## Abstract

Deflection angles of light rays passing through a refractive index field can be measured by the background-oriented schlieren (BOS) technique. Assuming that the deflection angle is sufficiently small and the paraxial approximation can apply to the light rays, a vector consisting of deflection angles in two orthogonal directions is shown to be derived from a gradient of a scalar potential. The scalar potential can be written as an integration of the refractive index field over the light ray path. Thus, a method to reconstruct an axisymmetric 3D refractive index field with the scalar potential is proposed here. An arbitrary measured deflection angle vector, however, is generally written not only with a scalar potential but with a vector potential. Thus, the Poisson’s equation is derived to extract a scalar potential from a measured deflection angle vector. The axisymmetric 3D refractive index field is able to be reconstructed using the Abel transformation [1] of the scalar potential derived by applying the 2D Fourier transformation to the Poisson’s equation. The scalar potential reconstruction method is validated by reconstructing a spherically symmetric refractive index field where a deflection angle vector field is able to be calculated accurately.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Refractive index fields of several media, including air, transparent liquids, and transparent solids, are often important for probing optical properties and determining some physical information. [1] Several physical quantities such as temperature, pressure, strain, and density of materials are related to refractive index [2,3]. Once a relationship between the refractive index and a quantity is known, the quantity can be obtained by measuring the refractive index.

Light rays passing through a medium with an inhomogeneous refractive index field will be deflected toward the area of higher refractive index. Thus, it should be possible to obtain information about the refractive index field from measurements of such deflection. Indeed, there are several methods for visualizing refractive index fields, including schlieren photography [4] and shadowgraphy [5]. Although these visualization methods are powerful tools for qualitatively analyzing the refractive index field, quantitative analysis requires additional methods. Rainbow schlieren photography is one candidate. In this method, the strength of deflection of light rays is obtained by using a rainbow aperture [6–8]. The quantitative phase-contrast imaging has also developed with incoherent-light [9,10], coherent-light [11,12], and asymmetric illumination [13]. There is also background-oriented schlieren (BOS) technique for measuring refractive index field quantitatively [14–20].

In the background-oriented schlieren (BOS) technique, the displacements of background dot patterns of a 2D image captured by a camera are measured. The displacement of the dot pattern is caused by the deflection of the light rays passing through an inhomogeneous refractive index field. Obtaining depth information of the refractive index field, however, is often difficult because the dot-pattern displacement results from the light ray deflected all along its path, requiring integration over the path. To address this, there are several methods of reconstructing the depth information [21–24].

Another method to reconstruct an axisymmetric 3D refractive index field is proposed here. Assuming that deflections of light rays passing through the refractive index field are sufficiently small and the paraxial approximation can be applicable to the light rays, the deflection angles are shown to be derived with a scalar potential. A deflection angle vector defined as a vector that consists of two components of deflection angles in orthogonal directions can be calculated with spatial gradient of the scalar potential. An arbitrary measured deflection angle vector, however, is generally written not only with a scalar potential but with a vector potential owing to the Helmholtz’s theorem [25]. Thus, the Poisson’s equation is derived to extract the scalar potential from a measured deflection angle vector. Using the Fourier transformation and Abel transformation with the Poisson’s equation, the axisymmetric 3D refractive index field is able to be reconstructed with the scalar potential. The scalar potential reconstruction method is validated by a reconstruction of a spherical refractive index fields where a deflection angle vector is accurately calculated.

## 2. Reconstruction method with scalar potential

#### 2.1 Small angle approximation and scalar potential for deflection angle vector

The path of a light ray passing through a refractive index field can be derived by the optical Lagrangian variational method [26–28]. The optical path length taken by the light ray passing from *A* to *B* can be written with refractive index *n* and infinitesimal line element *ds* along the optical path as

Here, the infinitesimal line element *ds* can be written in terms of the parameter *x* of (*x*, *y*, *z*) Cartesian coordinate system as

*x*. The Eq. (1) can be rewritten with the optical Lagrangian,

as

The optical light path remains stationary under variation within a family of nearby paths. To describe this, the following equations can be derived by applying the variational method to Eq. (4):

These equations are called the Euler–Lagrange equations, which can be used to calculate the light ray path.

Figure 1 shows a light ray path **q** from a far initial point **q**_{i} of (*x*_{i}, *y*_{i}, *z*_{i}) on a background to a far end point in the (*x*, *y*, *z*) Cartesian coordinate system. The light ray path **q** in an ambient homogenous refractive index field of *n*_{0} can be written with *y*-and *z*-direction angles of *α*_{y} and *α*_{z} respectively with respect to *x*-axis as

On the other hand, when the refractive index field is slightly deviated from the ambient refractive index *n*_{0} around the coordinate system origin within the radius of *R*_{c} as

**q**can be written with a deviation vector consisting of two components of Δ

*y*and Δ

*z*as

Differentiating **q** with respect to *x*, the following equation can be derived:

Here, the paraxial approximation is assumed to be applicable to the light ray path **q** where the direction angles of *α*_{y} and *α*_{z} are sufficiently smaller than 1. Thus, Eq. (10) can be approximated as

The second term of the right-hand of Eq. (11) is able to be written using deflection angles of *ε*_{y} and *ε*_{z} for *y*- and *z*-directions respectively with respect to *x*-axis as,

The deflection angles are assumed to be sufficiently smaller than 1, and Eq. (12) can be approximated as

Inserting Eq. (11) with Eq. (13) into Eqs. (5) and (6), the following equations can be derived:

Assuming that the second orders of *α _{y}*,

*α*,

_{z}*ε*

_{y}, and

*ε*

_{z}are negligible due to the small deflection angle assumption and the paraxial approximation, Eqs. (14) and (15) can be further approximated using Eq. (8) as

*nα*, Δ

_{y}*nα*, Δ

_{z}*nε*, and Δ

_{y}*nε*are assumed to be negligible in the derivations of the equations.

_{z}Integrating of Eqs. (16) and (17) over *x* respectively, the following equations can be derived as

*R*

_{c}<

*x*<

*R*

_{c}. The Eqs. (18) and (19) can be further approximated under the assumption of the small deviation angles and the paraxial approximation with (

*x*

_{0},

*y*

_{0},

*z*

_{0}) = (0,

*y*(0),

*z*(0)) as

Here, a scalar potential *ϕ* is introduced as

A deflection angle vector **ε** that consists of two components of deflection angles represented by Eqs. (20) and (21) in orthogonal directions can be written with the scalar potential *ϕ* as

Equation (23) indicates that the deflection angle vector is able to be derived from the scalar potential. Note that Eq. (23) is approximate equation under the assumption of small deviation angles and small direction angles of the light rays. In this work, applying the assumption of the small deviation angles with the paraxial approximation is called small angle approximation.

#### 2.2 Extracting scalar potential from measured deflection angle vector

Equation (23) indicates that the deflection angle vector can be derived with the scalar potential represented by Eq. (22) when the small angle approximation holds. An arbitrary vector, however, is written not only with a scalar potential but with a vector potential owing to the Helmholtz’s theorem assuming that the vector is twice-differentiable and decay sufficiently rapidly at infinity. Thus, a measured deflection angle vector **ε** that does not always obey the approximate equation of Eq. (23) is written not only with a scalar potential *ϕ* but with a vector potential **Ψ** as

For self-consistent use of the approximate equation of Eq. (23), it is necessary to extract the component consisting of the scalar potential. The Eq. (24), therefore, can be transformed as

Equation (25) is in the form of the Poisson’s equation, which indicates the scalar potential can be extracted from the measured deflection angle vector. The scalar potential can be calculated applying the 2D Fourier transformation method to Eq. (25). The 2D Fourier transformations of the scalar potential and the deflection angle vector are written respectively with Fourier-space quantities denoted with a tilde symbol as

Inserting Eqs. (26) and (27) into Eq. (25), the following equation can be derived:

Thus, the scalar potential can be extracted with the measured deflection angle vector inserting Eq. (28) into Eq. (26) as

#### 2.3 Axisymmetric assumption of refractive index field

The scalar potential represented by Eq. (22) can be written with a cylindrical coordinate system (*ρ*, *θ, z*) under the assumption of the axisymmetric refractive index field about the *z*-axis as

*n*(

*ρ*,

*z*) in the cylindrical coordinate system. The refractive index field is homogeneous outside of the sphere of the radius of

*R*c where the radius

*r*can be written as

Using the Abel transformation, Eq. (30) can be transformed inversely as

The differentiation of the scalar potential *ϕ* with respect to *y* can be derived using Eq. (29) as

Thus, the refractive index deviation field is able to be derived inserting Eq. (33) into Eq. (32) as

*ϕ*with respect to

*y*.

## 3. Reconstruction of axisymmetric 3D refractive index field

#### 3.1 Accurately calculated deflection angle vector field

The accuracy of the approximations of the scalar potential reconstruction method with Eq. (34) can be analyzed by applying the method to accurately calculated deflection angle vector. For the refractive index field that is spherically distributed with its center placed at the coordinate system origin *O*, the deflection angle vector can be accurately calculated as follows.

Figure 2 shows a light ray path on *x*-*Y* plane where the *Y*-axis is tilted with an angle Θ from the *y*-axis on the *y*-*z* plane. A cylindrical coordinates (*ρ*_{2}, *θ*_{2}) is taken for the light ray path **Q** on the *x*-*Y* plane with initial point **Q**_{0}. The refractive index field is axisymmetric about the origin *O* where the field is constant *n*_{0} with the radius over *R*_{c}. An angle *α* is defined as an angle between the light ray path and *x*-axis at the far initial point. A point **H** is an intersection of the light ray path with the *x* = 0 plane. The deflection angle *ε*_{2} defined as an angle between the light ray directions at the far initial point **Q**_{0} and the far end point is derived [24] as

where *e* is defined with *H* of a distance between the coordinate origin and the point **H** as

and *r*_{c} that represents a radius of a closet point of the light ray path to the coordinate system origin *O* satisfies the following equation as

Equation (35) can be transformed as

The deflection angle vector defined in the global Cartesian coordinate at the position ** H** in the

*x*= 0 plane can be derived with the tilt angle Θ in the limit of infinitesimal

*α*as

Note that the Eq. (39) is accurately derived without the small deflection angle approximation.

#### 3.2 Spherical refractive index field

As an example for performing the scalar potential reconstruction method, the spherical refractive index field is taken with the ambient refractive index *n*_{0} of 1 as

*r*is set to less than

*R*

_{c}. For the radius

*r*over than

*R*

_{c}, the refractive index field is defined as

The closest radius *r*_{c} is derived using Eq. (37) as

*R*

_{c}as

Here, the right-hand of Eq. (44) is derived with Eq. (36) in the limit of infinitesimal *α*.

## 4. Results and discussions

The deflection angle vector field in the *x* = 0 plane can be accurately calculated using Eqs. (38) and (39) with the refractive index represented by Eq. (40). Thus, the applicability of the scalar potential reconstruction method derived under the assumption of the small angular approximation is validated with the accurately calculated deflection angle vector field.

Figure 3 shows the two dimensional distributions of *y-* and *z*-component of the accurately calculated deflection angle vector with Δ*a* = 0.01 of Eq. (40) respectively. In each figure, the horizontal axis indicates *y* normalized by *R*_{c} and the vertical axis indicates *z* normalized by *R*_{c}. The *y*- and *z*-components of the deflection angle fields are contoured with a gray scale. As is expected, the *y*-component of the deflection angle is mirror-symmetric about the *z*-axis, and the *z-*component of the deflection angle is mirror-symmetric about the *y*-axis.

Using Eq. (29), a scalar potential *ϕ* is able to be derived from the 2D Fourier-transformation of the deflection angle vector field. Figure 4 shows the calculated scalar field in the *y*-*z* plane. The real part and the imaginary part of the scalar field are contoured with gray scale on the left-hand and the right-hand respectively. The 2D fast Fourier transformation (FFT) algorithm is applied to the deflection angle vector in an area of one side length normalized by *R*_{c} of 3.0 with spatial grids of 512x512. The real part of the scalar potential is shown to be spherically symmetric. The imaginary part is sufficiently small in comparison with the real part as is expected.

Using Eq. (34), the refractive index deviation is able to be reconstructed form the scalar potential. Figure 5 shows the reconstructed refractive index deviation fields on the *y*-axis with respect to *y* normalized by *R*_{c} are plotted with dashed lines for Δ*a* = 0.01, 0.1, and 1.0 respectively. The original refractive index deviation fields derived by Eq. (40) are also plotted with solid lines. The reconstructed refractive index deviation agrees well with the original profile for Δ*a* = 0.01 and 0.1. However, the reconstructed refractive index deviation differs from the original profile for Δ*a* = 1.0. This is caused by large deflection angles of the light rays which do not comply with the small angle assumption. Thus, the scalar potential reconstruction method is shown to work well when the small angle approximation is well applicable.

A cross-sectional view of the reconstructed axisymmetric 3D refractive index deviation field in the *x* = 0 plane for Δ*a* = 0.01 is contoured with a color scale in the left-hand of Fig. 6. The original refractive index profile is also shown in the right-hand of Fig. 6. The reconstructed 3D field agrees well with the original profile.

Figure 7 shows the reconstructed refractive index deviation fields with dashed lines with respect to the cylindrical radius *ρ* normalized by *R*_{c} for several constant *z* planes of normalized *z* by *R*_{c} of 0.0, 0.3, 0.4, 0.7 where the normalized quantity is denoted with a bar symbol. The original profiles are also plotted with solid lines. The reconstructed profiles agree well with the original profiles. There, however, are slight discrepancies between the reconstructed profiles and the original profiles. This is considered that the light ray paths derived by Eqs. (22) and (23) under the assumption of the small deflection angles differ slightly from the actual light ray paths. Indeed, as Δ*n* becomes smaller in Fig. 7, the accuracy of the reconstruction becomes higher.

Note that the scalar potential reconstruction method is able to be applied not only to the spherically symmetric field but any axisymmetric fields. The spherically symmetric field is used in this work because the deflection angle vector is able to be calculated accurately, which makes it possible to clarify the accuracy of the scalar potential reconstruction method.

In practice, a deflection angle vector field can be measured using the BOS technique with the optical flow algorithm [29–34]. With the measured deflection angle vector field, an axisymmetric 3D refractive index field is able to be reconstructed with the scalar potential reconstruction method.

## 5. Conclusions

A method to reconstruct an axisymmetric 3D refractive index field is proposed here. Assuming that deflections of light rays passing through a refractive index field are sufficiently small and the paraxial approximation can be applicable to the light rays, the deflection angles are shown to be derived with a scalar potential. A deflection angle vector defined as a vector that consists of two components of deflection angles in orthogonal directions can be calculated with spatial gradient of the scalar potential. An arbitrary measured deflection angle vector, however, is generally written not only with a scalar potential but with a vector potential owing to the Helmholtz’s theorem. Thus, the Poisson’s equation is derived to extract the scalar potential from the measured deflection angle vector. Applying the 2D Fourier transformation to the Poisson’s equation, the axisymmetric 3D refractive index field is able to be reconstructed with the Abel transformation of the scalar potential. The scalar potential reconstruction method is validated by a reconstruction of a spherical refractive index fields where a deflection angle vector is accurately calculated.

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