We validated the functionality of the proposed optical element as a mid-air imaging optical element through ray tracing simulations. The simulation focused on two main points: first, whether the mid-air image forms at the correct position and second, whether the proposed method effectively reduces stray light and expands the viewing range of the mid-air image compared to existing mid-air imaging optical elements. In line with the approach used by Kiuchi and Koizumi [8], ray tracing was employed to reproduce the appearance of both the mid-air image and stray light.

5.1 Modeling

Mitsuba 3, a ray tracing software, was used for the simulation of the mid-air imaging optical elements. Three models were created in Mitsuba 3: the proposed TCAP, the MMAP, and the previously proposed cylindrical optical element by Takenaka [5]. The modeling of MMAP was conducted following the method outlined by Kiuchi and Koizumi. In the simulations, a distance of 1 unit in Mitsuba 3 was assumed to correspond to 1 mm in physical space.

The parameters for modeling the optical element were set as follows:

The diameter of the cylinders and the spacing of the MMAP mirror array were set as 0.5 mm; the optical elements covered an area of 200 mm by 200 mm. As a result, 400 vertical by 400 horizontal cylindrical and truncated cylindrical elements were arranged.

The transmission surfaces of the cylindrical and truncated cylindrical optical elements employed a smooth dielectric material with a refractive index (\(n_2\)) of 1.7, determined using Eq. (1) for an incident angle (\(\alpha \)) of \(45^\circ \).

The angle of the cross-sections at both ends of the truncated cylinders was set to \(35^\circ \), in accordance with Eq. (2).

The length of the reflective parts for both the truncated cylinders and the comparative full cylinders was set to 0.23 mm, as specified by Eq. (4).

The length of the transmissive parts of the full cylinders was also set to 0.23 mm. The spacing between each cylinder of the comparative cylindrical optical elements was fixed at 0.345 mm, which is 1.5 times the length of the mirror.

The reflectivity of the mirrors used in the cylindrical optical elements and MMAP was set to 0.87, based on previous studies [8].

5.2 Evaluation of mid-air imaging position

To confirm that the mid-air image is formed at a position plane-symmetric to the light source with respect to the optical element, we measured the imaging position of the mid-air image. Following the approach used by Kiuchi and Koizumi [8], stereo matching was performed to measure the distance to the mid-air image using images captured by two cameras positioned at the back left and back right of the scene. A light source comprising flat disks arranged in a \(4\times 4\) grid was used. The positions of the disks were automatically detected using OpenCV’s image processing, and images from the left and right cameras were compared to determine the imaging position.

Figure 5 illustrates the setup of the display, mid-air imaging optical element, and cameras used for evaluating the mid-air imaging position. The distance between the center of the light source and the mid-air imaging optical element (\(L_A\)) was varied in increments of 1 mm, ranging from 50 to 75 mm. The distance between the optical element and the cameras (\(L_B\)) was fixed at 800 mm, and the distance between the two cameras was set to 50 mm. By measuring the distance between the cameras and the mid-air image using stereo matching and subtracting this value from \(L_B\), we were able to calculate the floating distance \(L_C\) of the mid-air image.

Fig. 5

Measurement setup for ray tracing simulation. TCAP indicates the truncated cylindrical array plate optical element used in the proposed method

From Fig. 6, the imaging position of the mid-air image aligns with the theoretical value. This result verifies that the mid-air image generated by the proposed optical element is formed at a position that is plane-symmetric to the light source with respect to the optical element.

Fig. 6

Experimental results of imaging position by stereo matching

5.3 Evaluation of viewing range and stray light

To compare the viewing range and stray light between the existing methods and the proposed method, various types of mid-air imaging optical elements were simulated and evaluated using ray tracing. The comparison included the MMAP modeled in Ref. [8] and the cylindrical optical element proposed by Takenaka [5].

In this experiment, the viewing angles in both azimuth and elevation directions were compared. Figure 7 illustrates the arrangement of the display, mid-air imaging optical element, and cameras for the simulation, with \(L_A = 50 \ \textrm\) and \(L_B = 200 \ \textrm\). The truncated cylinder in Fig. 7 is oriented such that the transmissive surface faces the camera when \(\theta _2 = 0^\circ \). To measure the azimuthal field of view, the angle \(\theta _1\) was fixed at \(45^\circ \), while \(\theta _2\) was varied from \(0^\circ \) to \(90^\circ \) in \(5^\circ \) increments. An 80 mm square image was used as the light source. For the elevation field of view, \(\theta _2\) was fixed at \(0^\circ \), and \(\theta _1\) was varied from \(30^\circ \) to \(60^\circ \) in \(5^\circ \) increments, using a 60 mm square image as the light source. We rendered images using a perspective pinhole camera in Mitsuba 3, setting the image sensor width to 36 mm, which matches the full-frame image sensor of a typical DSLR camera when using a standard 50 mm focal length lens. These settings were chosen to simulate conditions similar to human vision.

Fig. 7

Setup for field of view evaluation experiment. (a) Side view of experimental design (when \(\theta _2=0^\circ \)). (b) Top view of experimental design. TCAP is positioned such that the transmission surface of the truncated cylinder faces the camera when \(\theta _2=0^\circ \)

The experimental results for the azimuthal viewing range of mid-air images formed by MMAP, existing cylindrical elements, and truncated cylinders are shown in Fig. 8. The results indicated that in MMAP, stray light overlaps with the mid-air image, interfering with the observation of the image. In addition, the cylindrical elements produced dimmer mid-air images. In contrast, although the mid-air image formed by the truncated cylinders was slightly blurrier compared to the image formed by MMAP, it did not generate stray light that interfered with observation. Moreover, whereas MMAP produced a dark mid-air image with intense stray light around \(\theta _2=40^\circ \), the truncated cylinders generated a bright image from \(\theta _2=0^\circ \) to \(\theta _2=40^\circ \).

Fig. 8

Simulated mid-air images formed by (a) MMAP, (b) cylinder array plate, and (c) our proposed design (TCAP) at different angles of \(\theta _2\), when \(L_A=50 \ \textrm\). Red frame indicates mid-air image and green and blue frames indicate stray and transmitted light, respectively

The experimental results for the elevation viewing range of mid-air images formed by MMAP, existing cylindrical elements, and truncated cylinders are shown in Fig. 9. These results demonstrate that the mid-air image formed by the truncated cylinders becomes blurred or distorted as the angle \(\theta _1\) deviates from \(45^\circ \). This suggests that the truncated cylinders produce a mid-air image with a narrower visible range in the elevation angle compared to MMAP.

Fig. 9

Simulated mid-air images formed by (a) MMAP, (b) cylinder array plate and (c) our proposed design (TCAP) at different angles of \(\theta _1\), when \(L_A =50 \ \textrm\). Notably, images are magnified for better visualization

5.3.1 Luminance evaluation

We quantitatively evaluated the luminance at various elevation angles. Following the method proposed by Kiuchi and Koizumi [8], we calculated the luminance ratio between the display and the mid-air image. We used a 60-mm square light source and examined the ratio of the average grayscale value of the \(3\times 3\) pixels at the center of the rendered mid-air image relative to the light source. Following the approach of Kiuchi and Koizumi [8], the light source was positioned parallel to the TCAP, with a 100-mm distance between the light source and TCAP, and a 1000-mm distance between the mid-air image and camera.

Figure 10 shows the luminance ratio between the light source and the mid-air image at each elevation angle. The horizontal axis represents the elevation angle \(\theta _1\), and the vertical axis represents the luminance ratio. The results show that the luminance ratio follows this descending order of quality: MMAP, TCAP, and cylindrical array.

Fig. 10

Luminance ratio evaluation results. The luminance ratio results of the real MMAP were obtained from Tsuchiya and Koizumi [12]