Curved, Expanded and Inclined Diffraction Patterns of Grating--Rotating Grating Around Three Axes (1)

The orientation-dependence of the interference/diffraction patterns of the 1D-double slit/1D-grating and 2D-cross-double slit/2D-cross-grating experiments have been studied experimentally and theoretically. However, the above experiments were limited to certain orientations, namely rotating around either one axis or two axes. In this article, the 3-axis-rotation apparatus is proposed/made, which can rotate the 1D-double slit/2D-cross-double slit and 1D-grating/2D-cross-grating, CW and CCW respectively, 0 0 -360 0 around three axes independently and sequentially. By this apparatus, the orientation-dependence of the patterns is systematically studied. Moreover, the experiments can be performed easily. The complete phenomena of curved, expanded and inclined patterns are the orientation-dependent. Then we show that the photons, before landing on the detector/screen, behave as particles. The above observed phenomena provide the comprehensive information to theoretical study of the double slit/grating experiments. We suggest that the complete mathematical model should contain three rotation angles as parameters. Furthermore, the phenomena have potential applications.

To study the orientation-dependence completely, in this article, we introduce: (1) the three-axisrotation apparatus; and (2) a new coordinate system to describe the new apparatus. By which, the systematic study of the orientation-dependence of the patterns is achieved intuitively. The diaphragms of the double slit and cross-double slit, gratings and cross-gratings are rotated around three axes sequentially and respectively. New phenomena are observed.
Note that in this article, we only show the orientation-dependence of the diffraction patterns of the 1D-transmission grating. However, the same three-axis-rotation apparatus can be utilized in the 1Ddouble slit, 2D-cross-double slit and 2D-cross-grating experiments (to be continued).
The novel phenomena of the continuously curved, expanded and inclined patterns provide the comprehensive data/information to theoretically study the double slit/grating experiments. The phenomena would provide the comprehensive information/data for theorists. We suggest that a complete mathematical model should contains three rotation angles as parameters and should be able to describe all phenomena of the orientation-dependence of the patterns consistently.
The concept of the orientation-dependence of the patterns is importance for applications.

Three-Axis-Rotation Apparatus
To perform the experiments of studying the orientation-dependence of the patterns, it is practical convenience to keep the laser beam pointing to the same direction and rotate the grating. For this aim, we introduce the three-axis-rotation apparatus that can rotate the double slit/cross-double slit and grating/cross-grating around three axes to reach desired orientations ( Figure 1). Figure 1. Schematic of the 3-axis-rotation apparatus: the frame rotates around Z axis, the ring rotates around Y axis, the round grating rotates around X axis.
The shortcoming is that the thick ring may block light beam for certain orientation. To avoid it, one can make the ring as thin as possible to minimize the blocking. the grating/ring rotate around Y axis (b); the grating/ring/frame rotate around Z axis (b).
The Y and Z axes are perpendicular to each other always; The Y and X axes are perpendicular to each other always. Figure 2b shows that the grating, the ring with grating, and the frame with ring and grating rotate around the X axis, the Y axis and the Z axis respectively.
With this apparatus, the orientation-dependence of the patterns can be studied thoroughly and conveniently, and novel phenomena are shown.

Coordinate System and Direction of Rotation
To study the orientation-dependence of the patterns, we introduce the coordinate system and the original orientation of the grating ( Figure 1).

Coordinate System:
The rotation axis of the frame defines the Z axis; the rotation angles are either "Clockwise " (denoted as "CW ") or "Counterclockwise " (denoted as "CCW "). The frame can be rotated around Z axis between 0 0 ≤ ≤360 0 CW and CCW. The rotation axis of the ring defines the Y axis, the rotation angles are either "CW " or "CCW ". The ring can be rotated around Y axis between 0 0 ≤ ≤360 0 CW and CCW. The normal vector of the grating defines the X axis. The rotation angles are either "CW " or "CCW ". The grating can be rotated around X axis between 0 0 ≤ ≤360 0 CW and CCW.
The X axis and the Y axis are always perpendicular to each other. The Y axis and the Z axis are always perpendicular to each other. The X axis changes its direction when the ring rotates around the Y axis and when the frame rotates around the Z axis. The Y axis changes its direction when the frame rotates around the Z axis. The Z axis keeps the same direction. The laser beam keeps the same direction.
Direction of Rotation: to define the direction of the CW and CCW rotation, we introduce the righthand rule that the thumb of the right hand is pointed in the direction of the axis, the CCW rotation is given by the curl of the fingers (Figure 3).

Rotating Grating Around Three Axis
To study systematically the orientations-dependence of the interference patterns and diffraction patterns (hereafter denoted both as the "patterns") of the 1D-double slit/2D-cross-double slit/1Dgrating/2D-cross-grating (hereafter denoted as the "grating"), the effects of the orientations on the patterns need to be considered when the following rotations are performed: (A) First, let us start from the original orientation, then, rotating the grating around one axis only: When rotate the grating, the ring and the frame, we always rotate CCW and CW respectively.
In this article, as the first one of a series-articles, we only report the observations of the grating experiments for the orientations of rotating the grating around one axis and around two axes.

Rotating Grating Around One Axis
Let us start from the original orientation ( Figure 4). The slits of the grating are along the Y direction. Then, rotating the 1D-grating around one axis. Figure 5a shows the pattern due to the CW rotation of the grating, while Figure 5b shows the pattern due to the CCW rotation. The patterns of the grating have the rotation symmetry when the grating rotates around the X axis and the laser beam is on the normal vector/the X axis. Figure 5c shows the rotation symmetry of the patterns of the double slit/cross-double slit [19]. When the grating rotates around the X axis, the light is a normally incident beam. When the laser source is not on the normal vector of the grating, the rotation symmetry is no longer valid. Figure 6a shows the pattern when the grating is at the original orientation. We observed the expanded patterns: the larger the rotation angle, , the larger the expansion.

Experiment-A2: Rotating Grating/Ring Around Y axis
When the slits of the grating are along the Y axis, The directions of rotating around Y axis, either CW or CCW, have no effects on the expansion of the patterns.

Experiment-A3: Rotating Grating/ring/frame Around Z axis
We observed the curved patterns. The directions of the curves are determined by the direction of the rotations: the CW rotation makes the pattern curves towards to the left, which satisfy the right-hand rule (Figure 7b), while the CCW rotation makes the pattern curves towards to the right, which satisfy the left-hand rule (Figure 7c) [9]. The larger the rotation angle the larger the curvature of the curved pattern. In Experiment-A2, the incident light beam (Brown colored arrow in Figure 8) is in the X-Z plane with an incident angle to the X axis, which create the expanded pattern. In Experiment-A3, the incident light beam (Blue colored arrow in Figure 8) is in the X-Y plane with an incident angle to the X axis, which created the curved pattern.
If the light beam in the X-Z plane is incident at an arbitrary angle ! to the grating normal, the standard grating equation gives .
Where " is the diffraction angle, is the spacing between two adjacent slits. The difference between two diffraction angles of two adjacent diffraction order is constant, Namely the standard grating equation predicts the no-expansion patterns for the light beam with arbitrary incident angle.
The expanded patterns violate the grating equation.
To describe the expanded patterns of Experiment-A2, we extend the standard grating equation to: Now the difference between two diffraction angles of two adjacent diffraction order is incident-angledependent, i.e., the larger the incident angle, the larger the spacing between diffraction orders, The term ( ) ' shows that the expansions are independent with the directions, either CW or CCW , of the grating/ring's rotation.
Note that Equations (1) and (3) describe the patterns created by the oblique incident light beams that are in the X-Z plan ( Figure 8) and thus, no patten is curved. When the grating rotates around the Z axis, either CW or CCW, the different curved patterns are created as shown in Experiment-A3.
Experiment-A1, -A2 and -A3 indicate that the patterns due to the rotations around the different axes are completely different (Table 1). Those differences will guide us to understand the phenomena when the grating rotating around 2 axes and 3 axes.

Rotating Grating Around Two Axes Sequentially
In this section, we perform the following experiments. Note that since the direction of the curved patterns is determined by the direction of the rotation, so in the following experiments, we rotate the grating CW and CCW respectively. The plan of the experiments is the following.

Experiment-B1:
Rotate the grating around the X axis.

Grating Rotating Around X Axis CW, Then Rotating Around Y Axis and Z axis
Respectively Experiment-B1: Rotate the grating around the X axis  Figure 9b shows that the patterns expanded, curved towards the left (shown by the orange straight line) and the whole pattern is inclined to the vertical direction. Figure 9c shows that the patterns expanded, curved towards the right (shown by the orange straight line) and the whole pattern is inclined to the vertical direction.
Experiment-B1-1c: rotate the grating/ring/frame around the Z axis 30 0 CW (Figure 10b) Experiment-B1-1d: rotate the grating/ring/frame around the Z axis 30 0 CCW (Figure 10c) Figure 10. Grating rotates 30 0 CW around X axis (a), then rotates grating/ring/frame around Z axis: Patterns due to CW rotation (b) and CCW rotation (c) Figure 10b shows that the patterns expanded, curved upwards (right-hand rule) (shown by the orange straight line) and the whole pattern is inclined to the horizontal direction. Figure 10c shows that the patterns expanded, curved downwards (left-hand rule) (shown by the orange straight line) and the whole pattern is inclined to the horizontal direction. When rotating the tilt-grating around the Y axis an angle ! , the mathematic equation describing the expansion of the pattern of the tilt-grating is the extension of Equation (3), When rotating the tilt-grating around the Z axis an angle ! , the mathematic equation describing the expansion of the pattern of the grating is the extension of Equation (3), It is the challenge to describe the curved pattern. Patterns due to 60 0 CW rotation (b) and 75 0 CCW rotation (c) Figure 12b shows the patterns are curved upwards, expanded and inclined towards the horizontal axis. Figure 12c shows the patterns are curved downwards, expanded and inclined towards the horizontal axis.
Conclusion: the patterns are different when the tilt-grating rotates around the Y axis and around the Z axis.

Rotating Grating Around X Axis CCW, Then Rotating Around Y Axis and Z axis
respectively Experiment-B1-3: Rotate the grating around the X axis 50 0 CCW (Figure 13a), then perform the following 4 experiments respectively.
Experiment-B1-3a: rotate the grating/ring around Y axis 45 0 CW (Figure 13b) Experiment-B1-3b: rotate the grating/ring around Y axis 75 0 CCW (Figure 13c) Figure 13. Grating rotates 50 0 CCW around X axis (a), then rotates grating/ring around Y axis: Patterns due to 45 0 CW rotation (b) and 75 0 CCW rotation (c) Figure 13b shows the patterns are curved towards the left, expanded and inclined towards the vertical axis. Figure 13c shows the patterns are curved towards the left, expanded and inclined towards the vertical axis.
Experiment-B1-4a: rotate the grating/ring around Y axis CW 75 0 (Figure 15b) Experiment-B1-4b: rotate the grating/ring around Y axis CCW 60 0 (Figure 15c) Figure 15. Grating rotates 80 0 CCW around X axis (a), then rotates grating/ring/frame around Y axis: Patterns due to 75 0 CW rotation (b) and 60 0 CCW rotation (c) Figure 15b shows the patterns are curved upwards, expanded and inclined towards the vertical axis.  Figure 16c shows the patterns are curved upwards, expanded and inclined towards the horizontal axis.

Rotating Grating/ring Around Y Axis CW, Then Rotating Around X Axis and Z axis
respectively Experiment-B2: Rotate the grating/ring around the Y axis Experiment-B2-1: Rotate the grating/ring around the Y axis 60 0 CW first, the pattern is expanded (Figure 17a). Then perform the following 4 experiments respectively.
Experiment-B2-1a: rotate the grating around X 45 0 axis CW (Figure 17b We observe that the patterns are curved after the grating rotates around the X axis, which indicates a novel phenomenon that since the grating rotates around Y axis first, the grating has no longer the symmetry when it rotates around the X axis either CW or CCW. It is the challenge to explain why the patterns (Figure 17b and 17c) are curved. We observe that the patterns are expanded (due to the rotating around Y axis) and curved (due to the rotating around Z axis).

Experiment
It is the challenge to describe mathematically the curvatures of the patterns (Figure 18b and 18c). Rotating grating/ring/frame around Z axis CW (b) and CCW (c)

Rotating
It is the challenge to describe the curvature of the patterns.

Rotating Grating/ring/frame Around Z Axis CW, Then Rotating Around X Axis and Y axis respectively
Experiment-B3: Rotate the grating/ring/frame around the Z axis first Experiment-B3-1: Rotate the grating/ring/frame around the Z axis CW (Figure 21a), then perform the following 4 experiments respectively.
Experiment-B3-1a: rotate the grating around X axis CW (Figure 21b Grating around X axis CW (b) and rotating grating around X axis CCW (c) We observed that, due to the rotation around the X axis, the curvatures become smaller, while the patterns expanded.
(a) (b) (c) Figure 24. Rotate the grating/ring/frame around the Z axis CCW (Figure 20a), then rotate grating/ring around Y axis CW (b) and CCW (c) Figure 24 shows that the rotation of the grating/ring around the Y axis expands the patterns and reduces the curvature of the curved pattern.

Patterns of 2D-Cross-Grating Are Created by Photons that Behave as Particle
We have shown that the patterns of the double slit/cross-double slit/1D-grating experiments are formed by photons behaving as particle, but not as waves [16] [15].
Let us show that the curved, expanded and inclined patterns observed in the cross-grating experiments are created by the photons that behave as particle, but not as waves.

Experiment-4:
Placing a blocker between the cross-grating and the screen at different locations ( Figure   25a). Only propagation of particles can explain the above patterns. Namely the light propagates and creates the patterns as particles, but not as wave.

Discussion and Conclusion
We introduce the 3-axis-rotation apparatus, and associated coordinate system, that can rotate the double slit/cross-double slit/ grating/cross-grating to desired orientations. The apparatus makes it is possible to thoroughly study the orientation-dependence of the patterns of the grating without difficulty.
The experiments show that the patterns are curved, expanded and inclined differently, when the grating is at different orientations.