| = {max_diff:.2e}")

Stanford/El Gamal check (should be 0): 0
Fiete-Sieb (L_a -> inf) check (should be 0): 0

Shepherd & Day dead surface vs reversed C&L (L_D=0):
  MWIR 4µm: max |MTF_rev - MTF_SD| = 3.33e-16
  LWIR 9µm: max |MTF_rev - MTF_SD| = 1.06e-13

References

• Crowell, M. H. & Labuda, E. F., “The Silicon Diode Array Camera Tube”, Bell System Technical Journal, Vol. 48, No. 5, pp. 1481-1528, 1969.

• El Gamal, A. (Abbas), “EE392B Lecture 9: Spatial Resolution”, Stanford University.

• Seib, D. H., “CCarrier diffusion degradation of modulation transfer function in charge coupled imagers”, IEEE Trans. Electron Devices, Vol. 21, No. 3, 1974.

• Day, D. J. & Shepherd, T. J., “Transport in photo-conductors-I: Focal plane processing”, Solid-State Electronics, Vol. 25, No. 8, pp. 707-712, 1982.

• Fiete, R. D., “Modeling the Imaging Chain of Digital Cameras”, SPIE Tutorial Texts, 2010.

• Kopeika, N. S., “A System Engineering Approach to Imaging”, SPIE Press, 1998.