*2.4. Index Measurements*

Depth profiles of the refractive index was determined by two methods, i.e., the prism-coupling and the end-face coupling. In former, all the WG mode angles were consistently fitted. Instead of the mode angles θm, the effective refractive index *n*<sup>m</sup> is usually used in this community,

$$n\_{\mathbf{m}} = n\_1 \sin \theta\_{\mathbf{m}} \tag{3}$$

where *n*<sup>1</sup> denotes the index in the guiding layer.

After measuring the angles or the effective refractive indices where WG modes are excited, a depth profile of the index is determined in order to reproduce all the mode angles using the reflectivity calculation method (RCM). See Reference [3] for the RCM. The mode angles where WG modes are excited were measured by the prism-coupling method. Figure 3a schematically depicts a setup of the

where *n*1 denotes the index in the guiding layer.

prism-coupling apparatus. A laser line of 632.8 nm from a He-Ne laser was used as a light source. A polarizer was inserted to select a polarization plan of the WG mode from the transverse electric (TE) or the transverse magnetic (TM). In this paper, most of the data were detected in the TM configuration. A prism was attached on a part of the WG sample in order to couple with the light in air (a vacuum) and the light in the WG via evanescent wave in a gap between the prism and the WG. The prism, the sample, and a charge-coupled device (CCD) detector facing at the end-surface of the sample, were attached on a rotating table, in order to change an incident angle of the laser light to the WG layer without changing the configurations of the prism, the sample, and the CCD detector. The light intensity reflected at the boundary between the prism and the WG was monitored with inserting a beam splitter between the polarizer and the prism. The reflected light was transferred via the beam splitter and detected by "a dark-mode detector". When a mode is excited with adjusting the modal angle θm, the light intensity detected at the CCD detector facing the end-surface increases because a part of light is guided to the end-surface. Simultaneously, when the mode is matched, the incident light is efficiently introduced into the WG. Consequently, the reflection intensity at the prism-WG boundary decreases, i.e., the signal at the dark-mode detector decreases. After measuring the angles or the effective refractive indices where WG modes are excited, a depth profile of the index is determined in order to reproduce all the mode angles using the reflectivity calculation method (RCM). See Reference [3] for the RCM. The mode angles where WG modes are excited were measured by the prism-coupling method. Figure 3a schematically depicts a setup of the prism-coupling apparatus. A laser line of 632.8 nm from a He-Ne laser was used as a light source. A polarizer was inserted to select a polarization plan of the WG mode from the transverse electric (TE) or the transverse magnetic (TM). In this paper, most of the data were detected in the TM configuration. A prism was attached on a part of the WG sample in order to couple with the light in air (a vacuum) and the light in the WG via evanescent wave in a gap between the prism and the WG. The prism, the sample, and a charge-coupled device (CCD) detector facing at the endsurface of the sample, were attached on a rotating table, in order to change an incident angle of the laser light to the WG layer without changing the configurations of the prism, the sample, and the CCD detector. The light intensity reflected at the boundary between the prism and the WG was monitored with inserting a beam splitter between the polarizer and the prism. The reflected light was transferred via the beam splitter and detected by "a dark-mode detector". When a mode is excited with adjusting the modal angle *θ*m, the light intensity detected at the CCD detector facing the endsurface increases because a part of light is guided to the end-surface. Simultaneously, when the mode is matched, the incident light is efficiently introduced into the WG. Consequently, the reflection intensity at the prism-WG boundary decreases, i.e., the signal at the dark-mode detector decreases.

*Quantum Beam Sci.* **2020**, *4*, x FOR PEER REVIEW 5 of 13

Depth profiles of the refractive index was determined by two methods, i.e., the prism-coupling and the end-face coupling. In former, all the WG mode angles were consistently fitted. Instead of the

*n*m = *n*1 sin*θm* (3)

**Figure 3.** (**a**) a setup for detection of waveguide modes using prism coupling and the dark-mode detection. See text for details. (**b**) a typical data of the dark-mode detection from YAG sample irradiated with 200 MeV Xe ions in the TE mode. Each dip corresponds to a waveguides (WG) modes. (**c**) a setup of the end-face coupling. The polarization of the laser is controlled by a half-wave plate. **Figure 3.** (**a**) a setup for detection of waveguide modes using prism coupling and the dark-mode detection. See text for details. (**b**) a typical data of the dark-mode detection from YAG sample irradiated with 200 MeV Xe ions in the TE mode. Each dip corresponds to a waveguides (WG) modes. (**c**) a setup of the end-face coupling. The polarization of the laser is controlled by a half-wave plate.

provides sharper dips and more sensitive. We applied this method.

code (RSoft, Co. Ltd., Tokyo, Japan).

**3. Experimental Results** 

*3.1. X-Ray Diffraction* 

The WG modes were searched with scanning the WG angle or equivalently the effective index. Figure 3b shows an example of the effective index (i.e., the WG angle) dependence of the light

As schematically shown in Figure 3c, the end-face coupling method was also applied. With controlling the polarization of laser by a half-wave plate, the laser light was introduced by the objective lens 1. The guided light pattern was detected through the objective lens 2. The spatial intensity patterns at the end-surface were another important information, which were experimentally detected by a CCD camera and calculated by the beam propagation method (BPM) using BeamPROP

Figure 4 shows FIA-XRD patterns from undoped YAG crystals irradiated with 200 MeV Xe14+ ions to various fluences ranging from 0 to 5 × 1013 ions/cm2. Before the irradiation, relatively strong four peaks were observed, all of which were assigned to diffractions from the garnet structure: 55.8° for (4 2 2), 124.6° for (9 2 1), 135.7° for (9 3 2), and 150.3° for (7 7 2) [14]. Because the garnet structure includes many atoms in a unit cell, many peaks are reported in the powder diffraction patterns. However, our samples showed only limited peaks due to high crystallinity. After the irradiation to 1

The WG modes were searched with scanning the WG angle or equivalently the effective index. Figure 3b shows an example of the effective index (i.e., the WG angle) dependence of the light intensity detected by the dark-mode detector. In this case, dips in the signal correspond to the WG modes. In principle, the same modes should be detected also by the "bright-mode detection" using the CCD detector facing at the end-surface. However, as a rule of thumb, the dark-mode detection provides sharper dips and more sensitive. We applied this method.

As schematically shown in Figure 3c, the end-face coupling method was also applied. With controlling the polarization of laser by a half-wave plate, the laser light was introduced by the objective lens 1. The guided light pattern was detected through the objective lens 2. The spatial intensity patterns at the end-surface were another important information, which were experimentally detected by a CCD camera and calculated by the beam propagation method (BPM) using BeamPROP code (RSoft, Co. Ltd., Tokyo, Japan).
