## Abstract

An omnidirectional light collector consisting of an axisymmetric spatially gradient refractive index medium can almost perfectly absorb light rays, regardless of where they come from. Based on the conformal mapping with complex gradient-index medium, the omnidirectional light collector, which is here called a dark hole, is able to be designed with an exponential function. The dark hole, however, has a reflective boundary where the Fresnel reflection occurs, which might lessen the absorption efficiency. To design a dark hole with consideration of the Fresnel reflection loss, a method to estimate its absorptance is necessary. Therefore, a formula to calculate the absorptance of the dark hole is derived based on the Lagrangian optics with the etendue conservation. Absorptances calculated using the formula agree well with those calculated using the Mie scattering theory in refractive index small-difference limit, which validates the formula. Absorptance of a dark hole with a silicon core and another dark hole with a complex gradient-index intermediate medium are calculated using the formula to be more than 98.8%. A micro-size dark hole is also shown to efficiently collect light rays with an absorptance of more than 95% using FDTD (finite-difference time-domain) simulation.

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

## 1. Introduction

Propagation of an electromagnetic wave obeys the well-known Maxwell’s equations. The equations in tensor representations are form-invariant under coordinate transformations. Using the form-invariant feature, the coordinate transformation methods to obtain a new type of gradient-index (GRIN) medium to control the electromagnetic wave propagations have been developed [1]. As a special case of the coordinate transformation, conformal mapping provides a particular way to control the electromagnetic waves [2]. Several kinds of GRIN media have been developed using the conformal mapping [3–5].

An omnidirectional light collector to swallow light rays, regardless of where they come from, has the potential to be applied to several kinds of technologies. For example, a solar thermal power plant uses cylindrical tube absorbers for collecting the solar thermal energy [6–10]. The efficiency of the plant can be increased by widening the collecting view angle of the absorber. Thus, an omnidirectional cylindrical collector that absorbs a wide-view-angle incoming light rays would be useful. Another example is a spherical omnidirectional collector with a phosphorous medium placed at its central core that could be used for lighting apparatus with a high-intensity laser, which collects a laser beam on the phosphorous medium and emit down-converted incoherent light from the phosphor [11].

The omnidirectional light collectors that perform high-efficient light absorption have recently been developed using GRIN media [12–17]. In this paper, an omnidirectional light collector is shown to be constructed on the basis of the conformal mapping using an exponential function and a complex gradient-index medium. The light collector, however, has reflective boundaries where the Fresnel reflection occurs. Thus, the omnidirectional light collector is here called a dark hole instead of a black hole in order to note the Fresnel reflection loss. The Fresnel reflection loss depends on optical parameters and a structure of the dark hole. To design a dark hole with consideration of the Fresnel reflection loss, a method to estimate its absorptance is necessary. Therefore, a formula to calculate the absorptance of the dark hole is derived on the basis of the Lagrangian optics with the etendue conservation. Absorptances calculated using the formula agree well with those calculated using Mie scattering theory in refractive index small-difference limit, which validates the formula. An absorptance of a dark hole consisting of a silicon core and a complex gradient-index intermediate medium is shown to exceed that of a dark hole without the complex gradient-index intermediate medium. Although light ray paths inside the dark hole are calculated on the basis of the geometrical optics (i.e., the Lagrangian optics), a micro-size dark hole is also shown to efficiently collect light rays using FDTD (Finite-Difference Time-Domain) simulation.

The rest of the paper is as follows. In chapter 2, using conformal mapping, dark holes with and without gradient extinction coefficient are derived. Each dark hole has a central core inside it, and also has an extinction coefficient gap boundary. In chapter 3, the Fresnel reflection at the gap boundary is described. In chapter 4, a path of each light ray incident on the dark hole reaching the gap boundary can be calculated using the Lagrangian optics based on the Euler-Lagrangian equation. At the gap boundary, the Fresnel reflection occurs. Therefore, with consideration of the Fresnel reflection loss, a formula to calculate the absorptance of the dark hole is derived based on the etendue conservation. The formula is validated by comparing absorptances calculated by the formula with those calculated by the Mie scattering theory [18–20] in refractive index small-difference limit. Using the formula, an absorptance of a dark hole consisting of a silicon core can be calculated to be more than 98.8%. A dark hole with a complex gradient-index intermediate medium is shown to exceed that of a dark hole without it. A compact dark hole whose size is of the order of the wavelength of visible light is also proved to work as an omnidirectional absorber using FDTD (Finite-Difference Time-Domain) simulation.

## 2. Design of dark hole based on conformal mapping

Based on conformal mapping, a coordinate transformation from a virtual space to a physical space in a cross-sectional 2D plane can be accomplished using two complex planes of *z* = *x* + i*y* for the virtual plane and *w*(*z*) = *u* + i*v* for the physical plane. Here, an exponential transformation is used for the conformal mapping, which can be written as

*r*

_{0}is a real constant. Equation (1) can be transformed as

*r*indicates the radius from the coordinate origin of the physical complex plane. In the conformal transformation, an angle between two arbitrary vectors in the original complex plane

*z*can be preserved in the projected complex plane

*w*(

*z*). A normalized radius

*ρ*is defined aswhere

*ρ*is restricted to less than 1 in the conformal mapping.

An electromagnetic parameter in the Maxwell’s equations, namely, a complex refractive index *ϕ* defined in the physical plane with a refractive index *n* and an extinction coefficient *K*, can be derived as

*ϕ*

_{0}indicates a complex refractive index of the virtual space and can be written as

Figure 1 shows a schematic diagram in the original (virtual) complex plane. Horizontal axis indicates a real part *x* of *z*, and vertical axis indicates an imaginary part *y* of *z*. The original complex plane is set to a semi-infinite region where *y* is set to from 0 to *y*_{0} with the entire *x*-axis.

A schematic diagram of the projected (physical) plane *w*(z) is shown in Fig. 2. Horizontal axis indicates a real part *u* of *w*, and vertical axis indicates an imaginary part *v* of *w*. In the projected plane, lines of *x* = constant in the original plane are transformed into lines along radial directions. On the other hand, lines of *y* = constant in the original plane are transformed into circles. From this diagram, it can be found that light rays coming along the radial directions reach the center of the coordinate and the light rays passing in tangential directions with respect to the radial directions will turn round with constant radius.

Considering that light rays bend toward higher refractive index region, a light ray directing slightly inward with respect to the tangential direction (i.e., azimuth direction) falls into the coordinate origin. Thus, a dark hole that can collect any light rays regardless of where they come from can be realized by the medium having the complex refractive index represented by Eq. (4).

Equation (4), however, has a singularity at *ρ* = 0, which is not attainable in practice. Therefore, in a practical design of a dark hole, a material with high-refractive index is placed at the central core of the dark hole with the core radius of *ρ*_{core} to avoid the singularity. The refractive index is made to be continuously connected at the boundary of the central core that has a complex refractive index of *ϕ*_{core} = *n*_{core} + *iK*_{core}. Thus, the core radius *ρ*_{core} is related with the central core refractive index *n*_{core} using Eq. (4) as

*ϕ*, can be written in the physical plane aswhere the normalized radius

*ρ*is limited in range as

For example, the central core is assumed to be silicon. The complex refractive index of silicon is 4.94 + 0.23*i* for the optical wavelength of 0.42 μm [21]. When the dark hole is placed in an ambient of air that has a refractive index *n*_{0} of 1, the normalized central core radius *ρ*_{core} becomes 0.20.

Firstly, the original complex refractive index is set to *ϕ*_{0} = 1 + 0*i* with *K*_{0} = 0. The intermediate medium from the central core to the periphery of the dark hole has a gradient-index (GRIN) without an extinction coefficient. Thus, this type of the dark hole is named a GRIN dark hole. The calculated complex refractive index in the *u*-*v* plane using Eqs. (7) and (8) is shown in Fig. 3. The refractive index *n* is color contoured on the left, and the extinction coefficient *K* is color contoured on the right. It can be found on the right of the figure that there is a gap boundary where an extinction coefficient *K* has a gap at the central core boundary.

Secondly, the extinction coefficient *K*** _{0}** of the original complex refractive index is set as

*n*is color contoured on the left, and the extinction coefficient

*K*is color contoured on the right. As shown on the right of the figure, it can be found that the extinction coefficient is continuously connected at the central core boundary. On the other hand, there is a gap boundary where the extinction coefficient

*K*has a gap at the periphery of the dark hole.

## 3. Fresnel reflection loss

Energy loss of light rays occurs at the gap boundary because of the Fresnel reflection, even when it might be practically negligible. Figure 5 shows a light ray incident on a gap boundary where extinction coefficient has a gap. An incident angle on the surface of the gap boundary is Θ_{i}. The origin of the coordinate system is *O*. The complex refractive indexes across the gap boundary are *n* + 0*i* in the outer region and *n* + *iK* in the inner region respectively.

The Fresnel reflectances, *F _{s}* for

*s*-polarization and

*F*, for

_{p}*p*-polarization, can be written respectively with the incident angle Θ

_{i}[22] as

*K*to the refractive index

*n*(i.e.,

*K*/

*n*) and the incident angle Θ

_{i}.

Reflectances can be calculated using Eqs. (10) and (11) as shown in Fig. 6. Horizontal axis indicates incident angle, and vertical axis indicates reflectance. The ratios of *K*/*n* are set to 0.046, 0.6, 1.0, 1.5, and 2.0. The reflectances of *s*- and *p*-polarization are found to decrease with smaller ratio of the *K*/*n*. On the other hand, it can be found that the reflectances increase with larger ratio *K*/*n* and larger incident angle Θ that is close to 90 degrees, which implies that the Fresnel reflectance should be considered at the gap boundaries.

## 4. Absorptance based on etendue conservation

#### 4.1. Basic equation of light ray path based on the Lagrangian optics

The light ray path in the intermediate medium of the GRIN dark hole can be calculated based on the Lagrangian optics using the Euler-Lagrangian equation [23,24]. The refractive index *ϕ* represented by Eq. (4) with an extinction coefficient of zero can be written with a radius *r* of less than *r*_{0} and an ambient refractive index *n*_{0} as

A cylindrical coordinates (*r*, *θ*) with an origin *O* in the *u*-*v* plane are taken for a light ray path **Q** with initial point **Q**_{0} as shown in Fig. 7. The refractive index field is axisymmetric about the origin *O* where the field is constant *n*_{0} with the radius of more than *r*_{0}. An angle *β* is defined as an angle between the light ray path and *v*-axis at the far initial point. A point **H** is an intersection of an extrapolation along the initial light ray direction with the *v* = 0 line. The angle *β* can be set to zero without loss of generality, which means that light rays are incident on the dark hole in a direction parallel to the *v*-axis.

The Euler-Lagrangian equation with the radius *r* and the azimuth angle *θ* can be written as

*r*asUsing Eqs. (12) and (13), the following equation can be derived for

*r*of less than

*r*

_{0}asThe left-hand side of Eq. (15) can be written with an angle Θ between the light ray direction and an inward radial direction aswhere Θ is set in a range from 0 to π/2. The angle Θ can be considered as an internal incident angle on each thin shell that is taken inside the dark hole with radius

*r*.

An incident angle Θ_{0} of the light ray on the peripherical boundary of the dark hole (i.e., *r* = *r*_{0}) can be written as

From Eqs. (15), (16), and (17), the internal incident angle Θ on each thin shell with radius *r* is found to be always constant of Θ_{0} (i.e., Θ = Θ_{0}). Equation (16), therefore, can be solved analytically with the incident angle Θ_{0} as

#### 4.2. Absorptance formula

An absorptance of a dark hole can be derived based on the etendue conservation as follows. In an optical system, there is a conserved quantity, namely, an etendue. When an infinitesimal surface element, d*S*, that is immersed in a medium of refractive index *n* is crossed by the light rays confined to a solid angle dΩ with an incident angle Θ, the infinitesimal etendue, d*G*, can be written [25,26] as

An infinitesimal etendue of incident light rays that are assumed to be parallel beam is set to d*G*_{0} with an infinitesimal surface element d*S*_{0} where the incident angle is zero with a solid angle dΩ_{0}. Thus, the following relationship can be derived using the infinitesimal etendue d*G*_{0} and that of internal light rays, d*G*, at a normalized radius *ρ* with an incident angle Θ and a solid angle dΩ as

An infinitesimal reflectance, d*B*, on the infinitesimal surface element at the gap boundary with the normalized radius *ρ* can be written with a luminance *I*(*ρ*) using the Fresnel reflectance of *F _{s,p}* corresponding to Eq. (10) or Eq. (11) as

Assuming that there are not any light rays that transmit through the central core, an absorptance *A _{s}*,

*for*

_{p}*s*-polarization or

*p*-polarization can be written using Eq. (21) as

*I*

_{0}as

*S*

_{0}by the incident position of

*H*, assuming homogeneous incident rays as

*F*, is represented by Eq. (10) or Eq. (11), and the internal incident angle Θ that is constant of Θ

_{s,p}_{0}can be written with

*H*using Eq. (17) as

Equation (24) is a formula based on the etendue conservation to calculate the absorptance of the dark hole with consideration of the Fresnel reflection.

## 5. Results and discussions

#### 5.1. Light ray path inside GRIN dark hole

Figure 8 shows light ray paths calculated using Eq. (18) for a GRIN dark hole with a refractive index represented by Eq. (7) with *K* = 0. Light ray paths are plotted with solid lines in the *u*-*v* cross-sectional plane. Horizontal axis indicates *u*-axis normalized by *r*_{0}, and vertical axis indicates *v*-axis normalized by *r*_{0}. From the figure, each light ray is found to fall spirally into the central core, which validates that the dark hole can collect omnidirectional light rays, regardless of where they come from.

#### 5.2. Absorptance formula validation in refractive index small-difference limit

An absorptance of a dark hole with a central core of complex refractive index of *n*_{core} + *iK*_{core} can be calculated by the formula represented by Eq. (24). When the *n*_{core} is set to close to an ambient refractive index of *n*_{0}, the absorptance can also be calculated on the basis of the Mie scattering theory [18]. Thus, taking a small-difference limit of the refractive index between *n*_{core} and *n*_{0}, the absorptance can be calculated by the formula and is comparable to that calculated by the Mie scattering theory. Because the *s*- and *p*-polarization Fresnel reflectances represented by Eqs. (10) and (11) depend on the internal incident angle Θ* _{i}* and the ratio of

*K*/

*n*, the absorptances that are integrated over the incident angle depend only on the ratio of

*K*

_{core}/

*n*

_{core}in the refractive index small-difference limit.

Figure 9 shows *s*- and *p*-polarization absorptances calculated by both the formula (etendue conservation) and the Mie scattering theory (Mie theory) with respect to *K*_{core}/*n*_{core}. Horizontal axis indicates the ratio of *K*_{core}/*n*_{core}, and vertical axis indicates absorptance *Q*_{abs}. The solid line indicates absorptance calculated by the formula for *s*-polarization, and the dashed line indicates that for *p*-polarization. The circle mark indicates absorptance calculated by the Mie theory for *s*-polarization, and the cross mark indicates that for *p*-polarization. In the calculations based on the Mie theory, the diameter of the dark hole is set to 30 μm with a wavelength of light rays of 0.5 μm and the refractive index *n*_{core} of 5.0. In the refractive index small-difference limit (i.e., *n*_{core} = *n*_{0}), the diameter of the dark hole is the same as that of the core. The diameter of the dark hole is sufficiently larger than the wavelength, which ensures that the absorptances calculated by the Mie theory approximate those calculated on the basis of the geometrical optics. From the figure, it can be found that the absorptances calculated by the formula agree well with those calculated by the Mie theory for the ratio *K*_{core}/*n*_{core} of more than 0.01. On the other hand, the absorptances calculated by the Mie theory become smaller than those calculated by the formula for the ratio *K*_{core}/*n*_{core} of less than 0.001. This is because a light ray transmittance through the central core is not taken into account in the formula. The formula, therefore, will not decrease in the limit of small value of *K*_{core}/*n*_{core}. The transmittance, however, can be neglected when the diameter multiplied by the extinction coefficient is sufficiently larger than the wavelength.

Thus, the formula based on the etendue conservation agrees well with the Mie theory in the refractive index small-difference limit under the assumption of negligible transmittance of the core of the dark hole, which validates the formula.

#### 5.3. Absorptances of GRIN and CGRIN dark hole

The absorptances for the GRIN dark hole and for the CGRIN dark hole are able to be calculated with the formula represented by Eq. (24).

The GRIN dark hole with the refractive index previously shown in Fig. 3 has an extinction coefficient gap at the central core boundary. The diameter of the central core is assumed to be large enough that light rays will not transmit through the central core. The absorptances are calculated to be 98.8% for *s*-polarization and 99.1% for *p*-polarization.

The CGRIN dark hole with the complex refractive index previously shown in Fig. 4 has an extinction gap at the periphery of the dark hole. Light rays are assumed not to transmit through the dark hole once the light rays reach the gap boundary (i.e., periphery of the dark hole). The absorptances are calculated to be more than 99.9% for both *s*- and *p*-polarization. Thus, the CGRIN dark hole might be preferable for collecting solar thermal energy with high efficiency.

#### 5.4. Micro-size dark hole with silicon core

Although the light ray path represented by Eq. (18) are derived on the basis of the geometrical optics, a compact dark hole with size of the order of the wavelength of visible light is shown to work as the omnidirectional collector using FDTD (Finite-Difference Time-Domain) simulation [27] as follows.

Firstly, a GRIN dark hole that can collect incoming light rays into its central core and can absorb them by the central core is considered with an ambient refractive index of 1.0. The diameter of the dark hole, 2*r*_{0}, is set to 5.0 μm. The original complex refractive index, *ϕ*_{0}, is set to 1.0 + 0*i* where the *K*_{0} is set to zero. The central core material of the dark hole is set to silicon. The complex refractive index of silicon is 4.94 + 0.23*i* where the optical wavelength is set to 0.42 μm. The imaginary part of the complex refractive index in the dark hole has a gap at the central core boundary. Thus, it is possible for reflection to occur at the gap boundary. Figure 10 shows calculated steady-state electromagnetic wave propagation for *s*-polarization in the cross-sectional plane. The source of the electromagnetic wave is placed at (0.0 μm, −2.5 μm) with a width of 10.0 μm. The central core radius calculated using Eq. (6) is 0.5 μm. From the figure, the electromagnetic wave is found to fall spirally into the coordinate origin.

An absorption density *q*_{abs} is defined as a ratio of time-averaged absorbed power per unit area to total time-averaged absorbed power inside the dark hole, and can be written with complex electric filed **E** and complex magnetic field **H** [22] as

*u*and d

*v*indicate infinitesimal line elements, and the asterisk indicates complex conjugate. The absorption density

*q*

_{abs}of the GRIN dark hole calculated by the FDTD simulation is shown in Fig. 11 in the cross-sectional plane. The

*q*

_{abs}is color-contoured in log scale. The parameters are the same as those used in Fig. 10.

The absorption efficiency *η* is defined with an entire absorbed power, *Q* [W], and the incident intensity, *I*_{in} [W/μm], as

*η*of GRIN dark hole is calculated to be 0.95 whereas that of a silicon cylinder with the same diameter of 5.0 μm is calculated by the Mie theory to be 0.47 with the ambient refractive index of 1.0. As expected, the reflectance at the boundary of the central core of the GRIN dark hole is reduced enough.

Secondly, a CGRIN dark hole that gradually and weakly absorbs incoming light rays along their paths is considered. The diameter of the dark hole, 2*r*_{0}, is set to 5.0 μm. The original complex refractive index, *ϕ*_{0} = *n*_{0} + *iK*_{0}, is set to 1.0 + 0.001*i* where the imaginary part *K*_{0} is set to a small value to show the light ray paths inside the dark hole. The complex refractive index of the central core is set to 4.94 + 0.005*i*. A central core radius of the dark hole calculated using Eq. (6) is 0.5 μm. Figure 12 shows calculated electromagnetic wave propagation for *s*-polarization in the cross-sectional plane for several time durations. The source of the electromagnetic wave is placed at (2.0 μm, −2.5 μm) with a width of 3.0 μm. The optical wavelength is set to 0.42 μm. From the figure, the electromagnetic wave is found to fall spirally into the coordinate origin. The light ray intensities also decrease along the spiral paths without noticeable reflection.

Thirdly, a CGRIN dark hole that has a silicon core is considered. The diameter of the dark hole, 2*r*_{0}, is set to 5.0 μm. The central core has a complex refractive index of 4.94 + 0.23*i* where the optical wavelength is set to 0.42 μm. The original complex refractive index, *ϕ*_{0}, is set to 1.0 + 0.0465*i*. The central core radius calculated using Eq. (6) is 0.5 μm. The imaginary part of the complex refractive index in the dark hole has an extinction coefficient gap at the periphery of the dark hole. Thus, it is possible for reflection to occur at the boundary. Figure 13 shows calculated steady-state electromagnetic wave propagation for *s*-polarization in the cross-sectional plane. The source of the electromagnetic wave is placed at (0.0 μm, −2.5 μm) with a width of 10.0 μm. The absorption density *q*_{abs} of the CGRIN dark hole calculated by the FDTD simulation is shown in Fig. 14 in the cross-sectional plane. The *q*_{abs} is color-contoured in log scale. The absorption efficiency *η* calculated by FDTD simulation is 1.00874 with an accuracy of a few percent, which is larger than the GRIN dark hole.

## 6. Conclusions

A dark hole is shown to be constructed on the basis of the conformal mapping using an exponential function and a complex gradient-index medium. Based on the Lagrangian optics, a GRIN dark hole with a central core of silicon material is shown to collect omnidirectional light rays, regardless of where they come from.

A formula to calculate the absorptance of the dark hole is derived based on the etendue conservation. Absorptances calculated using the formula agree well with those calculated using Mie scattering theory in refractive index small-difference limit, which validates the formula. Absorptances of a GRIN dark hole that has an extinction coefficient gap at the central core boundary are calculated to be 98.8% for *s*-polarization and 99.1% for *p*-polarization. Absorptances of a CGRIN dark hole that has an extinction gap at the periphery of the dark hole are calculated to be more than 99.9% for both *s*- and *p*-polarization.

Although light ray paths inside the dark hole are derived on the basis of the geometrical optics (i.e., the Lagrangian optics), micro-size dark holes with size of the order of the wavelength of visible light are also shown to efficiently collect light rays using FDTD simulation. An absorptance of a CGRIN dark hole with a complex gradient-index intermediate medium is shown to exceed that of a GRIN dark hole.

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