# CICY3 Families of Nonsingular Codimension two K3

Geoffrey Mboya  Mathematical Institute, University of Oxford, United Kingdom
<EMAIL_ADDRESS>
###### Abstract.

We conduct a systematic search of codimension 2 Complete Intersection
Calabi–Yau threefolds (CICY3) in rank 2 toric ambient spaces and fibered by
complete intersection of a quadric and a cubic in $\mathbb{C}\mathbb{P}^{4}$.
We classify both the nonsingular ones as well as those with isolated
singularities.

##  1\. Introduction

Rational threefold Complete Intersection Isolated Singularities have been used
to construct new Super-Conformal Field Theory (SCFT) in [5]. From the database
of nonsingular Complete Intersection Calabi–Yau threefolds (CICY3) in [13, 9]
whose ambient spaces are products of projective spaces; those with elliptic or
K3 or a chain of elliptic and K3 fibration structure have been studied in [3].
In this paper, we construct CICY3 families of $S_{2,3}$ complete intersection
K3 with at most isolated singularities and whose ambient spaces are rank 2
toric varieties.
We prove the following result

######  Theorem 1.1.

Let $X=\mathbb{V}(f_{1},f_{2})$ be a threefold fibred over $\mathbb{P}^{1}$ by
a complete intersection $S_{2,3}\subset\mathbb{C}\mathbb{P}^{4}$ of a quadric
and a cubic. With integers $0=a_{1}\leq a_{2}\leq a_{3}\leq a_{4}\leq a_{5},$
let $X$ be embedded in a smooth fivefold rank 2 toric variety
$\mathbb{F}_{A}=\mathbb{F}(0,a_{2},a_{3},a_{4},a_{5})$ associated to the
weight matrix

$A=\begin{bmatrix}0&-a_{2}&\ldots&-a_{5}&1&1\\\ 1&1&\ldots&1&0&0\end{bmatrix}$

with a trivial canonical class
$K_{X}=(D_{1}+D_{2}+K_{\mathbb{F}_{A}})|_{X},\text{ for
}D_{1},D_{2},K_{F_{A}}\in\operatorname{Pic}(\mathbb{F}_{A})$ where

$f_{1}\in
H^{0}(\mathbb{F}_{A},D_{1})\cong\mathbb{C}[\mathbb{F}_{A}]_{\left(p,2\right)},f_{2}\in
H^{0}(\mathbb{F}_{A},D_{2})\cong\mathbb{C}[\mathbb{F}_{A}]_{\left(2-p-\sum
a_{j},3\right)}.$

There are 12 such families of Calabi–Yau threefolds $X$ with at most isolated
singularities along either of the base loci $B_{i}=Bs(|D_{i}|).$ Table 1 is a
list of these families.

$\text{No}.$ | $X\subset\mathbb{F}(0,a_{2},a_{3},a_{4},a_{5})=\mathbb{F}_{A}$ | $\text{Description of a General }X=\begin{bmatrix}p&2-p-\sum a_{j}\\\ 2&3\end{bmatrix}$
---|---|---
1 | $\begin{bmatrix}0&2\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,0,0)$ | $|D_{1}|,|D_{2}|\text{ are base-point-free }$
|  | $\text{General }X\text{ is nonsingular }CY\text{ threefold}.$
2 | $\begin{bmatrix}0&1\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,0,1)$ | $|D_{1}|,|D_{2}|\text{ are base-point-free}$
|  | $\text{General }X\text{ is nonsingular }CY\text{ threefold}.$
3 | $\begin{bmatrix}0&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,0,2)$ | $|D_{1}|,|D_{2}|\text{ are base-point-free}$
|  | $\text{General }X\text{ is nonsingular }CY\text{ threefold}.$
4 | $\begin{bmatrix}0&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,1,1)$ | $|D_{1}|,|D_{2}|\text{ are base-point-free}$
|  | $\text{General }X\text{ is nonsingular }CY\text{ threefold}.$
5 | $\begin{bmatrix}1&1\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,0,0)$ | $|D_{1}|,|D_{2}|\text{ are base-point-free}$
|  | $\text{General }X\text{ is nonsingular }CY\text{ threefold}.$
6 | $\begin{bmatrix}1&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,0,1)$ | $|D_{1}|,|D_{2}|\text{ are base-point-free}$
|  | $\text{General }X\text{ is nonsingular }CY\text{ threefold}.$
7 | $\begin{bmatrix}2&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,0,0)$ | $|D_{1}|,|D_{2}|\text{ are base-point-free}$
|  | $\text{General }X\text{ is nonsingular }CY\text{ threefold}.$
8 | $\begin{bmatrix}-3&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,1,2,2)$ | $|D_{1}|\text{ is base-point-free},\dim Bs(|D_{2}|)=3$
|  | $\text{General }X\text{ is nonsingular}$
9 | $\begin{bmatrix}-2&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,2,2)$ | $|D_{1}|\text{ is base-point-free},\dim Bs(|D_{2}|)=3$
|  | $\text{General }X\text{ has 6 isolated singular points}$
|  | $\text{along }Bs(|D_{2}|)$
10 | $\begin{bmatrix}-1&1\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,1,1)$ | $|D_{1}|\text{ is base-point-free},\dim Bs(|D_{2}|)=3$
|  | $\text{General }X\text{ is nonsingular}$
11 | $\begin{bmatrix}-1&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,1,2)$ | $|D_{1}|\text{ is base-point-free},\dim Bs(|D_{2}|)=3$
|  | $\text{General }X\text{ has 6 isolated singular points}$
|  | $\text{along }Bs(|D_{2}|)$
12 | $\begin{bmatrix}-1&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,1,1,1)$ | $|D_{1}|\text{ is base-point-free},\dim Bs(|D_{2}|)=2$
|  | $\text{General }X\text{ has 2 isolated singular points}$
|  | $\text{ along }Bs(|D_{2}|)\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$

Table 1. The data $(p,a_{j})$ for which a general codimension two Calabi–Yau
threefold in $|L_{p,2}|\cap|L_{2-p-\sum a_{j},3}|\subset\mathbb{F}$ is non
singular or has isolated singularities.

Acknowledgements This work is adapted from my DPhil thesis [7]. This work was
completed in the inter of 2022 through a generous support as a visiting
research student at University of Vienna. I am grateful for the guidance and
support of my supervisor, Balázs Szendrői, during and after my research. I
have also been supported by the Simons Foundation Grant $488625.$

##  2\. Preliminaries

Take $(a_{j},b_{j})\in\mathbb{Z}^{2}$ for $j=1,\ldots,n$ such that

$\displaystyle 0\leq a_{1}\leq\ldots\leq a_{n}\text{ and }$ $\displaystyle
1\leq b_{1}\leq\ldots\leq b_{n}.$

Consider the following $(\mathbb{C}^{*})^{2}$ action on the product
$(\mathbb{C}^{n}\setminus\\{0\\})\times(\mathbb{C}^{2}\setminus\\{0\\})$

(1)
$(\lambda,\mu)\colon(x_{1},\ldots,x_{n};x_{n+1},x_{n+2})\mapsto(\lambda^{-a_{1}}\mu^{b_{1}}x_{1},\ldots,\lambda^{-a_{n}}\mu^{b_{n}}x_{n};\lambda
x_{n+1},\lambda x_{n+2}).$

An $n$-dimensional rank 2 toric variety $\mathbb{F}_{A}$ associated to the
integral weight matrix

$A=\begin{bmatrix}-a_{1}&\ldots&-a_{n}&1&1\\\
b_{1}&\ldots&b_{n}&0&0\end{bmatrix}$

is the quotient

$\mathbb{F}_{A}=((\mathbb{C}^{n}\setminus\\{0\\})\times(\mathbb{C}^{2}\setminus\\{0\\}))/(\mathbb{C}^{*})^{2}.$

As in [12], we also have the toric projection

$\pi:\operatorname{Proj}\left(\bigoplus_{i=1}^{5}\mathcal{O}_{\mathbb{P}^{1}}(a_{i})^{\oplus
b_{i}}\right)\cong\mathbb{F}_{A}\xrightarrow{[x_{1},\ldots,x_{5};x_{6},x_{7}]\mapsto[x_{6},x_{7}]}\mathbb{P}^{1}\text{
with fibres }\mathbb{P}^{4}_{[x_{1}:\ldots:x_{5}]}[b_{i}].$

In this paper, the weight matrix will be

$A=\begin{bmatrix}-a_{1}&-a_{2}&\ldots&-a_{5}&1&1\\\
b_{1}&b_{2}&\ldots&b_{5}&0&0\end{bmatrix}=\begin{bmatrix}0&-a_{2}&\ldots&-a_{5}&1&1\\\
1&1&\ldots&1&0&0\end{bmatrix}.$

In particular, the quotient $\mathbb{F}_{A}$ has no stabilizers for any choice
of $(a_{j}).$
Since the action (1) gives an isomorphic quotient up to $S_{5}\times
GL(2,\mathbb{C})$ action on the weight matrix $A,$ we assumed that
$(a_{j})=(0,a_{2},\ldots,a_{n}).$ Further, for
$(b_{j})=(1,b_{2},\ldots,b_{n})$ we have, from [8], the following exact
sequence of abelian groups

(2) $0\longrightarrow\mathbb{Z}^{2}\xrightarrow[]{\begin{bmatrix}(1,0)\\\
(0,1)\end{bmatrix}\mapsto
A}\mathbb{Z}^{n+2}\longrightarrow\frac{\mathbb{Z}[\rho_{1},\ldots,\rho_{n},\sigma_{1},\sigma_{2}]}{\left(\sigma_{1}+\sigma_{2}-\sum_{j=1}^{n}a_{j}\rho_{j},~{}\sum_{j=1}^{n}b_{j}\rho_{j}\right)}=:N\longrightarrow
0$

for which $N$ is a free $\mathbb{Z}$-module. We define a fan $\Sigma\subset
N_{\mathbb{R}}\cong\mathbb{R}^{n}$ by taking

$\Sigma(1)=\\{\rho_{1},\ldots,\rho_{n},\sigma_{1},\sigma_{2}\\},~{}\Sigma(n)=\\{\tau_{i,j}={\rm
Span}(\rho_{1},\ldots,\rho_{i-1},\rho_{i+1}\ldots,\rho_{n},\sigma_{j})\\}$

so that, from [14], we have $X_{N,\Sigma}\cong\mathbb{F}_{A}.$
The $(\mathbb{C}^{*})^{2}$-action induces a bigrading on the Cox Ring
$S(\Sigma)=\mathbb{C}[\mathbb{F}_{A}]=\mathbb{C}[x_{1},\ldots,x_{n};x_{n+1},x_{n+2}]$
which assigns the weights $(-a_{i},b_{i})$ to the Cox variable $x_{i}$ as
described in [6].
From the group $Cl(\mathbb{F}_{A})$ of Weil divisors on $\mathbb{F}_{A}$ with
the torus invariant irreducible divisors
$D_{\rho_{i}}=\\{x_{i}=0\\}\subset\mathbb{F}_{A},$ a monomial

$\prod_{i=1}^{n+2}x_{i}^{q_{i}}\in
S(\Sigma)=\mathbb{C}[x_{1},\ldots,x_{n+2}]=\bigoplus_{(d_{1},d_{2})\in\mathbb{Z}^{2}}S_{(d_{1},d_{2})}$

has bi-degree

${d_{1}\choose
d_{2}}=\left[\sum_{i=1}^{n+2}q_{i}D_{\rho_{i}}\right]=q_{1}{-a_{1}\choose
b_{1}}+\ldots+q_{n+2}{-a_{n+2}\choose b_{n+2}}.$

It can also be computed that $-K_{\mathbb{F}_{A}}=\left(2-\sum
a_{j}\right)D_{\sigma_{1}}+(\sum b_{j})D_{\rho_{1}}$ so that

$H^{0}(\mathbb{F}_{A},-K_{\mathbb{F}_{A}})=\mathbb{C}[x_{1},\ldots,x_{5};x_{6},x_{7}]_{\left(2-\sum_{j=1}^{7}a_{j},\sum_{j=1}^{7}b_{j}\right)}=\mathbb{C}[x_{1},\ldots,x_{5};x_{6},x_{7}]_{\left(2-\sum_{j=1}^{5}a_{j},5\right)}.$

Now, let $D_{1},D_{2},K_{\mathbb{F}_{A}}\in\operatorname{Pic}(\mathbb{F}_{A})$
be such that $K_{X}=(D_{1}+D_{2}+K_{\mathbb{F}_{A}})|_{X}=\mathcal{O}_{X},$
with the general sections

$\displaystyle f_{1}\in H^{0}(\mathbb{F}_{A},D_{1})=$
$\displaystyle\mathbb{C}[x_{1},\ldots,x_{5};x_{6},x_{7}]_{\left(p,2\right)}\text{
and }$ $\displaystyle f_{2}\in H^{0}(\mathbb{F}_{A},D_{2})=$
$\displaystyle\mathbb{C}[x_{1},\ldots,x_{5};x_{6},x_{7}]_{\left(2-p-\sum_{j=1}^{5}a_{j},3\right)}.$

We can then define a codimension 2 threefold

$X=\mathbb{V}(f_{1},f_{2})\in|D_{1}|\cap|D_{2}|\subset\mathbb{F}_{A}=\mathbb{F}(0,a_{2},a_{3},a_{4},a_{5})$

fibred by complete intersection $S_{2,3}$ of a quadric and a cubic. We denote
$X$ by

$X=\begin{bmatrix}\deg{f_{1}}&\deg{f_{2}}\end{bmatrix}=\begin{bmatrix}p&2-p-\sum
a_{j}\\\ 2&3\end{bmatrix}.$

##  3\. Construction: Threefold families of K3 $X_{2,3}\subset\mathbb{P}^{4}$
in $\mathbb{F}_{A}$

We would like to solve this classification problem: Find the integral data
$(p,a_{i}),$ where $0=a_{1}\leq a_{2}\leq a_{3}\leq a_{4}\leq a_{5},$ for
which the codimension 2 threefold

$X\hookrightarrow\mathbb{F}_{A}=\mathbb{F}(0,a_{2},a_{3},a_{4},a_{5})$

defined by a generic choice of sections $f_{1}\in|L_{p,2}|$ and
$f_{2}\in|L_{2-p-\sum a_{j},3}|$ is either non singular or has isolated
singularities.

######  Lemma 3.1.

Let $|D_{1}|$ and $|D_{2}|$ be base point free linear systems on
$\mathbb{F}_{A}.$ The codimension two variety $X=\mathbb{V}(f_{1},f_{2})$ with
general sections $f_{i}$ of $|D_{i}|$ is nonsingular.

###### Proof.

By Bertini’s Theorem, since $|D_{1}|$ is base point free, a general section
$X_{1}:=\mathbb{V}(f_{1})\in|D_{1}|$ is a nonsingular hypersurface in
$\mathbb{F}_{A}.$ Since $|D_{2}|$ is base point free, the restriction
$D_{2}|_{X_{1}}$ is basepoint-free with at least as many sections as
$|D_{2}|.$ Therefore, with $f_{2}$ being the equation of a general section of
$|D_{2}|,$ the general section $X=\mathbb{V}(f_{1},f_{2})$ of $D_{2}|_{X_{1}}$
is also nonsingular. ∎

###  3.1. Proof of Theorem 1.1

The equations of general sections

$f_{1}(x_{i})=\sum_{\begin{subarray}{c}(q_{j})\vdash
2\end{subarray}}\alpha_{(q_{j})}(x_{6},x_{7})x^{q_{j}}\in
S_{(p,2)},~{}f_{2}(x_{i})=\sum_{\begin{subarray}{c}(q_{j})\vdash
3\end{subarray}}\beta_{(q_{j})}(x_{6},x_{7})x^{q_{j}}\in S_{(2-p-\sum
a_{j},3)}$

are such that the degrees of coefficients of monomial
$x^{q}=\prod_{j=1}^{5}x_{j}^{q_{j}}$ are

$\displaystyle\deg\alpha_{(q_{j})}(x_{6},x_{7})$
$\displaystyle=p+\sum_{j=2}^{5}q_{j}a_{j},$
$\displaystyle\deg\beta_{(q_{j})}(x_{6},x_{7})$
$\displaystyle=2-p+\sum_{j=2}^{5}(q_{j}-1)a_{j}.$

Consider the case when $|D_{1}|=|L_{p,2}|$ and $|D_{2}|=|L_{2-p-\sum
a_{j},3}|$ are both base point free on $\mathbb{F}$ which is implied by $2$
inequalities

(3) $\displaystyle p\geq 0$ and $\displaystyle 2-p-\sum a_{j}\geq 0.$

From inequalities (3) and $a_{5}\geq a_{4}\geq a_{3}\geq a_{2}\geq 0,$ we get
the following data of nonsingular Calabi–Yau threefold
$X\in|D_{1}|\cap|D_{2}|$ embedded in $5$-fold scrolls

$\displaystyle(p,a_{2},a_{3},a_{4},a_{5})=$
$\displaystyle(0,0,0,0,0),(0,0,0,0,1),(0,0,0,0,2),(0,0,0,1,1),(1,0,0,0,0)$
$\displaystyle(1,0,0,0,1)\text{ or }(2,0,0,0,0).$

By irreducibility of $X,$ both $B_{1}=Bs(|L_{p,2}|),B_{2}=Bs(|L_{2-p-\sum
a_{j},3}|)\subset\mathbb{F}$ are each of dimension at most three. Equivalently
$\alpha_{(00020)}(x_{6},x_{7})x_{4}^{2}\text{ appears in }f_{1}\text{ and
}\beta_{(00030)}(x_{6},x_{7})x_{4}^{3}$ appears in $f_{2}$

(4) $\displaystyle p+2a_{4}\geq 0$ and $\displaystyle
2-p-a_{2}-a_{3}+2a_{4}-a_{5}\geq 0.$

Now, depending on whether $|D_{1}|$ and $|D_{2}|$ are base point free or have
a base locus of dimension at most $3,$ there are at most $15$ more cases to
consider for potential data $(p,a_{2},\ldots,a_{5})$ for which $X$ has at most
isolated singularities.

$\dim B_{1}$ $\dim B_{2}$ | $\emptyset$ | 0 | 1 | 2 | 3
---|---|---|---|---|---
$\emptyset$ | $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}{\checkmark}7$ | ✗ | ✗ | ✗ | ✗
0 | ✓0 | ✗ | ✗ | ✗ | ✗
1 | ✓0 | ✓0 | ✗ | ✗ | ✗
2 | ✓1 | ✓0 | ✓0 | ✗ | ✗
3 | ✓4 | ✓0 | ✓0 | ✓0 | ✗
Table 2. Cases to check
$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}{\checkmark\\#}$
with the number
$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}{\\#}$ of
families found in each case and cases not to check ✗ for Potential nonsingular
(or isolated singularity on ) Calabi–Yau threefolds fibred in
$X_{2,3}\subset\mathbb{P}^{4}$

The case when $\dim B_{1}=\dim B_{2}=0$ is not possible because of the
symmetry of the base variables $x_{6},x_{7}$ and hence omitted. We also omit
the cases when

$1\leq\dim(B_{1})=\dim(B_{2})\leq 3$

since for these cases, there is at least a whole curve of singularity along
the base loci. Moreover, since $D_{2}$ has more sections than $D_{1},$ it
suffices to check cases where $\dim B_{2}<\dim B_{1}.$

  1. 1.

The cases when
$(\dim(B_{1}),\dim(B_{2}))=(1,0),(1,\emptyset),(2,1),(2,0),(3,2),(3,1)\text{
and }(3,0)$ do not result in Complete Intersection Calabi–Yau threefold data
$(p,a_{j})$.

  2. 2.

For the case when $(\dim(B_{1}),\dim(B_{2}))=(2,\emptyset),$ the Inequality
(4) implies that $\alpha_{(02000)}(x_{6},x_{7})x_{2}^{2}$ does not appear in
$f_{1},$ $\alpha_{(00200)}(x_{6},x_{7})x_{3}^{2}$ appears in $f_{1}$ and
$\beta_{(3000)}(x_{6},x_{7})x_{1}^{3}$ appears in $f_{2}.$ Equivalently

(5) $\displaystyle p+2a_{2}<0,$ $\displaystyle p+2a_{3}\geq 0$ and
$\displaystyle 2-p-a_{2}-a_{3}-a_{4}-a_{5}\geq 0.$

Define

$\Delta(x_{6},x_{7},x_{1},x_{2},0,0,0):=\bigwedge^{2}\begin{bmatrix}n_{11}&n_{12}&n_{13}\\\
n_{21}&n_{22}&n_{23}\end{bmatrix}$

where $n_{ij}:=\frac{\partial f_{i}}{\partial
x_{j+2}}\bigg{|}_{\\{x_{j+2}=x_{j+3}=x_{j+4}=0\\}}$

$\displaystyle n_{11}$
$\displaystyle=\alpha_{(10100)}(x_{6},x_{7})x_{1}+\alpha_{(01100)}(x_{6},x_{7})x_{2},$
$\displaystyle n_{12}$
$\displaystyle=\alpha_{(10010)}(x_{6},x_{7})x_{1}+\alpha_{(01010)}(x_{6},x_{7})x_{2},$
$\displaystyle n_{13}$
$\displaystyle=\alpha_{(10001)}(x_{6},x_{7})x_{1}+\alpha_{(01001)}(x_{6},x_{7})x_{2},$
$\displaystyle n_{21}$
$\displaystyle=\beta_{(20100)}(x_{6},x_{7})x_{1}^{2}+\beta_{(11100)}(x_{6},x_{7})x_{1}x_{2}+\beta_{(20100)}(x_{6},x_{7})x_{2}^{2},$
$\displaystyle n_{22}$
$\displaystyle=\beta_{(20010)}(x_{6},x_{7})x_{1}^{2}+\beta_{(11010)}(x_{6},x_{7})x_{1}x_{2}+\beta_{(20010)}(x_{6},x_{7})x_{2}^{2},$
$\displaystyle n_{23}$
$\displaystyle=\beta_{(20001)}(x_{6},x_{7})x_{1}^{2}+\beta_{(11001)}(x_{6},x_{7})x_{1}x_{2}+\beta_{(20001)}(x_{6},x_{7})x_{2}^{2}.$

Note that we need

(6) $\displaystyle\deg(\alpha_{(01010)})=p+a_{2}+a_{4}\geq 0$ and
$\displaystyle\deg(\beta_{(20001)})=2-p-a_{2}-a_{3}-a_{4}\geq 0,$

for an irreducible

$\displaystyle\text{Sing}(\widetilde{X})\cap\widetilde{\mathbb{F}(0,a_{2})}$
$\displaystyle=\mathbb{V}(f_{1},f_{2},\Delta(x_{6},x_{7},x_{1},x_{2},0,0,0),x_{3},x_{4},x_{5})$
$\displaystyle=\mathbb{V}(f_{2}(x_{6},x_{7},x_{1},x_{2},0,0,0),~{}\Delta(x_{6},x_{7},x_{1},x_{2},0,0,0),x_{3},x_{4},x_{5}).$

Hence, from Inequalities $a_{5}\geq a_{4}\geq a_{3}\geq a_{2}\geq 0,$ (5) and
(6) the Complete Intersection Calabi–Yau threefold

$X=\begin{bmatrix}-1&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,1,1,1)$

with 2 isolated singular points

$\displaystyle Sing(X_{2})\cap\mathbb{F}(0,0)$
$\displaystyle=\mathbb{V}(\beta_{(30000)}(x_{6},x_{7})x_{1}^{3}+\ldots+\beta_{(03000)}(x_{6},x_{7})x_{2}^{3},~{}\Delta(x_{6},x_{7},x_{1},x_{2},0,0,0))$
$\displaystyle=\mathbb{V}(|L_{0,2}|_{\mathbb{F}(0,0)}|,|L_{1,3}|_{\mathbb{F}(0,0)}|)\subset\mathbb{P}^{1}\times\mathbb{P}^{1}.$

  3. 3.

Consider the case when $(\dim(B_{1}),\dim(B_{2}))=(3,\emptyset),$ the
Inequality (4) implies that $\alpha_{(00200)}(x_{6},x_{7})x_{3}^{2}$ does not
appear in $f_{1},\alpha_{(00020)}(x_{6},x_{7})x_{4}^{2}\text{ appears in
}f_{1}$ and $\beta_{(3000)}(x_{6},x_{7})x_{1}^{3}$ appears in $f_{2}.$
Equivalently

(7) $\displaystyle p+2a_{3}<0,$ $\displaystyle p+2a_{4}\geq 0$ and
$\displaystyle 2-p-a_{2}-a_{3}-a_{4}-a_{5}\geq 0.$

Define $Sing(X)=\mathbb{V}(f_{1},f_{2},\Delta(x_{i}))$ with

$\Delta(x_{i})=\left(\text{rank}\begin{bmatrix}\nabla f_{1}\\\ \nabla
f_{2}\end{bmatrix}<2\right)=\left(\bigwedge^{2}\begin{bmatrix}\frac{\partial
f_{1}}{\partial x_{6}}&\frac{\partial f_{1}}{\partial x_{7}}&\frac{\partial
f_{1}}{\partial x_{1}}&\frac{\partial f_{1}}{\partial x_{2}}&\frac{\partial
f_{1}}{\partial x_{3}}&\frac{\partial f_{1}}{\partial x_{4}}&\frac{\partial
f_{1}}{\partial x_{5}}\\\ \frac{\partial f_{2}}{\partial x_{6}}&\frac{\partial
f_{2}}{\partial x_{7}}&\frac{\partial f_{2}}{\partial x_{1}}&\frac{\partial
f_{2}}{\partial x_{2}}&\frac{\partial f_{2}}{\partial x_{3}}&\frac{\partial
f_{2}}{\partial x_{4}}&\frac{\partial f_{2}}{\partial
x_{5}}\end{bmatrix}\right).$

Now, along $\\{x_{4}=x_{5}=0\\}$, the threefold $X$ has singular locus

$\displaystyle\text{Sing}(\widetilde{X})\cap\widetilde{\mathbb{F}(0,a_{2},a_{3})}$
$\displaystyle=\mathbb{V}(f_{1},f_{2},\Delta(x_{6},x_{7},x_{1},x_{2},x_{3},0,0),x_{4},x_{5})$
$\displaystyle=\mathbb{V}(f_{2}(x_{6},x_{7},x_{1},x_{2},x_{3},0,0),~{}\Delta(x_{6},x_{7},x_{1},x_{2},x_{3},0,0),x_{4},x_{5})$

where

$\widetilde{\mathbb{F}(0,a_{2},a_{3})}=q^{-1}(\mathbb{V}(x_{4},x_{5}))\subset\mathbb{C}^{5}_{x_{j}}\setminus\\{0\\}\times\mathbb{C}^{2}_{x_{6},x_{7}}\setminus\\{0\\}\xrightarrow{q}\mathbb{F}.$

The locus
$\text{Sing}(\widetilde{X})\cap\widetilde{\mathbb{F}(0,a_{2},a_{3})}$ are
isolated singularities if it is defined by

$f_{2}|_{\\{x_{4}=x_{5}=0\\}}\not\equiv 0$

and an irreducible non-constant quadric

$\Delta(x_{6},x_{7},x_{1},x_{2},x_{3},0,0)=m_{11}m_{22}-m_{12}m_{21}\not\equiv
0,\text{ where }m_{ij}:=\frac{\partial f_{i}}{\partial
x_{j+3}}\bigg{|}_{\\{x_{j+3}=x_{j+4}=0\\}}$ $\displaystyle m_{11}=$
$\displaystyle\alpha_{(10010)}(x_{6},x_{7})x_{1}+\alpha_{(01010)}(x_{6},x_{7})x_{2}+\alpha_{(00110)}(x_{6},x_{7})x_{3},$
$\displaystyle m_{12}=$
$\displaystyle\alpha_{(10001)}(x_{6},x_{7})x_{1}+\alpha_{(01001)}(x_{6},x_{7})x_{2}+\alpha_{(00101)}(x_{6},x_{7})x_{3},$
$\displaystyle m_{21}=$
$\displaystyle\beta_{(20010)}(x_{6},x_{7})x_{1}^{2}+\beta_{(11010)}(x_{6},x_{7})x_{1}x_{2}+\beta_{(10110)}(x_{6},x_{7})x_{1}x_{3}+$
$\displaystyle\beta_{(02010)}(x_{6},x_{7})x_{2}^{2}+\beta_{(01110)}(x_{6},x_{7})x_{2}x_{3}+\beta_{(00210)}(x_{6},x_{7})x_{3}^{2},$
$\displaystyle m_{22}=$
$\displaystyle\beta_{(20001)}(x_{6},x_{7})x_{1}^{2}+\beta_{(11001)}(x_{6},x_{7})x_{1}x_{2}+\beta_{(10101)}(x_{6},x_{7})x_{1}x_{3}+$
$\displaystyle\beta_{(02001)}(x_{6},x_{7})x_{2}^{2}+\beta_{(01101)}(x_{6},x_{7})x_{2}x_{3}+\beta_{(00201)}(x_{6},x_{7})x_{3}^{2}.$

Equivalently

(8) $\displaystyle\deg(\alpha_{(00110)})=p+a_{3}+a_{4}\geq 0$ and
$\displaystyle\deg(\beta_{(00210)})=2-p-a_{2}+a_{3}-a_{5}\geq 0.$

We then get, from Inequalities $a_{5}\geq a_{4}\geq a_{3}\geq a_{2}\geq 0,$
(8) and (7), the following data on Complete Intersection Calabi–Yau threefold
in 5-fold scrolls

$(p,a_{2},a_{3},a_{4},a_{5})=(-3,0,1,2,2),(-2,0,0,2,2),(-1,0,0,1,1)\text{ and
}(-1,0,0,1,2).$

We check for isolated singularities in each case above:

    1. i)

$X_{1}=\begin{bmatrix}-3&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,1,2,2)$
with

$\displaystyle Sing(X_{1})\cap\mathbb{F}(0,0,1)$
$\displaystyle=\mathbb{V}(\beta_{(30000)}(x_{6},x_{7})x_{1}^{3}+\ldots+\beta_{(00300)}(x_{6},x_{7})x_{3}^{3},~{}\Delta(x_{6},x_{7},x_{1},x_{2},x_{3},0,0))$
$\displaystyle=\mathbb{V}(|L_{0,3}|_{\mathbb{F}(0,0,1)}|,|L_{4,3}|_{\mathbb{F}(0,0,1)}|)=\emptyset\subset\mathbb{F}(0,0,1).$

    2. ii)

$X_{2}=\begin{bmatrix}-2&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,2,2)$
with

$\displaystyle Sing(X_{2})\cap\mathbb{F}(0,0,0)$
$\displaystyle=\mathbb{V}(\beta_{(30000)}(x_{6},x_{7})x_{1}^{3}+\ldots+\beta_{(00300)}(x_{6},x_{7})x_{3}^{3},~{}\Delta(x_{6},x_{7},x_{1},x_{2},x_{3},0,0))$
$\displaystyle=\mathbb{V}(q_{3}(x_{1},x_{2},x_{3}))\cap|L_{2,3}|_{\mathbb{F}(0,0,0)}|\subset\mathbb{P}^{1}_{[x_{6}:x_{7}]}\times\mathbb{P}^{2}_{[x_{1}:x_{2}:x_{3}]}$

where
$q_{3}=\beta_{(30000)}(x_{6},x_{7})x_{1}^{3}+\ldots+\beta_{(00300)}(x_{6},x_{7})x_{3}^{3}.$
Here we are intersecting a general $(0,3)$\- and a general $(2,3)$-form on
$\mathbb{F}(0,0,0)\cong\mathbb{P}^{1}\times\mathbb{P}^{2}.$ Now, with at most
2 vertical rulings and at most 3 horizontal rulings on
$\mathbb{P}^{1}\times\mathbb{P}^{2}$, we conclude that there are $6=2\times 3$
isolated singular points
$\\{[\gamma_{1i}:\gamma_{2i};k_{1i}:k_{2i}:k_{3i}:0:0]:i=1,2,3\\}$ on $X.$

    3. iii)

$X_{3}=\begin{bmatrix}-1&1\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,1,1)$
with

$\displaystyle Sing(X_{3})\cap\mathbb{F}(0,0,0)$
$\displaystyle=\mathbb{V}(\beta_{(30000)}(x_{6},x_{7})x_{1}^{3}+\ldots+\beta_{(00300)}(x_{6},x_{7})x_{3}^{3},~{}\Delta(x_{6},x_{7},x_{1},x_{2},x_{3},0,0))$
$\displaystyle=\mathbb{V}(|L_{1,3}|_{\mathbb{F}(0,0,0)}|,|L_{2,3}|_{\mathbb{F}(0,0,0)})=\emptyset\subset\mathbb{P}^{1}\times\mathbb{P}^{2}.$

    4. iv)

$X_{4}=\begin{bmatrix}-1&0\\\ 2&3\end{bmatrix}\subset\mathbb{F}(0,0,0,1,2)$
with 6 isolated singular points

$\displaystyle Sing(X_{2})\cap\mathbb{F}(0,0,0)$
$\displaystyle=\mathbb{V}(\beta_{(30000)}(x_{6},x_{7})x_{1}^{3}+\ldots+\beta_{(00300)}(x_{6},x_{7})x_{3}^{3},~{}\Delta(x_{6},x_{7},x_{1},x_{2},x_{3},0,0))$
$\displaystyle=\mathbb{V}(r_{3}(x_{1},x_{2},x_{3}),|L_{2,3}|_{\mathbb{F}(0,0,0)}|)\subset\mathbb{P}^{1}\times\mathbb{P}^{2}$

where
$r_{3}=\beta_{(30000)}(x_{6},x_{7})x_{1}^{3}+\ldots+\beta_{(00300)}(x_{6},x_{7})x_{3}^{3}.$

##  4\. Cohomology of Codimension Two CICY3 in $\mathbb{F}_{A}$

In this section we explicitly compute the cohomology for the first $7$
families in terms of the data $(p,a_{j})$ using the Cayley trick in [2,
Section 2].
Denote by $i:X\hookrightarrow\mathbb{F}^{5}(a_{j})$ the inclusion for the
first $7$ threefold families in the Table 1. For the cohomology in the middle
dimension $3,$ we use [4, Definition 10.9] adapted to this case

######  Definition 4.1.

The variable cohomology

$H^{3}_{\text{var}}(X):={\rm
Cokernel}(H^{3}(\mathbb{F},\mathbb{C})\xrightarrow[1:1]{i^{*}}H^{3}(X,\mathbb{C})).$

######  Proposition 4.2.

Let
$X=\mathbb{V}(f_{1})\cap\mathbb{V}(f_{2})\subset\mathbb{F}_{A}=\mathbb{F}^{5}(a_{j})$
where $\deg(f_{1})=(p,2)$ and $\deg(f_{2})=(2-p-\sum a_{j},3)$ as in the first
$7$ entries of Table 1.1. Consider a rank 3 toric variety

$\mathbb{G}_{B}=((\mathbb{C}^{5}_{x_{1},\ldots,x_{5}}\setminus\\{0\\})\times(\mathbb{C}^{2}_{x_{6},x_{7}}\setminus\\{0\\}\times\mathbb{C}^{2}_{y_{1},y_{2}}\setminus\\{0\\})/(\mathbb{C}^{*})^{3},\text{
with }B=\begin{bmatrix}A&-\deg{f_{1}}&-\deg{f_{2}}\\\ 0&1&1\end{bmatrix}$

where

$(\lambda,\mu,\nu).(x_{j},x_{6},x_{7},y_{1},y_{2})=(\lambda^{-a_{j}}\mu^{b_{j}}x_{j},\lambda
x_{6},\lambda x_{7},\mu^{-p}\nu y_{1},\mu^{-(2-p-\sum a_{j})}\nu y_{2}).$

The Cox ring
$S=\mathbb{C}[\mathbb{G}_{B}]=\bigoplus_{(d_{i})\in\mathbb{Z}^{3}}S_{(d_{1},d_{2},d_{3})}$
with

$\displaystyle S_{(d_{1},d_{2},d_{3})}\cong$
$\displaystyle\bigoplus_{(n_{1},n_{2})\vdash
d_{3}}[\operatorname{Sym}^{2n_{1}+3n_{2}+d_{2}}(x_{1},\ldots,x_{5})\otimes$
$\displaystyle\left(\bigoplus_{(q_{j})\vdash
2n_{1}+3n_{2}+d_{2}}\operatorname{Sym}^{pn_{1}+(2-p-\sum
a_{j})n_{2}+d_{1}+\sum
a_{j}q_{j}}(x_{6},x_{7})\right)\otimes\operatorname{Sym}^{d_{3}}(y_{1},y_{2})]$

Taking
$F(x_{j},x_{6},x_{7},y_{1},y_{2}):=y_{1}f_{1}+y_{2}f_{2}\in\mathbb{C}[G_{B}]_{(0,0,1)}$
and $Y=\mathbb{V}(F)\subset\mathbb{G}_{B},$ let

$J(F)=\left(\partial F/\partial x_{j},\partial F/\partial x_{6},\partial
F/\partial x_{7},\partial F/\partial y_{1},\partial F/\partial
y_{2}\right)\triangleleft\mathbb{C}[G_{B}]$

and the Jacobian ring $R(F)=\mathbb{C}[G_{B}]/J(F)$ with an induced tri-
grading from $\mathbb{C}[\mathbb{G}_{B}].$ There is a cannonical isomorphism

$R(F)_{(0,0,9-q)}\cong H^{q-2,5-q}_{\text{var}}(X),\text{ where }q\neq 3.$

###### Proof.

The Cayley trick in codimension $s=2$ of dimension $d=5$ involves taking the
rank 2 vector bundle $\oplus L_{\deg f_{i}}$ whose fibre is
$\mathbb{C}^{2}_{y_{1},y_{2}}$ where the line bundles $L_{\deg f_{i}}$ are
associated to $f_{i};$ then look at $\mathbb{G}_{B}$ as a
$\mathbb{P}^{1}$-bundle

$\mathbb{G}_{B}=\operatorname{Proj}^{5+2-1}(L_{(p,2)}\oplus L_{(2-p-\sum
a_{j},3)})$

over $\mathbb{F}_{A}.$

${Y}$${\mathbb{G}_{B}}$${\mathbb{F}_{A}}$${X}$$\scriptstyle{\varphi}$$\scriptstyle{j}$$\scriptstyle{p}$$\scriptstyle{i}$
${H^{2}(\mathbb{F}_{A})}$${H^{2}(X)}$${H^{6}(\mathbb{F}_{A})}$${H^{6}(\mathbb{F}_{A}\setminus
X)}$${H^{3}_{\text{var}}(X)}$${0}$${H^{4}(\mathbb{G}_{B})}$${H^{4}(Y)}$${H^{6}(\mathbb{G}_{B})}$${H^{6}(\mathbb{G}_{B}\setminus
Y)}$${H^{5}(\mathbb{G}_{B}\setminus
Y)}$${H^{5}(\mathbb{G}_{B})}$$\scriptstyle{\cup[X]}$$\scriptstyle{i^{*}}$$\scriptstyle{i_{!}}$$\scriptstyle{\cup[Y]}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$

Since $X$ is chosen to be the intersection of sections of base point free line
bundles and is nonsingular, $Y$ is nonsingular by [2, Lemma 2.2]. The grading
on the Jacobian ring is much simpler due to the Calabi-Yau condition; we have
that

$\beta=\deg F=\deg(f_{j})+\deg(y_{j})=(0,0,1)$

whereas

$\beta_{0}=\deg(x_{1}.\ldots.x_{5}.x_{6}.x_{7}.y_{1}.y_{2})=\deg(x_{1})+\ldots+\deg(y_{2})=(0,0,2).$

The result then follows from [2, Theorem 3.6];

$R(F)_{(5+2-q)\beta-\beta_{0}}=R(F)_{(0,0,9-q)}\cong
H^{q-2,5-q}_{\text{var}}(X),\text{ where }q\neq 3.$

∎

######  Corollary 4.3.

As in [2, Theorem 4.6], we use a uniformly graded ideal

$J_{1}(F)=\left(x_{j}\partial F/\partial x_{j},x_{6}\partial F/\partial
x_{6},x_{7}\partial F/\partial x_{7},y_{1}\partial F/\partial
y_{1},y_{2}\partial F/\partial y_{2}\right)\subset\mathbb{C}[G_{B}]_{(0,0,1)}$

instead of $J(F)$ to get
$R_{1}(F)=\mathbb{C}[\mathbb{G}_{B}]/J_{1}(F)\cong\mathbb{C}[\mathbb{G}_{B}]/J(F)=R(F).$
Therefore for $q\neq 3$ we have

$\displaystyle
h^{q-2,5-q}(X)=\dim\left(R_{1}(F)_{(0,0,9-q)}\right)=\dim\left(S_{(0,0,9-q)}/J_{1}(F)\cap
S_{(0,0,9-q)}\right).$

For the first $7$ Calabi–Yau threefolds in Table 1, we get the following Hodge
data

$\begin{tabular}[]{cccccccccc}&&&$h^{3,3}$&&&\\\ &&$h^{3,2}$&&$h^{2,3}$&&\\\
&$h^{3,1}$&&$h^{2,2}$&&$h^{1,3}$&\\\
$h^{3,0}$&&$h^{2,1}$&&$h^{1,2}$&&$h^{0,3}$&\end{tabular}=\begin{tabular}[]{ccccccccc}&&&$1$&&&\\\
&&$0$&&$0$&&\\\ &$0$&&$\rho(\mathbb{G}_{B})$&&$0$&\\\ $1$&&$\dim
R_{1}(F)_{(0,0,5)}$&&$\dim R_{1}(F)_{(0,0,5)}$&&$1$&\end{tabular}$

##  5\. Other CICY3 families fibered by Complete Intersection K3s

Using computer algebra inspired by [10, Appendix A], one can find other 83
finite Lists of codimension $2$ CICY3 similar to Table 1 using the same set
up:
Decide how much singularity you want, say quasi-smooth, and consider
threefolds $X$ fibred by the other $83$ codimension $2$ K3s

$S_{(d_{1},d_{2})}\subset\mathbb{P}[b_{1},\ldots,b_{5}]$

from Fletcher’s List [1, Section 13.8, Table 2]. With integers $0=a_{1}\leq
a_{2}\leq a_{3}\leq a_{4}\leq a_{5},$ we have that

$X=\mathbb{V}(f_{1},f_{2}):=\begin{bmatrix}p&2-p-\sum a_{j}\\\
d_{1}&d_{2}\end{bmatrix}\hookrightarrow\mathbb{F}_{A}.$

Here, the singular fivefold rank 2 toric variety $\mathbb{F}_{A}$ has
associated weight matrix

$A=\begin{bmatrix}0&-a_{2}&\ldots&-a_{5}&1&1\\\
b_{1}&b_{2}&\ldots&b_{2}&0&0\end{bmatrix}.$

We take $D_{1},D_{2},K_{F_{A}}\in\operatorname{Pic}(\mathbb{F}_{A})$ such that
the canonical bundle $K_{X}=(D_{1}+D_{2}+K_{\mathbb{F}_{A}})|_{X}$ is trivial
and $X:=\mathbb{V}(f_{1},f_{2})$ where

$f_{1}\in
H^{0}(\mathbb{F}_{A},D_{1})\cong\mathbb{C}[\mathbb{F}_{A}]_{\left(p,d_{1}\right)},f_{2}\in
H^{0}(\mathbb{F}_{A},D_{2})\cong\mathbb{C}[\mathbb{F}_{A}]_{\left(2-p-\sum
a_{j},d_{2}\right)}.$

For singular $X,$ the singularities live on either of the base loci
$B_{i}=Bs(D_{i})$ whenever

$1\leq\dim B_{2}\leq\dim B_{1}\leq 3.$

######  Remark 5.1.

A stronger claim would be that of existence of a weight matrix $A$ for which
an arbitrary CICY3 $X$ fibred over $\mathbb{P}^{1}$ in codimension $2$ K3 $S$
would be embedded in $\mathbb{F}_{A}.$

## References

  * [1] A.R. Iano-Fletcher, Working with weighted complete intersections, in: Explicit birational geometry of 3-folds, 101–173, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000.
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[figure]font=footnotesize

# Sensitivity fields and parameter estimation
from dielectric objects

George Barbastathis  †,‡
$\rule[0.0pt]{0.0pt}{12.91663pt}{}^{\dagger}$Department of Mechanical
Engineering
Massachusetts Institute of Technology
$\rule[0.0pt]{0.0pt}{12.91663pt}{}^{\ddagger}$Singapore-MIT Alliance for
Research & Technology
Massachusetts Institute of Technology

(June 30, 2024)

## Abstract

The quantitative phase image formation process is posed as a problem of
parameter estimation from intensity measurements. This approach is inclusive
of traditional pixel-oriented imaging, where the sought parameters are the
pixel values. The resulting optimization process to find the parameters is
then seen to depend on the Sensitivity Field: this is the gradient of the
scattered field with respect to the parameters, and it turns out to obey a
scattering relationship that is analogous to that of the original scattered
field. Examples are given from several regimes of scattering strength.

## 1 Introduction

“Quantitative Phase Imaging” has become, not least thanks to the tireless work
of our late colleague and friend Gabi Popescu [1], whose memory this special
issue honors, a standard and popular sub-discipline of Imaging Science. It is
the topic of dedicated technical meetings and special sessions within bigger
conferences, and of course numerous theses and government grants. Taken at
face value, the topical term suggests as its ultimate goal the numerically
accurate reconstruction of the spatial details of electromagnetic field’s
phase.

The question of phase, and of its meaning and utility for imaging, dates back
to at least eighty years ago. This was when the origins of present optical,
x-ray and electron microscopy methods were being set by Lawrence Bragg and
others. In the words of Denis Gabor [2]:

> “[…] Bragg’s method, in which a lattice is reconstructed by diffraction from
> an X-ray diffraction pattern, can be applied only to a rather exceptional
> class of periodic structures. It is customary to explain this by saying that
> diffraction diagrams contain information on the intensities only, but not on
> the phases. The formulation is somewhat unlucky, as it suggests at once that
> since phases are unobservables, this state of affairs must be accepted. In
> fact, not only that part of the phase which is unobservable drops out of
> conventional diffraction patterns, but also the part which corresponds to
> geometrical and optical properties of the object, and which in principle
> could be determined by comparison with a standard reference wave.”

Already in this introductory paragraph, and many times subsequently in this
remarkable paper, Gabor makes the point that the phase is not a goal in its
own right, but rather as the means to recover more information about the
object of interest.

Of course, one important difference between Gabor’s time and ours is that then
image interpretation was done entirely manually; therefore, it was reassuring
for the human processors, scientists and their assistants alike, to be
operating on visually detailed images whose fidelity to the true scene would
be somehow guaranteed. Nowadays, image exploitation is largely automated, and
yet we still treat the processing algorithms “anthropomorphically”; that is,
in equal need of spatial richness and accuracy. Even more importantly, the
intervening development of Information Theory—with considerable contributions
by Gabor himself—has taught us that the information content of an object is
limited by its statistical variation. Spatial variability that is irrelevant
to a specific information retrieval task is, at best, unnecessarily resource
demanding and, in some cases, it may be outright harmful.

As an example, consider the task of localizing a blob-like object whose blob
shape is already known. The anthropomorphic way would be to image the blob as
accurately as possible, and then find its center of mass. In the 1960’s this
would have involved calculators and rulers. Nowadays, with a computer one
might first try to “improve” the image as much as possible with regularizing
filters and the like; and then apply a centroid or similar algorithm. This
seems like a lot of work just to get out a single coordinate—the center of
mass. Moreover, even though my students and I followed this exact same
approach before and showed that at least our specific instance was accurate
[3, 4], in general there is no guarantee that the image “improvement
algorithm” is bias-free.

On the other hand, it is possible to imagine methods of determining the center
of mass that are impervious to shape specifics. In fact, one of the earliest
“optical computing” methods, the van der Lugt correlator [5, sec. 8.4], does
just that. Thus, it is evident that a gap exists between the traditional
imaging mentality that prioritizes exquisite spatial detail even if only a few
parameters are to be extracted from it; and the information theoretic-driven
view that acknowledges the sparsity of the sought-after description and
orients the retrieval method accordingly.

My purpose in the present paper is to bridge this gap by recasting spatial
imaging as a problem of parameter estimation. This includes fully spatially
resolved images as special cases. When the parameters of interest are
significantly fewer than the number of pixels, certain efficiencies become
possible for imaging system design: it is no accident that the parametric view
is popular in modalities deprived of the benefit of high pixel-count spatial
reconstructions, e.g. the geosciences.

The core development of the parametric approach is in Section 2. It consists
of formulating an optimization functional on the detected intensity—which is
always the case, even if the estimate of a phase-like quantity is to be sought
after later as an intermediate product. Taking gradients on the functional
reveals the need for a Sensitivity Field, which is simply defined as the
gradient of the detected field to the sought-after parameters. The Sensitivity
Field is an artificial, mathematical construct; however, it turns out to obey
propagation rules similar to the physical fields whose gradients it
represents.

Section 3 derives the Sensitivity Fields for three regimes that are commonly
assumed in light-matter interaction: the thin-film approximation, the first
Born (or Wolf) kernel, and strong scattering approaches such as the Lippmann-
Schwinger equation and the Beam Propagation Method. Interestingly, the
Sensitivity Fields themselves scatter similar to and driven by the
electromagnetic fields they originated from. These results are analogous,
albeit not entirely identical to sensitivity and adjoint analysis of dynamical
systems. [6, 7]

Before diving into the technical details, let me note what the paper is not
attempting to do. First, it is not meant to be an exhaustive analysis of
electromagnetic scattering models, and it could not credibly be in such a
short space. Several excellent textbooks cover this topic in detail [8]. The
purpose of using a few such scattering models is to show how their Sensitivity
Functions are derived.

Second, my goal here is not to compare the validity or fidelity of different
parametric descriptions or scattering models. The former requires a careful
formulation of detection noise statistics, which is beyond the scope of the
present paper. The basics of the latter are in the aforementioned textbooks,
and still subject to active research. [9] Lastly, in Section 2 I only use a
simple quadratic—that is, optimal for additive Gaussian statistics only—loss
function to derive the sensitivity. Examining the behavior of the Sensitivity
Field with alternative loss functions, e.g. adversarial, and more generally
adoption of modern optimization methods would be interesting topics for
follow-up work.

## 2 Parameter estimation from spatial images

###  2.1 Intensity detection and the sensitivity field

Figure 1: General scattering geometry with a single intensity detection on a
photoelectric transducer.

The geometry to be used in the subsequent analysis is shown in Figure
[$\rightarrow$ Make.] Let ${\psi}_{\text{i}}(\boldsymbol{r})$ denote the
incident electromagnetic field as function of the Cartesian spatial coordinate
$\boldsymbol{r}$; and let $n(\boldsymbol{r};\bm{\theta})$ denote the
refractive index of the object whose parameters $\bm{\theta}$ are of interest.
The refractive index is assumed to be real, i.e. signifying a pure dielectric
object. The parameters $\bm{\theta}$ may be thought of as “controlling” the
spatial distribution of the refractive index; some examples are given in
Section 2.5 below.

The resulting scattered field is denoted as
${\psi}_{\text{s}}(\boldsymbol{r};\bm{\theta})$ and transduced at $M$ pixel
locations $\boldsymbol{r}_{m},$ $m=1,\ldots,M,$ where the intensities

$g_{m}(\bm{\theta})=\left|{\psi}_{\text{s}}(\boldsymbol{r}_{m};\bm{\theta})\right|^{2}$
(1)

are recorded. The vector

$\bm{\mathrm{g}}(\bm{\theta})=\left(\rule[-2.15277pt]{0.0pt}{12.91663pt}\begin{array}[]{ccccc}g_{1}(\bm{\theta})&\cdots&g_{m}(\bm{\theta})&\cdots&g_{M}(\bm{\theta})\end{array}\right)^{\dagger}$
(2)

is “the ideal measurement,” where $\dagger$ denotes the transpose. The right-
hand side in (1) is the the physical model for light propagation through the
object for any choice of parameters $\bm{\theta}$. It is also known as the
“forward operator.” In optical systems usually the $\boldsymbol{r}_{m}$’s are
arranged on a planar 2D lattice of pixels, but in other modalities, e.g. in
geo-imaging, they may be randomly placed.

In a noiseless measurement and with perfect knowledge of $\bm{\theta}$, eq. 1
holds with strict equality. In practice, a noisy measurement
$\bm{\mathrm{\tilde{g}}}$ will be instead recorded; and the parameter vector
$\bm{\theta}$ is to be determined. To carry out this latter task, define a
loss function such as

$E(\bm{\theta})=\frac{1}{4M}\>\sum_{m=1}^{M}\left[\rule[-2.15277pt]{0.0pt}{10.76385pt}\tilde{g}_{m}-\left|{\psi}_{\text{s}}(\boldsymbol{r}_{m};\bm{\theta})\right|^{2}\right]^{2}.$
(3)

It is useful to reiterate that inside the square bracket the first term is the
noisy measurement, whereas the second term is the forward model, parameterized
according to $\bm{\theta}$. The quadratic loss function is optimal in the
maximum likelihood sense for additive Gaussian detection statistics. Other
classical choices, e.g. for Poisson (shot-noise) statistics, or more modern
ones such as adversarial loss are possible, but I will not work them out in
detail in this paper.

The optimal parameter vector is found from solving the minimization problem

$\bm{\hat{\theta}}=\>\stackrel{{\scriptstyle[}}{{\bm{}}}{\theta}]{}{\text{argmin}}E(\bm{\theta}).$
(4)

If additional prior information on the parameters can be captured in the form
or a regularizer $\Phi(\bm{\theta})$, then the minimization problem is
modified as the unconstrained

$\bm{\hat{\theta}}=\>\stackrel{{\scriptstyle[}}{{\bm{}}}{\theta}]{}{\text{argmin}}\left\\{\rule[-2.15277pt]{0.0pt}{10.76385pt}E(\bm{\theta})+\kappa\>\Phi(\bm{\theta})\right\\};$
(5)

or the inequality-constrained

$\bm{\hat{\theta}}=\>\stackrel{{\scriptstyle[}}{{\bm{}}}{\theta}]{}{\text{argmin}}\Phi(\bm{\theta})\quad\text{subject
to}\quad E(\bm{\theta})<\kappa.$ (6)

In either option, $\kappa$ is acting as a regularization parameter. In the
interest of brevity, the regularized forms (5-6) are largely left unexploited
in this paper.

Existence and uniqueness of minima need to be analyzed on a case-by-case
basis, depending on the forward problem and its parameterization. If a minimum
does exist, it may be found in a number of ways: e.g. genetic algorithms,
gradient-based methods, etc. The simplest among the latter, the gradient
descent, proceeds iteratively as

$\bm{\hat{\theta}}[t+1]=\bm{\hat{\theta}}[t]-\eta[t]\,\frac{\partial
E\left(\bm{\hat{\theta}}[t]\right)}{\partial\bm{\theta}}.$ (7)

Here, $t$ denotes the iteration index and $\eta$ is the learning rate. The
gradient term is obtained from (3) as

$\frac{\partial
E\left(\bm{\hat{\theta}}[t]\right)}{\partial\bm{\theta}}=\frac{1}{M}\sum_{m=1}^{M}\text{Re}\left\\{\,{\psi}_{\text{s}}^{*}(\boldsymbol{r}_{m})\>\frac{\partial{\psi}_{\text{s}}(\boldsymbol{r}_{m})}{\partial\bm{\theta}}\,\right\\}\>\left[\rule[-2.15277pt]{0.0pt}{10.76385pt}\tilde{g}_{m}-\left|{\psi}_{\text{s}}(\boldsymbol{r}_{m};\bm{\theta})\right|^{2}\right].$
(8)

It is evidently convenient at this point to introduce the Sensitivity Field

${\chi}_{\text{s}}(\boldsymbol{r};\bm{\theta})\equiv\frac{\partial{\psi}_{\text{s}}(\boldsymbol{r};\bm{\theta})}{\partial\bm{\theta}},$
(9)

which is simply the gradient of the scattered field with respect to the chosen
parameterization; and the local error term

$\Delta_{m}(\bm{\theta})\equiv\tilde{g}_{m}-\left|{\psi}_{\text{s}}(\boldsymbol{r}_{m};\bm{\theta})\right|^{2}.$
(10)

In terms of these two newly defined quantities, the loss function gradient is
written more compactly as

$\frac{\partial
E}{\partial\bm{\theta}}=\frac{1}{M}\sum_{m=1}^{M}\text{Re}\left\\{\,\rule[-2.15277pt]{0.0pt}{10.76385pt}{\psi}_{\text{s}}^{*}(\boldsymbol{r}_{m};\bm{\theta})\>{\chi}_{\text{s}}(\boldsymbol{r}_{m};\bm{\theta})\,\right\\}\>\Delta_{m}(\bm{\theta}).$
(11)

###  2.2 Multiple exposures

Figure 2: General scattering geometry with several intensity detection steps,
each time with the illumination and photoelectric transducer at a different
orientation or rotation.

For the sake of completeness, I develop this notation a little bit further for
the numerous methods that employ multiple exposures onto the same object.
Popular examples are confocal microscopy, transport of intensity,
ptychography, computed (Radon) tomography, optical diffraction tomography,
optical coherence tomography, etc. A drawing suggestive of rotations between
exposures, as in tomography, is shown in Figure 2. Let $l$ denote the counter
of exposures, entered as superscript in the other notations. Thus,
$\boldsymbol{r}_{m}^{(l)}$ is the coordinate of the $m$-th detector pixel at
the $l$-th exposure, ${\psi}_{\text{i}}^{(l)}(\boldsymbol{r})$ is the
illuminatioun at the $l$-th exposure, etc. The loss function is then expressed
as

$E(\bm{\theta})=\frac{1}{4LM}\sum_{l=1}^{L}w^{2}_{l}\sum_{m=1}^{M}\left[\rule[-2.15277pt]{0.0pt}{10.76385pt}\tilde{g}^{(l)}_{m}-\left|{\psi}_{\text{s}}^{(l)}(\boldsymbol{r}^{(l)}_{m};\bm{\theta})\right|^{2}\right]^{2}.$
(12)

Here, for full generality we have weighed different exposures with the
positive numbers $w^{2}_{l}$. These express relative confidence, for example
noisier exposures might receive lower weights.

All the derivations from the previous section follow similarly, finally
obtaining

$\frac{\partial
E}{\partial\bm{\theta}}=\frac{1}{LM}\sum_{l=1}^{L}w^{2}_{l}\sum_{m=1}^{M}\text{Re}\left\\{\,\rule[-2.15277pt]{0.0pt}{10.76385pt}{\psi}_{\text{s}}^{(l)\,*}(\boldsymbol{r}^{(l)}_{m};\bm{\theta})\>{\chi}_{\text{s}}^{(l)}(\boldsymbol{r}^{(l)}_{m};\bm{\theta})\,\right\\}\>\Delta^{(l)}_{m}(\bm{\theta}).$
(13)

This formulation is appealing because it is compatible with tensorial
formulations in computing platforms such as Python and Julia.

###  2.3 Holographic detection and the sensitivity field

Figure 3: General scattering geometry with holographic detection.

In the holographic configuration, rather than detecting the intensity
scattered field itself, one interferes it with a reference field
${\psi}_{\text{r}}(\boldsymbol{r})$. Thus, the intensity at the $m$-th pixel
is the interferogram

$g_{m}=\left|{\psi}_{\text{r}}(\boldsymbol{r})+{\psi}_{\text{s}}(\boldsymbol{r};\bm{\theta})\right|^{2}$
(14)

It is also common in experiments for the reference power to be much stronger
than the scattered field; that is, $|{\psi}_{\text{r}}|\gg|{\psi}_{\text{s}}|$
everywhere. We then obtain

$\frac{\partial
E}{\partial\bm{\theta}}\approx\frac{1}{M}\sum_{m=1}^{M}\text{Re}\left\\{\,\rule[-2.15277pt]{0.0pt}{10.76385pt}{\psi}_{\text{r}}^{*}(\boldsymbol{r}_{m};\bm{\theta})\>{\chi}_{\text{s}}(\boldsymbol{r}_{m};\bm{\theta})\,\right\\}\>\Delta_{m}(\bm{\theta}).$
(15)

Whether (15) is an improvement over the reference-free version (11) depends on
the nature of the scattered field. For slowly-varying fields, the reference
term acts as an amplification factor on the sensitivity, which should be
beneficial. On the other hand, the high frequencies in the scattered field
would make the interference vary too rapidly, and that might lead to local
minima in the loss function. A detailed comparison of direct vs. holographic
intensity detection is beyond the scope of this paper, but certainly very
interesting for future work.

###  2.4 Partially coherent illumination

So far the illumination has been assumed to be temporally and spatially
coherent. To account for spatial partial coherence, one should replace the
field in (1) with

$g_{m}(\bm{\theta})={\Gamma}_{\text{I}}(\boldsymbol{r}_{m},\boldsymbol{r}_{m};\bm{\theta}).$
(16)

Here, ${\Gamma}_{\text{I}}(\boldsymbol{r}_{1},\boldsymbol{r}_{2})$ is the
spatial correlation function of the scattered field, more usually referred to
as mutual intensity, at a general pair of spatial coordinates
$\boldsymbol{r}_{1}$, $\boldsymbol{r}_{2}$. This may also be expressed in
terms of the coherent modes [$\rightarrow$ Ref] as

${\Gamma}_{\text{I}}(\boldsymbol{r}_{1},\boldsymbol{r}_{2};\bm{\theta})=\sum_{q=1}^{\infty}\beta_{q}(\bm{\theta})\>\zeta_{q}(\boldsymbol{r}_{1};\bm{\theta})\>\zeta^{*}_{q}(\boldsymbol{r}_{2};\bm{\theta}).$
(17)

The ideal measurement then is

$g_{m}(\bm{\theta})=\sum_{q=1}^{\infty}\beta_{q}(\bm{\theta})\left|\zeta_{q}(\boldsymbol{r}_{m};\bm{\theta})\right|^{2}.$
(18)

Expression (16) or (18) is then to be substituted in the loss function (3) and
subsequent formulae. In the interest of keeping the present paper confined, a
full analysis of parameter estimation under partially coherent illumination is
left for future work.

###  2.5 On the choice of parameterization

The following three examples elucidate how the parameterization choice
influences the formulation of the estimation problem.

  * 1)

Traditional pixel-oriented spatial imaging. Conceding that at best one may
only hope to recover a bandlimited version of the refractive index, let

$\left\\{\boldsymbol{r}_{j}\right\\}_{j=1,\ldots,J}$

denote a sampling lattice at the object space, and

$n(\boldsymbol{r})=\sum_{j=1}^{J}f_{j}\>\delta(\boldsymbol{r}-\bm{\mathrm{r}}_{j}).$
(19)

The coefficients $f_{j}$ are what traditionally one might call the spatial
pixel values of the (quantitative) phase in 2D, or of the refractive index
reconstruction in 3D. By assembling these values into the parameter vector

$\bm{\theta}=\left(\rule[-2.15277pt]{0.0pt}{12.91663pt}\begin{array}[]{ccccc}f_{1}&\cdots&f_{j}&\cdots&f_{J}\end{array}\right)^{\dagger},$

one can see that the minimum of (3), if it exists, will correspond to the
traditional spatial “quantitative phase image.”

Figure 4: Geometry for estimating the displacement $\boldsymbol{r}_{0}$ of a
known blob shape $f(\boldsymbol{r})$.

Note that if $LM>J$, then the problem is overdetermined, i.e. one has more
measurements than unknowns; whereas, if $LM<J$ then it is ill-posed, or
underdetermined. Alternatively, (3-4) may be thought of as solving a set of
$M$ nonlinear equations with $J$ unknowns.

  * 2)

Localization of a known shape. Returning to the localization example from the
Introduction section, suppose that

$n(\boldsymbol{r};\boldsymbol{r}_{0})=f(\bm{\mathrm{r}}-\boldsymbol{r}_{0}),$
(20)

where $f(\boldsymbol{r})$ is a known shape, and it is desired to localize it;
i.e., the only unknown is the displacement $\boldsymbol{r}_{0}$. This geometry
is shown in Figure 4. The parameterization with respect to displacement is

$\bm{\theta}=\boldsymbol{r}_{0}.$

If, moreover, the imaging system is spatially shift-invariant, then (11)
reduces to

$\frac{\partial
E}{\partial\bm{\theta}}=\frac{1}{2M}\sum_{m=1}^{M}\frac{\partial\left|{\psi}_{\text{s}}(\boldsymbol{r}-\bm{\theta})\right|^{2}}{\partial\boldsymbol{r}}\>\Delta_{m}(\bm{\theta}).$
(21)

The problem has become one of finding the best fit displacement coordinate
vector to the forward model for the $LM$ intensity measurements. As long as
$LM>3$ (or $2$ for planar blobs) this estimation problem is overdetermined.

  * 3)

Spatial representation in terms of basis functions. Compressive Sensing relies
on the existence of a set of basis functions $c_{j}(\boldsymbol{r})$ where the
object can be represented efficiently. For the refractive index as object,
this is expressed as

$n(\boldsymbol{r})=\sum_{j=1}^{J}f_{j}\,c_{j}(\boldsymbol{r})$ (22)

where most of the coefficients $f_{j}$ should turn out to be zero. Such sparse
bases are discovered by unsupervised learning, e.g. dictionaries
[$\rightarrow$ Ref] The pixel-sampled parameterization (19) is seen to be a
special albeit not necessarily sparse version of (22). The parameter vector is
similarly chosen as

$\bm{\theta}=\left(\rule[-2.15277pt]{0.0pt}{12.91663pt}\begin{array}[]{ccccc}f_{1}&\cdots&f_{j}&\cdots&f_{J}\end{array}\right)^{\dagger}.$

Now a regularized approach such as (5) or (6) suggests itself, with a suitable
choice of sparsity-promoting regularizer, e.g.
$\Phi(\bm{\theta})=\left|\\!\left|\bm{\theta}\right|\\!\right|_{1}.$

## 3 Sensitivity Fields for some scattering models

The Sensitivity Field (9) in general may be obtained numerically by automatic
differentiation. Instead, it is worthwhile to derive explicit expressions for
a few common cases of scattering strengths: the thin-film object, the single-
event scatterer or First Born approximation, and the multiple scattering
regime. The derivations are simple and yield results with a certain intuitive
appeal. Moreover, they may provide insights about convergence and fidelity in
future real-life applications.

The starting point is the spatially modulated scalar Helmholtz wave equation

$\nabla^{2}\psi(\boldsymbol{r};\bm{\theta})+k^{2}\,n^{2}(\boldsymbol{r};\bm{\theta})\,\psi(\boldsymbol{r};\bm{\theta})=0,$
(23)

where $k$ is the free-space wavenumber and $\psi(\boldsymbol{r})$ the
appropriate element of the electric field vector. Eq. 23 already neglects
polarization effects due to propagation at large numerical apertures and
optical anisotropy.

###  3.1 Point-spread functions

Before developing the Sensitivity Fields, one important detail needs to be
taken care of. In the following, ${\psi}_{\text{G}}(\boldsymbol{r})$ denotes
the optical system’s Green function, also known as impulse response and point-
spread function (PSF). Some typical Green’s functions are listed below and may
be replaced in any of the scattering formulae of subsequent Sections, as
appropriate.

  * •

The free-space Green’s function

${\psi}_{\text{G}}(\boldsymbol{r})=\frac{\exp\left\\{ik\left|\boldsymbol{r}\right|\right\\}}{ik\left|\boldsymbol{r}\right|},$
(24)

which is an ideal spherical wavefront. Since an ideal point source does not
exist, this expression does not represent a physically realizable solution to
the Helmholtz equation (23) for electromagnetic fields; closest to a realistic
solution is the dipole field [$\rightarrow$ Ref.] Nevertheless, for numerical
apertures less than $60^{\circ}$ the ideal spherical wave approximation works
quite well.

  * •

The paraxial free-space Green’s function, also referred to as the Fresnel
kernel

${\psi}_{\text{G}}(\boldsymbol{r})=\exp\left\\{ik\,z+ik\,\frac{x^{2}+y^{2}}{2z}\right\\}.$
(25)

  * •

The ideal in-focus diffraction-limited system PSF

${\psi}_{\text{G}}(x,y)=\text{jinc}\left(\frac{\sqrt{x^{2}+y^{2}}}{{R}_{\text{DL}}}\right),$
(26)

where ${R}_{\text{DL}}$ is the radius of the diffraction-limited spot and
$\text{jinc}(\cdot)$ is the same function as in [$\rightarrow$ Ref specific
equation in Goodman].

###  3.2 “Phase objects” and the thin-film approximation

Loosely speaking, a phase object is generally understood as a spatial material
distribution that interacts with light in such a way as to impart phase delay
only on the wavefield; no attenuation is to be imparted anywhere. A good
approximation for such an object is an infinitessimally thin film yet with a
thickness along the optical axis that is modulated as $d(x,y;\bm{\theta})$ as
function of the lateral spatial coordinates $x,y$; and whose index of
refraction $n(x,y;\bm{\theta})$ is purely real, and also modulated. This
configuration is depicted in Figure 5.

It is customary to define the optical path difference (OPD)

$\phi(x,y;\bm{\theta})=k\,\left[\rule[-2.15277pt]{0.0pt}{10.76385pt}n(x,y;\bm{\theta})-1\right]\,d(x,y;\bm{\theta})$
(27)

and say that the wavefront itself is modulated along the lateral dimensions
$(x,y)$ according to the OPD. That is, the field emanated directly after the
rightmost edge of the thin transparency, at the vertical dashed line in Figure
5, is

${\psi}_{\text{s}}(\boldsymbol{r};\bm{\theta})={\psi}_{\text{i}}(\boldsymbol{r})\,\exp\left\\{\rule[-2.15277pt]{0.0pt}{10.76385pt}i\,\phi(x,y;\bm{\theta})\right\\}.$
(28)

A brief proof of (28) including the list of conditions that have to be met for
it to be acceptable is in the Appendix.

Figure 5: The thin transparency is an object that imparts pure phase
modulation onto an incident field.

Past the sample, the field propagates either in free space or through an
optical system till it reaches the detector. In both cases, we denote the
point-spread function as ${\psi}_{\text{G}}(\boldsymbol{r})$, as discussed in
Section 3.1. The final field on the detector is

${\psi}_{\text{d}}(\boldsymbol{r};\bm{\theta})=\iint{\psi}_{\text{i}}(x^{\prime},y^{\prime},z)\,\exp\left\\{\rule[-2.15277pt]{0.0pt}{10.76385pt}i\,\phi(x,y;\bm{\theta})\right\\}{\psi}_{\text{G}}(x-x^{\prime},y-y^{\prime})\>\text{d}x^{\prime}\text{d}y^{\prime}.$
(29)

Note that for applications utilizing the thin film approach, such as imaging
of biological cells, it may be necessary to parameterize both refractive index
and thickness. This is even though they cannot be decoupled from a single
intensity measurement and in the absence of other priors, as is evident from
(27).

Leaving this issue aside, the Sensitivity Field is readily obtained as

${\chi}_{\text{d}}(\boldsymbol{r};\bm{\theta})=\iint{\chi}_{\text{i}}(x^{\prime},y^{\prime},z;\bm{\theta})\,\exp\left\\{\rule[-2.15277pt]{0.0pt}{10.76385pt}i\,\phi(x,y;\bm{\theta})\right\\}{\psi}_{\text{G}}(x-x^{\prime},y-y^{\prime})\>\text{d}x^{\prime}\text{d}y^{\prime},$
(30)

where the “Sensivity illumination” field is

$\displaystyle{\chi}_{\text{i}}(\boldsymbol{r};\bm{\theta})$
$\displaystyle={\psi}_{\text{i}}(\boldsymbol{r})\>\frac{\partial\phi(x,y;\bm{\theta})}{\partial\bm{\theta}}$
(31)
$\displaystyle={\psi}_{\text{i}}(\boldsymbol{r})\>\left\\{\rule[-4.30554pt]{0.0pt}{12.91663pt}\frac{\partial
n(x,y;\bm{\theta})}{\partial\bm{\theta}}\,d(x,y;\bm{\theta})+\left[\rule[-2.15277pt]{0.0pt}{10.76385pt}n(x,y;\bm{\theta})-1\right]\frac{\partial
d(x,y;\bm{\theta})}{\partial\bm{\theta}}\right\\}.$ (32)

Evidently, the Sensitivity Field is the same as the field produced by the same
object when the illumination is modulated by the OPD sensitivity
$\partial\phi/\partial\bm{\theta}$. This interesting symmetry will recur in
Section 3.3 with the first Rytov approximation (43) and in Section 3.4 with
the Lippmann-Schwinger integral equation (45).

Alternatively, the terms within the integral may be regrouped as

${\chi}_{\text{d}}(\boldsymbol{r};\bm{\theta})=\iint{\psi}_{\text{i}}(x^{\prime},y^{\prime},z;\bm{\theta})\,\exp\left\\{\rule[-2.15277pt]{0.0pt}{10.76385pt}i\,{\chi}_{\text{t}}(x,y;\bm{\theta})\right\\}{\psi}_{\text{G}}(x-x^{\prime},y-y^{\prime})\>\text{d}x^{\prime}\text{d}y^{\prime},$
(33)

where now the phase transmissivity is

${\chi}_{\text{t}}(x,y;\bm{\theta})=\phi(x,y;\bm{\theta})-i\log\left(\frac{\partial\phi(x,y;\bm{\theta})}{\partial\bm{\theta}}\right).$
(34)

Due to the imaginary part in this expression, this is no longer a pure “phase
object;” it includes an amplitude modulation. An analogous result occurs in
Section 3.3 with the first Born approximation (38).

###  3.3 First Born and First Rytov approximations

The Born series is a rigorous, if not necessarily stable numerically method to
obtain a solution for the scattered field in (23). Keeping the first term only
implies that multiple scattering is negligible. This results from the
substitution

$\psi(\boldsymbol{r};\bm{\theta})={\psi}_{\text{i}}(\boldsymbol{r})+{\psi}_{\text{s}}(\boldsymbol{r};\bm{\theta})$
(35)

for the total field, and the approximation

${\psi}_{\text{s}}(\boldsymbol{r};\bm{\theta})\approx\int{\psi}_{\text{i}}(\boldsymbol{r})\,v(\boldsymbol{r^{\prime}};\bm{\theta})\,{\psi}_{\text{G}}(\boldsymbol{r}-\boldsymbol{r^{\prime}})\,\text{d}\boldsymbol{r^{\prime}}.$
(36)

Here,

$v(\boldsymbol{r};\bm{\theta})\equiv\frac{1}{2}\left[n^{2}(\boldsymbol{r};\bm{\theta})-1\right]$
(37)

is the scattering potential. Result (36) is known as the First Born
approximation or Born integral and more recently it has started being referred
to as the Wolf transform. [$\rightarrow$ Ref Wolf 1967]

It is straightforward to obtain the Sensitivity Field for the First Born
approximation as

${\chi}_{\text{s}}(\boldsymbol{r};\bm{\theta})=\int{\psi}_{\text{i}}(\boldsymbol{r^{\prime}})\>{\chi}_{\text{v}}(\boldsymbol{r^{\prime}};\bm{\theta})\>{\psi}_{\text{G}}(\boldsymbol{r}-\boldsymbol{r^{\prime}})\,\text{d}\boldsymbol{r^{\prime}},$
(38)

where the modified potential

${\chi}_{\text{v}}(\boldsymbol{r};\bm{\theta})=\frac{\partial
n(\boldsymbol{r};\bm{\theta})}{\partial\bm{\theta}}\>n(\boldsymbol{r};\bm{\theta}).$
(39)

Here, as in (33), the Sensitivity Field is produced by the same illumination
as the original scattered field but with a modified scattering potential.

One alternative to the Born series is the Rytov perturbation expansion, which
starts by the transformation ${\psi}_{\text{s}}=\text{e}^{{\phi}_{\text{s}}}$.
To a first-order approximation then the scattered field is obtained as

$\displaystyle\psi(\boldsymbol{r})$
$\displaystyle\approx{\psi}_{\text{i}}(\boldsymbol{r})\>\exp\left\\{\rule[-2.15277pt]{0.0pt}{10.76385pt}\phi_{1}(\boldsymbol{r})\right\\},\qquad\text{where}\qquad$
(40) $\displaystyle\phi_{1}(\boldsymbol{r};\bm{\theta})$
$\displaystyle=\int\frac{{\psi}_{\text{i}}(\boldsymbol{r^{\prime}})}{{\psi}_{\text{i}}(\boldsymbol{r})}\>v(\boldsymbol{r^{\prime}};\bm{\theta})\>{\psi}_{\text{G}}(\boldsymbol{r}-\boldsymbol{r^{\prime}})\>\text{d}\boldsymbol{r^{\prime}}.$
(41)

This result is known as the Rytov approximation or Rytov integral.
Consequently, its Sensitivity Field is

$\displaystyle{\chi}_{\text{s}}(\boldsymbol{r};\bm{\theta})$
$\displaystyle={\chi}_{\text{i}}(\boldsymbol{r};\bm{\theta})\>\exp\left\\{\rule[-2.15277pt]{0.0pt}{10.76385pt}\phi_{1}(\boldsymbol{r})\right\\},\qquad\text{where}\qquad$
(42) $\displaystyle{\chi}_{\text{i}}(\boldsymbol{r};\bm{\theta})$
$\displaystyle=\int\frac{{\psi}_{\text{i}}(\boldsymbol{r^{\prime}})}{{\psi}_{\text{i}}(\boldsymbol{r})}\>{\chi}_{\text{v}}(\boldsymbol{r^{\prime}};\bm{\theta})\>{\psi}_{\text{G}}(\boldsymbol{r}-\boldsymbol{r^{\prime}})\>\text{d}\boldsymbol{r^{\prime}}$
(43)

is the sensitivity illumination field and ${\chi}_{\text{v}}$ is again given
by (39). In this case, as in (30), the Sensitivity Field results from a
modified illumination acting on the same scattering potential as in the
original scattered field.

###  3.4 Multiple scattering via the Lippmann-Schwinger equation

The Lippmann-Schwinger integral equation captures multiple scattering orders
and is formally equivalent with, albeit usually better behaved numerically
than the Born series. The scattered field is obtained from

${\psi}_{\text{s}}(\boldsymbol{r})={\psi}_{\text{i}}(\boldsymbol{r})+\int{\psi}_{\text{s}}(\boldsymbol{r^{\prime}})\,v(\boldsymbol{r^{\prime}})\,{\psi}_{\text{G}}(\boldsymbol{r}-\boldsymbol{r^{\prime}})\,\text{d}\boldsymbol{r^{\prime}}.$
(44)

The scattering potential $v(\boldsymbol{r})$ is the same as in (37).

Taking gradients on both sides, the Sensitivity Field for the Lippmann-
Schwinger integral equation is found to satisfy

${\chi}_{\text{s}}(\boldsymbol{r};\bm{\theta})={\chi}_{\text{i}}(\boldsymbol{r})+\int{\chi}_{\text{s}}(\boldsymbol{r^{\prime}};\bm{\theta})\,v(\boldsymbol{r^{\prime}};\bm{\theta})\,{\psi}_{\text{G}}(\boldsymbol{r}-\boldsymbol{r^{\prime}})\,\text{d}\boldsymbol{r^{\prime}},$
(45)

where the Sensitivity illumination field is

${\chi}_{\text{i}}(\boldsymbol{r};\bm{\theta})=\int{\psi}_{\text{s}}(\boldsymbol{r^{\prime}};\bm{\theta})\>{\chi}_{\text{v}}(\boldsymbol{r};\bm{\theta})\>{\psi}_{\text{G}}(\boldsymbol{r}-\boldsymbol{r^{\prime}})\,\text{d}\boldsymbol{r^{\prime}}.$
(46)

Once again one arrives at a situation like (32) and (43), where the
Sensitivity Field is scattered by the same potential as the original scattered
field with an illumination that results from the original scattered field
modulated by the gradient of the scattering potential.

###  3.5 Multiple scattering via the Beam Propagation Method

If back scattering may be neglected, then a popular alternative is the Beam
Propagation Method. In [9] it was shown that the Beam Propagation Method
approximates the Lippmann-Schwinger equation solution well for weakly
scattering objects, and the error grows as the fourth power of the maximum
scattering angle, i.e. the spatial bandwidth of the scatterer.

In BPM, the object is divided into multiple slices oriented perpendicular to
the directional axis $z$. Each slice is treated like a thin transparency, and
subsequently the field is forward-propagated as if through free space to the
next slice. That is,

$\psi(x,y,z_{l+1})=\int\psi(x^{\prime},y^{\prime},z_{l})\,\exp\left\\{\rule[-2.15277pt]{0.0pt}{10.76385pt}ik\,\varepsilon(x^{\prime},y^{\prime};z_{l})\right\\}\\\
{\psi}_{\text{G}}(x^{\prime}-x,y^{\prime}-y,z^{\prime}-z)\,\text{d}x^{\prime}\text{d}y^{\prime}.$
(47)

The recursion is initialized with

$\psi(x,y,z_{1})={\psi}_{\text{i}}(x,y,z_{1}),$ (48)

the incident illumination. Since the field from a given slice can reach the
next slice at a considerably large angle, usually the ideal spherical wave
kernel (24) is used in BPM.

The sensitivity analysis on (47) was carried out in [$\rightarrow$ Ref
Kamilov] leading to an elegant analogy between the sensitivity equation (11)
for BPM and backpropagation in neural networks. Even though these papers
considered only the pixel-oriented spatial imaging parameterization, #1 in
Section 2.5, it is straightforward to generalize them to other
parameterizations.

## 4 Conclusions

There are two main messages from this paper. The first is that treating the
quantitative phase imaging problem as one of parameter estimation allows some
interesting generealizations that are less “anthropomorphic.” Thus, perhaps
accuracy improvements and efficiency savings may result. The parametric
approach is inclusive of the traditional spatial imaging way of thinking, as
the reader has seen in case #1 of Section 2.5.

The second main point is the Sensitivity Field. It results in straightforward
fashion from taking gradients on the loss function, eqs. 3-9, with respect to
the sought parameters. Superficially, thanks to automatic differentiation it
may appear that there is no reason to carry out all the derivations of
Sensitivity Fields in Section 3. I beg to disagree. The results reveal an
interesting intuition: the Sensitivity Fields are either produced by the same
given scattering potential but with a modified Sensitivity Illumination field
that is computed from the original scattered field, as in eqs. 30, 42, 45; or
by the same illumination field but instead with a scattering potential that is
the gradient of the original potential, as in eqs. 33 and 38. Such analogies
are well known in the community of feedback control [6, 7] and have led to
interesting innovations. [10]

### Acknowledgments

This research was funded by the U. S. Air Force Office of Scientific Research
through the Multi-University Research Initiative (MURI) “Searching for what’s
new: the systematic development of dynamic x-ray microscopy,” grant No.
FA9550-23-1-0284; and by Singapore’s National Research Foundation through the
Intra-Create thematic grant “Retinal Analytics through Machine learning aiding
Physics” (RAMP), grant No. NRF2019-THE002-0006. The author is grateful to
Steven G. Johnson of the MIT Mathematics Department; Chris Jacobsen of the
Northwester University Physics Department; Shen Zuowei and Ji Hui of the
National University of Singapore’s Department of Mathematical Sciences; Fabian
Bräu, Bingyao Tan, Michael Girard, Leopold Schmetterer and Aung Tin of the
Singapore Eye Research Institute; Sandip Mondal of SMART; and Qihang Zhang of
Tsinghua University for helpful discussions.

## Appendix: Derivation of the thin film approximation

Starting with the space-variant Helmholtz equation (23), we elect a
directional optical beam along the $z$ axis as

$\psi(\boldsymbol{r})=\tilde{\psi}(\boldsymbol{r})\>\text{e}^{ikz}.$ (49)

Here, $\tilde{\psi}(\boldsymbol{r})$ is a slow-varying modulation , i.e.

$\left|\frac{\partial\tilde{\psi}}{\partial w}\right|\ll k\quad\text{for\
}w=x,y,z.$ (50)

Substituting (49) into (23) we obtain

$\frac{\partial^{2}\tilde{\psi}}{\partial{x}^{2}}+\frac{\partial^{2}\tilde{\psi}}{\partial{y}^{2}}+\frac{\partial^{2}\tilde{\psi}}{\partial{z}^{2}}+2ik\,\frac{\partial\tilde{\psi}}{\partial
z}+k^{2}\,\left[\rule[-2.15277pt]{0.0pt}{10.76385pt}n^{2}(\boldsymbol{r})-1\right]\,\tilde{\psi}(\boldsymbol{r})=0.$
(51)

We now make the strong assumption

$\displaystyle\left|\frac{\partial^{2}\tilde{\psi}}{\partial{w}^{2}}\right|$
$\displaystyle\ll k\>\frac{\partial\tilde{\psi}}{\partial z},\quad\text{for\
}w=x,y,z;\qquad\text{and}\qquad$ (52) $\displaystyle\frac{\partial n}{\partial
z}$ $\displaystyle=0.$ (53)

In Geometrical and Hamiltonian optics, condition (53) is also known as a
“guide medium.” The remaining terms in (51) then have the solution

$\tilde{\psi}(x,y,z)=\tilde{\psi}_{0}(x,y)\,\exp\left\\{ik\,\frac{n^{2}(x,y)-1}{2}\,z\right\\},$
(54)

where $\tilde{\psi}_{0}(x,y)\equiv\tilde{\psi}(x,y,0)$ is the incident field.
To bring this into an even more recognizable form, we further assume a weak
refractive index modulation

$n(x,y)=1+\varepsilon(x,y),\qquad\text{where}\qquad\left|\varepsilon(x,y)\right|\ll
1.$ (55)

Then (54) becomes

$\tilde{\psi}(x,y,z)=\tilde{\psi}_{0}(x,y)\,\exp\left\\{\rule[-2.15277pt]{0.0pt}{10.76385pt}ik\,\varepsilon(x,y)\,z\right\\}.$
(56)

In the exponent, we recognize the optical path difference

$\text{OPD}(x,y)=\varepsilon(x,y)\,z.$ (57)

Clearly, the thin dielectric object under these approximations imparts a pure
phase modulation on the incident field. This is a key consequence and
justification for the term “phase object.”

To summarize, the phase object or thin film approximation (56) relies on all
of the following assumptions to be true:

  * (i)

The refractive index is independent of the axial variable $z$, according to
(53).

  * (ii)

The refractive index modulation is small relative to the surrounding medium,
according to (55).

  * (iii)

The following three conditions that are necessary for (56) to be consistent
with (52):

$\displaystyle\left|\frac{\partial^{2}\tilde{\psi}_{0}(x,y)}{\partial{w}^{2}}\right|$
$\displaystyle\ll 2k^{2}\varepsilon(x,y),$ (58)
$\displaystyle\left|\frac{\partial\tilde{\psi}_{0}}{\partial
x}(x,y)\frac{\partial\varepsilon(x,y)}{\partial w}z\right|$ $\displaystyle\ll
k\,\varepsilon(x,y),$ (59)
$\displaystyle\left|\tilde{\psi}_{0}(x,y)\frac{\partial^{2}\varepsilon(x,y)}{\partial{w}^{2}}z\right|$
$\displaystyle\ll 2k\,\varepsilon(x,y),$ (60)

where the spatial derivatives are with respect to $w=x,y$. Without going into
a full analysis of these expressions, one might just point out that as the
depth $z$ increases, so does the potential of (58-60) becoming violated. The
same holds for the lateral bandwidth of the incident illumination
$\tilde{\psi}_{0}(x,y)$ and the modulation $\varepsilon(x,y)$.

Remarkably, biological cells seem to satisfy these smoothness conditions most
of the time. This is probably why QPI has been fairly succesful in visualizing
them and identifying correlates to biological functions. On the other hand, it
is also cutomary to apply the thin film approximation to certain artificial
objects that seem to break it, e.g. if they include sharp edges. This is the
case for almost all binary optics and 3D-printed dielectric objects, for
example. Even more remarkably, the thin-film approximation even then seems to
predict the output of the overall optical system in good agreement with
experiments.

It is possible that such objects in their physical realization are not
actually as sharp as their models assume. Moreover, usually the light
scattered from the highly sloped edges is directed outside the aperture of the
optical system. Its absence from the detected intensity would appear to cause
no damage other than smoothening; which may just be ignored or corrected by
use of prior knowledge, i.e. an edge-enhancing regularizer such as Total
Variation (TV).

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