1. Introduction

Can we shut all of the light of any direction up using transparent material? It is well known that even in transparent materials we can repel light rays by using total internal reflection. In practice, total internal reflection is used for confining light in a micro-dielectric sphere [1] and a silicon ring resonator [2]. They are already used as high Q optical resonators. However, when a light ray propagates in normal direction to the surfaces of these resonators, it can pass through into or from them. After all, these resonators cannot confine all of the light rays having arbitrary directions. Furthermore, these resonators have no empty spaces in themselves. Two-dimensional photonic crystal cavities with high Q [3,4] have also same problems. On the other hand, three-dimensional photonic crystal cavity has possibilities of three-dimensional confinement but applicable wavelength region is limited more or less due to their periodic structure. Furthermore, it requires complicated structures and ultra-high grade technique with high accuracy.

By using simple structure, can we completely confine light rays with arbitrary directions in wide wavelength region?

If all light rays that enter into the one side of the wall made of transparent materials are totally reflected always at the opposite side of the wall, even after undergoing multi-reflection, we can completely shut the light by this wall. By covering a certain space with this wall, we can produce the completely closed optical cavity from where any light cannot escape. In 2010, we reported [5] that it is theoretically possible by using the wall with specially arranged inner and outer surfaces and with the refractive index of larger than $\sqrt{4+2\sqrt{2}}=\mathrm{2.6132......}$

Recently, we have succeeded in the experimental confirmation of our theories. In this paper we will demonstrate the experimental results using a prototype cavity composed of GaP.

2. The summary of our previous work

In order to confine light rays with arbitrary directions completely in a cave, the following two necessary requirements must be satisfied.

1) Surface of one side of the wall consists of the planes belonging to one of the following two plane groups; (A) and (B) and the surface of the opposite side consists of the planes of the other plane group.

(A) Principal-plane group (Fig. 1(a) ): it includes the 6 planes. Their normal vectors are parallel to the one of the principal axes x, y, and z. Accordingly, their normal vectors are

 

Fig. 1 The specially arranged planes required to compose “completely closed-optical-cavities”. (a) Principal planes and (b) 45-planes.

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(B) 45-plane group (Fig. 1(b)): it includes 12 planes. Their normal vectors are perpendicular to the one principal axis and are at the 45° angle to the other two principal axes. Then, their normal vectors are given by

2) The relative refractive index of the wall-material $n_r$ has to satisfy the following conditions.

If the two requirements mentioned above are satisfied, all of light rays which enter the wall through all of the principal-planes reflect totally at all of the 45-planes and can never pass through 45-planes. It is seen from the reciprocity of light that no light ray that enters the wall through 45-planes can pass through the principal-plane.

Consequently, we can compose a completely closed optical cavity by using the wall with the principal planes for inner surface and with the 45-planes for outer surface. If we exchange the inner surfaces for the outer surfaces, we can also compose a completely closed optical cavity.

Typical example of the optical cavity is shown in Figs. 2(a) and 2(b). In (a), the outer surface shapes a pair of pyramids with 45° slopes, where one pyramid is placed upside up and the other pyramid is placed upside down. It has an empty cubic cave inside. In (a), the inner surface is principal plane surface and the outer surface is 45-plane surface. In (b), the inner surface is exchanged for the outer surface. The light box (c) is suitable for a big cavity.

 

Fig. 2 Typical examples of completely closed optical cavities. (a) A pyramid-like cavity with Cubic cave inside, (b) the inner and the outer surface in (a) are exchanged mutually, and (c) a light box type cavity suitable for big-size cavity.

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Materials suitable for composing cavities

Many semiconductors like Si(n = 3.4), Ge(4.0), GaAs(3.3), and InP(3.2), which are useful for electronics and opto-electronics, have high-indices that satisfy$n>\sqrt{4+2\sqrt{2}}$. They are transparent in the infrared region but not in the visible region. A few materials like GaP(n = 3.5), ZnTe(2.92), and SiC(2.63) are partially transparent in visible regions. For the infrared region, Si has the high potentialities in photonics because it has excellent optical features and high process technologies for optoelectronic integrated circuits. Although the applicable wavelength is limited to the infrared region Si is a powerful candidate for practical use of our cavity in the future.

3. Experimental studies and results

In order to confirm experimentally our previous theoretical work we constructed a prototype optical cavity using a GaP crystal. A GaP is transparent in the red-yellow wavelength region. In the visible region we can directly observe with our eyes (wide area light sensors with better sensitivity and dynamic range) which become powerful tool for confirmation of function.

Beforehand, the refractive index of GaP was measured to be$3.5\pm 0.1$from the experiment of refraction using a GaP plate (thickness 2cm) and HeNe 633nm laser. This plate and the prototype optical cavity were made using the same GaP rod. The net propagation loss (Fresnel reflection is subtracted) of GaP was measured to be 25% per cm from the transmission experiment using the same plate and same laser.

Figure 3(a) shows the structures of the cavity and Fig. 3(b) is the photograph of the GaP cavity we constructed. Here, the outer surface is composed of 8 kinds of 45-planes and the inner surface is composed of six kinds of principal planes. Cavity block is composed of 3 parts as shown in Fig. 3(a).

 

Fig. 3 Optical cavity constructed using a transparent GaP crystal. (a) Outline figure, A and B are caps and C is a trunk part, (b) photograph of the cavity where the cap A is removed off.

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Two parts of them are caps A and B. Each cap shapes the quadratic prism whose one end is sharpened to the wedge type. The other part is a square pillar C (trunk part) having the square tunnel inside. The trunk part is constructed by pasting a pair of triangle pillars and a pair of home-base-like pentagon pillars.

Since we can observe nothing in the cave if we cover the trunk with both caps, we take the cap A off as shown in the photograph Fig. 3(b) in our experiment. In this photograph, we can see slightly the reflected light from the contact surfaces inside the crystal block.

Firstly, a HeNe 633 nm laser beam was irradiated to the 45-plane surface at various angles from the outside of the block. Although the trace of laser beam was observed inside of the GaP crystal (this shows that the scattered lights exist in the crystal), no light beam was observed in the cave.

From the fact mentioned above, it is presumed that laser beam at any incident angle reflects totally from the inner surface as been expected by our theory. On the other hand, the direction of the scattered lights along the beam in the crystal are random, and as a result, a part of the scattered light may not be totally reflected and may leak in the cave, although there are very few total amounts of the leaking light. Actually, Q of 10⁶ or more is estimated as described later even if such scattering exists.

Observation from the outside of the cavity in the case when the LED is placed in the cave

A laser beam is not convenient since the direction of light is limited by the wall. Thus, we used red LEDs (λ = 610 & 640nm 10mcd) for light sources. The light rays from LED spread widely as shown in Fig. 4(a) . We inserted this LED into the cave deeply as shown in Fig. 4(b).

 

Fig. 4 Side view of optical cavity block having a brilliant LED in its inside. (a) LED (610nm, 640nm), (b) outline of the optical cavity block (the left side cap is removed), and (c) side view photograph of optical cavity block with the LED inside.

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Figure 4(c) is the photograph which views the side surface of the block. Except for the open end no bright area is found in the directions of the side surface and the closed end (right). The measured light power in the case that LED is inserted in a cave was very weak (<<10⁻³) as compared with the light power in the case that the cavity is removed. This fact shows that any light ray from the LED in the cave reflects totally at the outer 45-plane-surface and never go out from the block through the 45-plane surface. If we cover the cap on the open end, the LED light cannot pass through this cap as same as the closed end of opposite side. As a result no light from the cave can escape out from the block. This means the completely closed optical cavity can be established by the one layer transparent material.

Observation of leaking light in the cave from the outside

Figure 5 shows the cavity block irradiated by LED lights from the outside. The light of 640nm LEDs are scattered on the white surface of the surrounding paper. The direct light rays and scattered light rays are irradiated to the outer surface of the cavity block from every direction except for open end side. The deep darkness in the cave is in contrast to the brightness of the outer surface. The optical powers of the cave and the outer surface were measured using a photo-detector with a fiber-light guide (3.2mm$\varphi$, NA = 0.55) without a cap at open end side. Measured power ratio is nearly 10³ (710~3000, which depends upon the places of the outer surface). This ratio corresponds to the ratio of the power density. In order to know the ratio of the total power an integrating sphere may be required even in the cave. Anyway this fact shows that the light rays from the outside of the block are totally reflected at the inner surface of the block and that only a part of weak scattered light passes through into the cave. This ratio is limited by scattering effect in GaP crystal. To get higher extinction ratio, it requires higher optical quality to the crystal.

 

Fig. 5 Photograph of the case of external irradiation of light rays. Although light rays are irradiated to the block from the outside, no light ray reaches the cave inside of the block (a perfect darkroom composed by a transparent material). Here, weak white light is irradiated from the front so that the open side with the cave may be in sight.

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As mentioned above, Fig. 4, and Fig. 5 prove that our theoretical results are right.

4. Conclusions and discussions

We constructed the cavity whose inner surface consists of principal planes and whose outer surface consists of 45-planes using a GaP crystal. Using 610-640nm LED we confirmed experimentally that light rays do not pass regardless of their directions from the outside to the inside and also from the inside to outside of the cavity. It is concluded that this cavity has function of completely closed optical cavity as predicted by our theory.

The refractive index of GaP (3.5) is fairly larger than the limit value of 2.613. Additionally, light rays decay to nearly zero after several reflections because of high absorption loss. These facts loosened, I think, the allowable margin of angular error of the cavity.

The quality of the cavity is usually represented by a Q-factor. Q is defined as $Q=2\pi c\tau /\lambda$ where $\tau$ is a photon life time (the decay time of the light energy in a cavity) and c and λ are the velocity and the wavelength of the light in vacuum, respectively. The photon lifetime is determined by the imperfectness of the cavity mirrors, the propagation loss in the cavity, and the size of the cavity. In present case, since the optical medium GaP crystal has large absorption loss of 25%/cm, this absorption loss is dominant factor which determines photon life time. If the light propagates a distance of $1/\alpha$ in the optical medium, light energy decays to value of 1/e of the original value and the propagation time is $n/(\alpha c)$ where $\alpha$is an absorption coefficient of the medium. Accordingly, for the case that most of area of the cavity is occupied by this absorbent medium, Q-factor is almost independent on the cavity-modes and the ray-paths and even on a cavity size. For this case Q is approximately given by

The lengthened rate corresponds to the ratio of lengths of light's stay time in a hollow part and in the medium. If we let $\delta$denote the lengthened rate, Q becomes

Comparison of various cavities

A Fabry-Perot cavity has the highest Q (~10^8~12) [6] because most of areas of cavity are empty. MDS with high quality materials [1] have 10⁷ of Q. PCC [3,4] and our new cavity have also similar Q (~10^6~7). As shown in Eq. (5), Q value depends strongly on materials. PCC and our new cavity may soon surpass MDS. While a big size cavity is possible using our method, a small size (wavelength size) cavity is impossible because light passes through the cave area by tunneling effect. On the other hand, PCC is suitable for a small size cavity but it requires high technology. The feature of our cavity is that light is confined in all the directions. If we put a light-emitting element in a cave part, all spontaneous emission lights will also be confined. It will be possible to apply this cavity to light-emitting devices in which spontaneous emission is controlled [7,8].