Experimental Characterization of High Tolerance to Beam Irradiation Conditions of Light Beam Power Receiving Module for Optical Wireless Power Transmission Equipped with a Fly-Eye Lens System

Abstract

This paper is an experimental characterization of a light-receiving module containing a fly-eye lens system with high tolerance to beam irradiation conditions. The fly-eye lens system, which is tolerant to fluctuations in beam shape, beam size, number of beams, beam incident position, and beam incident direction, was proposed, a light receiver module with a fly-eye lens system was constructed, and its characteristics were evaluated. The effect of the beam size on the fly-eye lens system was evaluated and the tolerance to misalignment of beam incident position was measured. When a GaAs solar cell was irradiated with a laser beam of 450 nm wavelength and 6 W light output through a 90 cm long water tank with tap water, a maximum output of 0.755 W was obtained as underwater OWPT. In addition, a fly-eye lens system with mirrors applied to four surfaces was proposed and fabricated as a light-receiving side module that can receive high incident angles from any direction of up, down, left, and right and its effectiveness was clarified through experiments.

1. Introduction

Almost all current electronic equipment requires a direct wired power supply or battery recharging using a wire. This situation strongly limits the function and convenience of equipment. Therefore, wireless power transmission (WPT) will greatly improve convenience by simplifying the charging process. For example, in particular, since underwater environments, such as the sea, where, currently, only limited wireless technology of sound waves is used almost exclusively, have been called the ‘last frontier’, the development of underwater high-performance wireless technology is highly desired [1]. Although underwater equipment is either wired or battery powered, wire limits the operation distance and continuous operation of equipment is difficult with batteries; recharging work and battery replacement are also necessary [2,3]. However, the work of pulling it out of the water and connecting it under wet conditions is a large load. The problems and limitations are the same on the ground and in the air as well as in the water. In particular, the use of electric vehicles, robots, drones, and other forms of electric mobility is expanding; however, there are challenges in increasing battery capacity, easy/automatic recharging work, and reducing the charging time for long-distance moving and autonomous systems. If a large number of power source systems are prepared as infrastructure facilities for dynamic charging, batteries are not needed or can be decreased to a small capacity as well as removing the recharging process.

Various WPT methods, such as electromagnetic induction and microwave, are already in practical use [4,5,6]. However, the electromagnetic induction method has high power transfer efficiency at a very short distance of a few centimeters and the efficiency decreases exponentially with increasing distance. Therefore, it cannot be used even at a distance of only a few meters. The microwave method can transmit power to remote locations because it uses beam radiation. However, its long wavelength results in large diffraction and the beam area expands quadratically with distance, resulting in lower power transmission efficiency due to a lower beam collection rate at the receiving equipment. For this reason, using 10 m or less may be realistic. Furthermore, these existing methods use electromagnetic waves in the high-frequency band, which may cause electromagnetic noise interference (EMI) to other equipment. In addition, in the water, transmission of conventional electromagnetic waves is known as very difficult in nature due to large transmission loss.

Therefore, an optical wireless power transmission (OWPT) method is expected [7,8]. The wavelength of the light is about five digits shorter than that of microwaves, resulting in less diffraction of the optical beam and enabling a long-distance power transmission in a compact configuration due to high beam collection rate capability. It also has the feature of generating no EMI. In addition, due to the large loss of electromagnetic waves other than visible light, mainly green and blue, OWPT is considered as practically the only remote WPT method underwater. OWPT is, thus, a promising method. Although the concept was proposed around 1970, its research and development are a new technological field that has only recently begun to emerge. In fact, various studies have been conducted on the various elemental technologies of OWPT systems [9,10,11,12]. Among these, efficient operation of the solar cell, which is the light-receiving device, is an important part of the OWPT. Thus, verification of the operation characteristics and development of improved devices on the light-receiving side has been investigated [13,14,15]. While improving the photoelectric conversion efficiency of solar cells is important, uniform light irradiation to the solar cell module without light leakage from the solar cell is an important issue, especially for OWPT that irradiates with “light beams”. Such light irradiation conditions can be easily designed for passive optics if the distance and the positional relationship between the light source and the solar cell, as well as the angle of incidence, are fixed. However, when the positional relationship and beam angle are arbitrary or variable, high tolerance for changes in beam shape, beam size, beam misalignment, and a wide range of incident beam angles is required [16,17]. There are many limitations in OWPT for realizing such various functionalities and tolerances. Therefore, this is a challenge in OWPT. As one attractive solution, the application of a fly-eye lens system is promising [18].

The purpose of this study is to experimentally clarify the effectiveness of the fly-eye lens system in OWPT. In addition, improvements in the functionality of the fly-eye lens system are also investigated. Although the fly-eye lens system can be applied in a variety of environments and applications, this study is performed mainly for the application of the underwater OWPT, and a configuration using a blue-color laser was prepared and its characteristics were evaluated.

The structure of this paper is as follows: Section 2 explains the required functions, basic operating mechanism, and conditions for the fly-eye lens for better understanding of this report, Section 3 describes the construction of the basic configuration of the fly-eye lens system and experimental verification of its operation, Section 4 reports characteristics of underwater OWPT using the fly-eye lens system, Section 5 proposes a method for expanding the beam incidence angle on the fly-eye lens system and its construction and characterization of the module are reported, and Section 6 summarizes and concludes this report.

2. Requirements and Fundamental Operation Principle of Fly-Eye Lens System

2.1. Issues of Light Irradiation on Solar Cells

OWPT can operate more efficiently, even using the same solar cells when generating electricity from a monochromatic light beam than from a broad spectrum of sunlight. In fact, by using monochromatic light, such as laser light, photovoltaic conversion efficiencies of more than 40% have been reported for Si solar cells [19], and GaAs solar cells have been reported to have high photovoltaic conversion efficiencies of nearly 70% [20].

In general, the output voltage of a single material semiconductor solar cell is 0.2–0.4 V lower than the voltage corresponding to its bandgap energy [21], even under ideal conditions, and 0.4–0.6 V lower in practice. The output voltage is about 0.5–0.7 V for Si solar cells and 0.7–1.1 V for GaAs solar cells. Efficient operation is often difficult if this voltage is used, as in load circuits and voltage conversion circuits. For this reason, solar cell modules generally consist of multiple cells arranged in a plane and connected in series to achieve high-output voltage. Existing applications of solar cells, such as power generation under sunlight and indoor lighting, typically operate as expected properly, because the entire module is irradiated by the light of nearly uniform intensity. On the other hand, in OWPT that uses light beams with a finite beam size, it is assumed that only a part of the module of the solar cell is irradiated by a beam with light intensity distribution or that light leaks outside the solar cell module. In the former case, such light irradiation significantly reduces the operating efficiency of the solar cell module due to the series-connected configuration of the cells [22]. The latter reduces the system operating efficiency due to the waste of unavailable light. Furthermore, while OWPT has the advantage of allowing for arbitrary variety in light beam intensity, due to the low conversion efficiency, solar modules locally generate more heat under high-power light beam irradiation [23].

There are many possible solutions to these problems in light beam irradiation to the solar cell module, such as avoiding the effects by circuit-configuration-based arrangement of solar cell connections, applying bypass diodes, and actively controlling the irradiation beam shape; however, there are also many issues, such as the complexity of the system configuration and the limitations of improving efficiency. Therefore, a fly-eye lens system is the most attractive to irradiate the entire solar cell module with a uniform intensity beam without any light leakage to the outside of the solar cell module, using only passive optics with no loss in principle.

2.2. Uniform Irradiation by Fly-Eye Lens System

Figure 1 shows a conceptual diagram of the operation of a fly-eye lens. A fly-eye lens is an array lens of elemental lenses in a specific shape. The use of two of these lens array plates and one imaging lens with an appropriate focal length enables uniform irradiation with a zoomed elemental lens shape [24,25]. In this paper, this lens set is called a “fly-eye lens system”. The uniform irradiation principle is based on the spatial division of the light beam by the first lens array and its superposition by the imaging lens. Spatial division of the light beam and its super-position do not always result in perfectly uniform irradiation. Uniformity also strongly depends on the spatial distribution of the beam. However, especially in beam irradiation to series-connected solar cell modules, direct incident of strong intensity non-uniformity beam causes a large output degradation. The fly-eye lens system can reduce such non-uniformity and provide effective uniform irradiation. The beam irradiation distance, that is, the position of the solar cell, is the focal length of the imaging lens. The second fly-eye lens has the same focal length as the first and the final beam irradiation position can be fixed; even if the incident angle changes over some range of incident angles, that maximum angle is called “allowable incident angle”. The operating principle of the fly-eye lens system, which consists of beam splitting and superposition, causes certain constraints; however, the incident beam size is arbitrary and the beam shape is arbitrary as well [26].

This fly-eye lens system has been used or is being considered for applications, such as backlighting in optical disks, stereo microscopes, EUV lithography, solar cell measuring instrument, and 3D displays [27,28,29,30,31].

The irradiation beam shape of the fly-eye lens system is similar to that of an elemental lens; however, the elemental lens shape is usually square, rectangular, hexagonal, or circular to ensure a sufficiently high lens occupancy. In this study, a rectangle is selected, assuming a general solar module shape. A square is selected in the experiment.

For an elemental lens height $h$, elemental lens focal length $f$, and imaging lens focal length $F$, the irradiation beam height $H$ is calculated by Equation (1).

$$H=\frac{h}{f}·F$$

(1)

In existing applications using fly-eye lens systems, the fly-eye lens system is mounted as part of the light source module and the distance between the light source and fly-eye lens system is usually fixed. Light-source position and light irradiation position on the fly-eye lens system are also usually fixed. On the other hand, in OWPT, the positional relationship between the light source and the light-receiving device is highly arbitrary. Therefore, conditions in which light is emitted from an arbitrary distance and the position of the light source fluctuates with respect to the receiver equipment will be occurred. Therefore, this study was designed to install a fly-eye lens system as a part of the light-receiving module. This makes the beam propagation distance from the light source to the fly-eye lens system arbitrary. The light beam can enter anywhere in the fly-eye lens system because the beam entering each elemental lens forms an image at the designed position, that is, the solar cell distance. Therefore, there is a degree of freedom in the positional relationship between the light source and the fly-eye lens system. Furthermore, multiple light sources can be irradiated at arbitrary positions as needed, because the intensity of each light source is superimposed on the solar cell surface.

2.3. Operation Limitations of Fly-Eye Lens

Because the fly-eye lens is constructed by arranging elemental lenses in a plane, the fly-eye lens plate itself can be prepared as infinitely large. However, the light irradiation of the fly-eye lens system is in the designed area through the imaging lens. Therefore, the effective size of the fly-eye lens plate or the misalignment tolerance of the light beam is limited by the size of the imaging lens.

The diameter of the imaging lens with focal length $F$ is assumed to be $2R$. Here, the aperture size of the lens is described by the F-number $(F_{\#})$ or numerical aperture ($NA$).

$$F_{\#}=\frac{F}{2R}$$

(2)

On the other hand, since the sine condition holds for ideal image formation

$$NA=\frac{1}{2F_{\#}}=\frac{R}{F}$$

(3)

NA is defined in Equation (4) by the medium refractive index $n$ between the lens and the image formation point and the elevation angle $\theta$ when looking at the lens from the focal point.

$$NA=n\sin \theta$$

(4)

Here, $0<\sin \theta <1$ and the refractive index of air are assumed to be approximately 1, relation Expression (5) is valid.

$$0<NA<1$$

(5)

As a result, relation Expression (6) holds for an imaging lens diameter of $2R$.

$$2R<2F$$

(6)

Thus, the theoretical upper limit for the diameter $2R$ of the imaging lens is twice the focal length $F$ of the imaging lens. This is the upper limit of the effective size limit of the fly-eye lens system, that is, misalignment tolerance of the fly-eye lens system. To achieve the largest misalignment tolerance, the imaging lens diameter $2R$ is set as the same as $2F$. This condition is obtained under the condition that $NA$ is 1. Usually, larger $NA$ causes difficulty in lens design and deterioration in image formation characteristics. Fortunately, OWPT does not require high image formation characteristics and, thus, the high-$NA$ condition can be applied easily.

The fly-eye lens system requires the distance of the focal length of the imaging lens. This causes a relatively thick light-receiving module. This thick module is one of the drawbacks of the fly-eye lens system in OWPT. To reduce the thickness, a reduction in the focal length of the imaging lens is effective; however, this causes a reduction in the misalignment tolerance. The optimal design of the fly-eye lens system and module will be changed for different applications.

3. Experimental Characterization of the Fly-Eye Lens System

3.1. Beam Size Dependence

Experimental characterization of the effect of beam size on the fly-eye lens system was performed. The fly-eye lens utilized was formed using an acrylic lens by a cutting-type numerically controlled machine with a narrow needle, a size of 1 cm square elemental lens, and 10 × 10 arrays with a focal length of 20 mm. The imaging lens was a 10 cm square and 200 mm focal length plastic Fresnel lens (NTKJ Co., Ltd., Tokyo, Japan, model name: CF200). $NA$ of the imaging lens is calculated as 0.35. Anti-reflection (AR) coating for 450 nm wavelength is applied in both surfaces of each lens.

The light source is a 450 nm wavelength laser. The applied light source module output is a collimated beam with a maximum power of 6 W and slightly elliptical and the average full width at half maximum (FWHM) is about 1.5 mm through the collimating optics. To change the size of the light beam entering the fly-eye lens system, the beam size was widened by a concave lens (focal length −25 mm, lens diameter 20 mm) with a beam spread angle of several degrees.

Figure 2 shows the experimental results of the relationship between the electricity power output and the incident beam size (FWHM of the average of the vertical and horizontal values) at the fly-eye lens surface. A 10 cm square 12-cell series-connected Si solar cell (SUNYOOO solar Limited., Changzhou, China, model name: SY-M2W) was installed at the irradiation surface of the fly-eye lens system. The laser power was set at 500 mW. The electric power output was plotted for the maximum power condition from a solar cell I-V characterization instrument (ADC CORPORATION, Saitama, Japan, model name: 6244).

The results show that the solar cell output increases with increasing beam size in a beam size range below 20 mm and saturates at beam sizes above 20 mm. This characteristic is based on the fact that when the beam size is smaller than about two lenses for the 1 cm square elemental lens size used in this study, the light intensity distribution on the solar cell surface becomes non-uniform and the solar cell output is affected by the uniformity of the irradiated light. Since the beam size must span multiple elemental lenses, it is necessary to increase the beam size to 20 mm or more for the present experimental configuration.

When the beam size is sufficiently large, as shown in this section, a high output can be obtained due to the effective uniform irradiation of the solar cell. Since the mechanism of the fly-eye lens system is spatially dividing and overlapping the intensity distribution of the beam, similar uniform illumination characteristics can be obtained for various beam shapes with a sufficiently large beam size. In addition, when using multiple beams, if the beams are incident on different fly-eye lens positions and the total incident beam area of all the beams is sufficiently large, the same uniform irradiation characteristics can be obtained. Therefore, a fly-eye lens module is effective for beam size, beam shape, and multiple beams.

3.2. Beam Position Tolerance of Fly-Eye Lens System

The tolerance of the fly-eye lens system to beam incidence misalignment was evaluated. In this experiment, an acrylic fly-eye lens is a 5 mm square elemental lens with a focal length of 10 mm and a 16 × 16 array, and a plastic Fresnel lens with a focal length of 160 mm and an 8 cm square was used for the imaging lens. In this configuration, the light irradiation size is 8 cm square. In this experiment, a GaAs solar cell module (ATI, 5 cm × 8 cm) with series consisting of five cells was applied as an 8 cm square by placing two modules. The beam emitted from the light source was magnified with a concave lens and the FWHM of the beam was about 15 mm in the moving direction of beam incident position. This width satisfies over twice the elemental lens size. The beam width is approximately 50 mm in the direction perpendicular to the direction of movement. The laser output power was set at 1.0 W.

Figure 3 shows a conceptual front view of the fly-eye lens. Figure 4 shows the measured results of the incident beam position and solar cell output. As a result, when the beam was injected at around the center of the fly-eye lens system within a certain range, the solar cell output power was nearly uniform at around 90 mW, and when the beam began to deviate out of the fly-eye lens, the solar cell output power dropped steeply. In this experiment, output power fluctuation of about 10% was observed, even when the beam entered the fly-eye lens system. This is believed to be because the solar cells were not irradiated completely uniformly due to the intensity distribution of the incident beam itself and errors in the installation position of the lens system. However, the results confirm the tolerance of the fly-eye lens system to misalignment of about 5 cm for output of similar to the peak and about 7 cm for relatively high output, even when the beam size is taken into account in this experimental system.

4. Experimental Investigation of Underwater OWPT

4.1. Beam Expander Configurations

A basic evaluation of underwater OWPT was conducted using a fly-eye lens system. Beam characteristic tolerance, such as beam size, beam shape, and beam incident position, is particularly important underwater, because changes in utilization conditions and the occurrence of misalignment are likely to occur [32].

In this experiment, at first, two methods for generating the beam size incident on the fly-eye lens system were configurated. In this experiment, the light beam was passed in the water in the water tank. The light source system and light-receiving system were outside the water tank. Even in the realistic systems, both light source and light receiver systems will be installed in the isolated module from the water.

Figure 5a shows the configuration with a concave lens for beam expansion just before the fly-eye lens on the left side. The beam from the light source is collimated in the light source module with an FWHM of 1–2 mm. Figure 5b shows a beam propagation configuration with a beam expander, in which the beam size is enlarged by a concave lens on the light source side and a collimated beam is generated by a collimating lens. In the experiment, a focal length of −25 mm of the concave lens for beam expansion was prepared. The irradiated beam FWHM was about 20 mm. The fly-eye lens system is the same as in the evaluation of beam size dependence, as shown in Figure 2.

The irradiation beam shape projected on the screen at the left end of Figure 5 shows that the fly-eye lens system produces a 10 cm square beam of nearly uniform intensity in both configurations.

Figure 5a uses a narrow collimated beam from a light source module and enlarges it to the required size just before the fly-eye lens system. The overall configuration is simplified because a lens for collimating the expanded beam is not required. In addition, when a fly-eye lens system with different characteristics is used, the system has a high degree of freedom because the necessary beam size is generated on the receiver equipment side. However, the beam must be injected precisely into the concave lens installed just before the fly-eye lens system and the beam incidence misalignment tolerance feature of the fly-eye lens system cannot be used.

In the configuration shown in Figure 5b, the beam size is generated by the beam expander system on the light source side, so the configuration on the solar cell side is simplified to only a fly-eye lens system. This configuration is effective when multiple same light-receiving equipment is targeted. This configuration is useful to tolerate beam incidence misalignment to the fly-eye lens system. However, when different beam sizes are required for different fly-eye lens systems, necessary beams must be prepared at the light source side each time.

4.2. Power Transmission Efficiency in Underwater OWPT

The characteristics of optical power output and efficiency by underwater OWPT were evaluated using Figure 5a as the beam magnification system. The fly-eye lens is an elemental lens with a focal length of 20 mm and a 10 mm square and 8 × 8 array. The imaging lens is an 8 cm square plastic Fresnel lens with a focal length of 160 mm. In this configuration, the irradiation size is 8 cm square and the 8 cm square GaAs solar cell module was used as before.

The same as in Figure 5, a 90 cm long water tank was applied and filled with tap water and a 90 cm long underwater OWPT was performed. It is noted that there are only several reports about underwater OWPT. Currently, the output power of these reports is less than 100 mW because of the initial stage of the evaluation of the underwater power transmission characteristics [33,34,35]. In this study, a practically attractive receiver module is prepared and a relatively higher output power condition is examined because the light beam irradiation area can be increased to several centimeters square and the temperature rise can be suppressed.

Figure 6 shows the measured and estimated characteristics of the optical output and loss at each device setting position. The GaAs solar cell output of 755 mW was obtained for a laser output of 5.95 W. The efficiency of the power output relative to the laser output of the entire experimental system was 12.7%. This efficiency includes the loss of light propagation and the water tank.

Just after the laser power was passed through the water tank, the beam power was 4.90 W and it was a decrease of 17.6%. The four surface reflections of the acrylic tank and the 90 cm length of tap water caused losses. The reflection losses between the air and the outer surface of the water tank are estimated in principle to be about 8% for the two surfaces. On the other hand, the reflection loss on the inside of the tank is low due to the small difference in refractive index between water and acrylic. However, there is also scattering on both surfaces due to imperfect flatness. As a result, the loss due to the water tank is considered to be 9.4% in total. On the other hand, for light with a wavelength of 450 nm, the loss coefficient is less than 0.1 m⁻¹ [36,37] for clean ocean or lake water and about 0.01 m⁻¹ [38,39] for pure water. Even though the loss of tap water is considered to vary depending on the conditions, the preliminary evaluation of this study based on experiments with tank lengths of 30 cm, 45 cm, and 90 cm showed that the loss coefficient was 0.095 m⁻¹, which is close to that of water from the sea or a lake. This results in an estimated loss of 8.2% for a 90 cm length of tap water.

Ideally, the passive lens loss, including the fly-eye lens system, would be regarded to be almost zero if appropriate AR coating is applied. However, there is a lens system loss, mainly due to the scattering loss at the elemental lens boundary of the fly-eye lens and the scattering of the Fresnel lens of the imaging lens. In particular, the fly-eye lens used in this study is a cutting with a needle, which generates an unnecessary width of 0.5 mm at the elemental lens boundary due to the needle width. For an elemental lens size of 1 cm square, approximately 10% of the area has no lensing function, resulting in scattering loss. Including the briefly measured scattering and reflection loss of the imaging lens of the Fresnel lens, about 17% is lost in the lens system. Reflection loss of 1% was measured on the solar cell surface.

As a result, based on these measured and estimated losses, the light incident on the solar cells is estimated to be 4.07 W. Unfortunately, direct incident light measurements were not possible due to the large area of the 8 cm square irradiation beam. For this incident light, the solar cells produced a power output of 755 mW, with an estimated efficiency of 18.6% for the solar cells themselves.

The efficiency of the applied GaAs solar cells in sunlight is around 24% in the specification. In our previous study, efficiency of about 40% for near-infrared irradiation at 808 nm was confirmed [18]. Near-infrared monochromatic light irradiation operates at nearly 1.5-times or more of the efficiency of sunlight. However, the monochromatic light with a wavelength of 450 nm used in this experiment showed much lower efficiency than that of sunlight. This is because photon energy of 2.755 eV is much higher than the solar cell band gap energy of 1.42 eV, resulting in a large heat loss. The light irradiation intensity of the experiment is slightly weaker than that of sunlight and it will also decrease the efficiency. In addition, in the case of light with a short wavelength of 450 nm, the generation of large surface recombination due to large absorption on the solar cell surface may also have an influence. In the future, the realization of solar cells with a wider bandgap, such as those suitable for blue 450 nm, is expected to more than double the current efficiency.

5. Oblique Incidence Characteristics of Fly-Eye Lens System

5.1. Principle of Angle Dependence in Fly-Eye Lens System

To obtain the appropriate operation of the function of the fly-eye lens system, not only the irradiation shape but also the irradiation position should be maintained for different incident beam conditions. Due to the function of the second fly-eye lens, the fly-eye lens does not affect the light irradiation position, even if the angle of incidence changes. However, this condition is limited to cases where the angle of incidence is below an allowable incident angle. The allowable incident angle ${\theta }_c$ is given as Equation (7) [40].

$$\tan {\theta }_c=\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$2f$}\right.$$

(7)

It is determined by the elemental lens height $h$ and the elemental lens focal length $f$. Figure 7a,b show schematic drawings for the cases within and above the allowable incident angle, respectively. When oblique incidence occurs at an angle greater than the allowable incident angle, the light of the same shape is irradiated at a position adjacent to the irradiated area formed within the allowable incident angle.

As shown in Figure 7, the allowable incident angle is the case where the beam after the first fly-eye lens enters an elemental lens of the second fly-eye lens in the same position of the elemental lens of the first fly-eye lens. If the beam enters an adjacent elemental lens of the second fly-eye lens, the position of the final irradiated area changes. The situation of this adjacent irradiation is confirmed by a numerical formula. Figure 8 shows the relationship between lenses and beams.

Consider the case that the incident angle is sufficiently large. At the focal plane of the first fly-eye lens, that is, the distance of the second fly-eye lens, the height $b$ of the beam incident on the center of the elemental lens of the first fly-eye lens with the incident angle ${\theta }_1$, relation of Equation (8), is held using the focal length $f$ due to the ineffective nature of the lens shape to the beam passing the center.

$$\tan {\theta }_1=\frac{b}{f}$$

(8)

The output angle ${\theta }_2$ of the beam incident on the height $a$ of the first fly-eye lens with the incident angle ${\theta }_1$ reaches the same height $b$ due to focusing and, thus, the relation is given as Equation (9) using Equation (8).

$$\tan {\theta }_2=\frac{b-a}{f}=\frac{f\tan {\theta }_1-a}{f}$$

(9)

The important parameter is the output angle ${\theta }_3$ of the second lens. The height $c$ at the focal plane of the second lens is not important; however, the height $c$ of the incident angle ${\theta }_2$ is expressed using the lens height $h$ as Equation (10) from the focal position of Figure 8.

$$\tan {\theta }_2=\frac{c-h}{f}$$

(10)

Therefore, using Equation (9),

$$c=f\tan {\theta }_1-a+h$$

(11)

Thus, the important output angle ${\theta }_3$ of the second lens of the incident angle ${\theta }_2$ is

$$\tan {\theta }_3=\frac{c-b}{f}=\frac{f\tan {\theta }_1-a+h-b}{f}=-\frac{a-h}{f}$$

(12)

After transmission through an imaging lens of focal length $F$, the height $x$ at the focal plane is expressed as Equation (13). Only the output angle ${\theta }_3$ is important and that is expressed as incident height $a$ and focal length $f$.

$$x=F\tan {\theta }_3=-\frac{a-h}{f}·F$$

(13)

Since the range of $a$ is $-\frac{h}{2}\le a\le \frac{h}{2}$, the range of the irradiated surface $x$ is

$$\frac{h}{2f}·F\le x\le \frac{3h}{2f}·F$$

(14)

On the other hand, the range of $x$ in the case of incidence within the allowable incident angle considered in the same way is

$$-\frac{h}{2f}·F\le x\le \frac{h}{2f}·F$$

(15)

Comparing Equation (14) with Equation (15), as the incident angle increases and the light beam enters the adjacent elemental lens of the second fly-eye lens, the size of the irradiated surface remains the same and the area adjacent to the irradiated surface for the allowable incident angle is irradiated.

5.2. Increase in the Allowable Incidence Angle for Fly-Eye Lens by Mirrors

The expansion of the functionality in the fly-eye lens system was examined to increase the incident angle.

The fly-eye lens system can generate a rectangular beam with uniform intensity; however, if the incident angle is larger than the allowable incident angle ${\theta }_c$, the beam will not be able to irradiate the solar cell. This allowable incident angle is, for example, 14.04° for the elemental lens used in the experiment in Figure 6, with a focal length of 20 mm and a size of 10 mm square. This light beam incidence condition is not a problem for applications where the light beam is incident almost perpendicular to the fly-eye lens system. However, under various application conditions of OWPT, including underwater applications, the positional relationship between the light source and the light-receiving equipment is not always fixed perpendicularly with high precision. Changes in the positional relationship may cause a shift in the beam incident angle. Therefore, a method is needed to widen the allowable incident angle.

Therefore, a configuration that folds back the irradiated light using a mirror is proposed, as shown in Figure 9, taking advantage of the fact that the fly-eye lens system irradiates light in the same shape at adjacent positions of the original irradiation position when the incident angle increases. Expanding on the previous equations of principle, up to twice the allowable incident angle, the original irradiated surface is irradiated by once folding reflection of the mirror. If the angle of incidence is further increased, irradiation is made to the next adjacent position. In such a case, two reflections by the two mirrors allow for proper turnaround. Even at high angles of incidence, the multiple reflections are expected to work properly. This configuration requires that the fly-eye lens, imaging lens, and solar cells be the same size.

5.3. Fabrication of the Light-Receiving Module Applying Mirrors

A fly-eye lens module with mirrors on four surfaces was fabricated as a light-receiving module that can cover high-angle oblique incidence and can be used from either the top, bottom, left, or right direction. Figure 10 shows the photograph of the light-receiving module fabricated by a 3D printer. On the left side of the photograph, two fly-eye lenses and one imaging lens are installed and the inside of the white area behind them is mirrors for folding back. A solar cell is installed at the rear.

The fly-eye lens is an elemental lens with a focal length of 20 mm, size of 1 cm square, and an 8 × 8 array. The imaging lens is an 8 cm square Fresnel lens with a focal length of 160 mm. Each lens is coated with AR coating for the 450 nm band on both sides. The mirrors used in this experiment are commercially available aluminum mirrors for the initial experiment and their reflectivity is around 80%. The solar cells were installed as 8 cm square using flexible GaAs solar cell modules. The light source was a 450 nm laser and no water tank was used for measurement.

5.4. Angular Dependence of the Light-Receiving Module

Figure 11 shows the measured angular dependence of the fabricated module. X-axis of the graph shows the angle when the incidence from the front of the fly-eye lens module is 0 degrees. As a basic mechanism of this module, it will show the same characteristics for horizontal or vertical angle change; however, in the experiment, the angle was changed in the horizontal direction. Experimental evaluation was performed with a light source output of 1.0 W. The light beam was injected into the fly-eye lens module as a beam with an FWHM of 15 mm on the short axis and 50 mm on the long axis by a beam expander consisting of a concave lens and a convex lens while changing the angle. Output power of over 100 mW was obtained for the perpendicular beam incident to the module. The output was 10% of efficiency for 1 W of irradiation light power, which is lower than the efficiency in the case of the water tank experiment, as shown in Figure 6. This is due to the fact that the light intensity density on the solar cells affects the power conversion efficiency and weak intensity causes lower power conversion efficiency due to the nature of the solar cell.

As a control experiment, when only the fly-eye lens system was installed on the optical bench without a mirror, the solar cell output decreased rapidly near the allowable incident angle of 14°, as shown in the graph. On the other hand, the module with four mirrors was confirmed to produce relatively high output, even at high incident angles of 14° or more. However, although the output was assumed to be close to that of perpendicular incident, even at high angles, the actual output was reduced to about half. Four reasons are assumed to be responsible for this.

First, the reflectivity of the applied mirror is assumed to be around 80% and the non-reflected light is due to an absorption loss. Therefore, the beam intensity reaching the solar cell surface is assumed to be decreased. The application of multilayer mirrors with high reflectivity will improve the output by 10–20%.

Second, in this experiment, mirrors were not installed between the two fly-eye lenses and in the area up to the imaging lens. Installing mirrors in these areas may be effective in improving efficiency, depending on the incident conditions.

Third, the beam from the first fly-eye lens is usually formed on the second fly-eye lens; however, at high incident angles, as shown in Figure 12a, the difference between the optical path length L to the second fly-eye lens and the focal length f of the fly-eye lens becomes larger. The image formation to the second fly-eye lens is no longer properly performed and the uniformity of the irradiation pattern on the solar cell surface may be degraded. This influence is thought to occur even when the angle of incidence is less than the allowable incident angle and is thought to be the cause of the gradual decrease in solar cell output in a range of 0° to 14° in Figure 11 in both experiments. This effect can be suppressed by improving the image formation characteristics of the lens according to the oblique incidence characteristics.

Fourth, as shown in Figure 12b, when the incident angle is near the allowable incident angle, the focusing position of the first fly-eye lens becomes the boundary of the array of the second fly-eye lens. The lens array boundary does not function as a lens. Furthermore, the fly-eye lens applied in this study is fabricated by cutting with a needle and the 0.5 mm width of the lens boundary does not function as a lens. The light incident on this boundary becomes scattered light. This is one of the reasons for the steep decrease in output near 14° in both modules, with and without mirrors. In the case without a mirror, the output decrease was clear; however, the decrease is not abrupt at the allowable incident angle of 14° in detail. The decrease in output occurs over an angular range of about 2°. This is because, for a boundary width of 0.5 mm, which is 1/10 of the half lens size of 5 mm, the angular range of 1/10 for allowable incident angle in one direction is considered not to show lens function. It means that scattering occurs in a range of about 1/10 of the allowable incident angle of 14° and the solar cell output is affected by the scattered light in a range of about 2°.

Not all of the light scattering, including the scattered light at the lens boundary, affects the output reduction. Some of the scattered light is directed to the solar cell. In Figure 11, the output power of the module with mirrors is slightly higher than that without mirrors in the vicinity of perpendicular incident. This may be due to the fact that a part of scattered light at the lens boundary of the first fly-eye lens and scattering at the Fresnel lens of the imaging lens reach the solar cells by the function of the mirrors. However, while uniform irradiation can effectively extract the output power, its effectiveness is greatly reduced if the irradiation is not sufficiently uniform.

5.5. Comparison of Light-Receiving Module

Table 1. Comparison of light-receiving module.

	Irradiation Uniformity by Narrow Beam	Misalignment Tolerance for Appropriate Beam Size and Normal Incident	Angular Tolerance for Appropriate Beam Size	Module Thickness
Only solar sell	Low	Small	Narrow	Very Thin
Fly-eye-lens system on solar cell [18]	High	Wide	Narrow	Very thick
Fly-eye lens system with a mirror on solar cell (this work)	High	Wide	Wide	Very thick

shows a comparative table of light-receiving modules. The verification results indicate that the fly-eye lens module with a mirror confirms the tolerance to misalignment and that it has a wider range of acceptable incident angles. These results indicate that the fly-eye lens module with a mirror has a high tolerance to beam irradiation conditions.

6. Conclusions

Beam irradiation characteristics of fly-eye lens modules for OWPT were experimentally tested. A solar cell uniform irradiation system using a fly-eye lens system that is tolerant of fluctuations in beam shape, number of beams, incident position, and incident direction was constructed for use in various usage situations of OWPT systems and its characteristics were evaluated. A blue-light-based configuration was constructed for application to underwater OWPT. An evaluation of the effect of the irradiation beam size and beam incident position on the fly-eye lens was performed as the basic characterization of the fly-eye lens system. A structure to expand the incident angle range using mirrors was proposed and its effectiveness was clarified through experimental investigation. A laser beam of 450 nm wavelength with an output power of 6 W was irradiated through a 90 cm long water tank to a GaAs solar cell with a fly-eye lens system to obtain a maximum output power of 0.755 W. The obtained output power underwater is one of the benchmark results of current underwater OWPT. As the conclusion, the proposed and verified fly-eye lens module with mirrors will be useful for various OWPT applications.