Non-Uniform Diffraction Pattern of Grating Experiments---Rotating Grating Around Three Axes (3)

The orientation-dependence of the interference/diffraction patterns of the 1D-double slit, 2D-cross-double slit, 1D-grating and 2D-cross-grating experiments have been studied experimentally and theoretically. The universal phenomena of the curved, expanded, inclined and mirror-symmetric pattens were reported. However, the above experiments were limited to certain orientations, namely rotating the diaphragm/grating around either one axis or two axes. Here we show that, for certain orientations by rotating the grating around 3 axes, the spacings between adjacent diffraction orders are non-uniform. Namely, from “one end” of the pattern to “other end”, the spacings between adjacent diffraction orders gradually increase. This extraordinary phenomenon challenges the existing theories. The above observed phenomena provide the comprehensive phenomena to theoretical study of the grating experiments. We suggest that the complete mathematical model should contain three rotation angles as parameters and should be able to describe the phenomena of the non-uniform-spacing, curved, expanded, inclined and mirror-symmetric diffraction patterns consistently.

The orientation-dependence, for certain orientations, of the interference patterns of the 1D-double slit and 2D-cross-double slit experiments have been studied experimentally and theoretically [1] [2] [3] [4]. The universal phenomena of the curved, expanded and inclined patterns were reported.
Recently, we have introduced the 3-axis-rotation apparatus, by which, the orientation-dependence of the diffraction patterns of the 1D-grating and 2D-cross-grating experiments have been studied [5] [6]. The simultaneously curved, expanded, inclined and mirror-symmetric diffraction patterns (hereafter denoted as "pattern") emerged in the grating experiments. The new phenomena show that the characteristics of the patterns depend on the orientations of the grating. However, the study only coves the orientations by rotating the 1D-grating and 2D-cross-grating around two axes.
In this article, we study the orientation-dependence of the patterns completely by utilizing the 3axis-rotation apparatus. The 1D-grating is rotated around three axes sequentially and respectively. The systematic study of the patterns is achieved intuitively.
We report, the very first time, the new phenomena that for certain orientations by rotating the grating around 3 axes respectively, the spacings between adjacent diffraction orders are non-uniform.
The phenomena challenge both the standard theory of the 1D-grating for the normal incident light beam and the theories of the 1D-grating for the oblique incident light beam [7] [8] [9] [10].
The novel phenomena of the continuously curved, expanded, inclined, mirror-symmetric and nonuniform spacing patterns provide the comprehensive phenomena/information to theoretically study the double slit/grating experiments. We suggest that a complete mathematical model should contains three rotation angles as parameters and should be able to describe all phenomena of the orientationdependence of the patterns consistently.

Experiments: Rotating 1D-grating Around 3 Axes
In the previous articles [5] [6], the grating has been rotated around 2 axes. In this article, we rotate the 1D-grating around 3 axes. By rotating the grating clockwise (CW) and counterclockwise (CCW) around X axis, CW and CCW around Y axis, and CW and CCW around Z axis, besides the phenomena of the curved, expanded, inclined and mirror-symmetric patterns, we observed the non-uniform-spacing patterns as shown by Experiment-1 to 8 in Figure 1 to Figure 8. Although in Experiment-1 to 8, we use the specific discrete angles for rotating around each axis, one can rotate the grating either with any other discrete angles or continuously rotating the grating around any axis.
We perform Experiment-1 to 4 starting with rotating the grating CW 45 0 around the X axis ( Figure A(b)), and Experiment-5 to 8 starting with rotating the grating CCW 45 0 around the X axis ( Figure A(c)). Figure A(a) shows the pattern before rotating the grating.    .
(a) (b) Figure 1b shows the pattern of the grating after rotating around 3 axes, the X axis, Y axis and the Z axis. The third rotation around the Z axis cause expanding more.
Experiment-1 shows, the first time, the most significant novel phenomena that the pattern is expanded non-uniformly. More specifically, the expansion of the spacing between the adjacent diffraction orders increase gradually from the bottom-portion (below the zero order) to the top portion (above the zero order) of the pattern.  Observation: Experiment-4 shows that the pattern is expanded non-uniformly. Namely, the expansions of the spacing between the adjacent diffraction orders increase gradually from the top-portion to the bottom-portion of the pattern.

Axes Sequentially
Starting with rotating the grating CCW 45 0 around the X axis ( Figure A(c)) first, then performing the following four Experiment-5 to -8, respectively: rotating the grating CW and CCW around the Y axis, and CW and CCW around Z axis sequentially.

Discussion
The preliminary experiments of this article show, the first time: (1) the non-uniform spacings between adjacent diffraction orders. More specifically, the spacings are gradually increase either from the top-portion to the bottom-portion of the pattern, or from the bottomportion to the top-portion of the pattern.
(2) the phenomena of the non-uniform spacings exist regularly.

Recent Developments of Research on Double Slit and Grating Experiments
Feynman stated that Young's double slit experiment was the mystery of the quantum mechanics. It is interesting to show how the study of the mystery of the double slit experiments has been progressed recently.

A1. 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 cross-grating. For this aim, we use the three-axis-rotation apparatus [5] [6] that can rotate the double slit/cross-double slit and grating/cross-grating around 3 axes to reach desired orientations ( Figure A1). grating can rotate around Z axis; around Y axis; around X axis.
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 the original orientation of the grating. Figure 2c shows that the grating rotates around the X axis. Figure 2d shows that the cross-grating, the ring/cross-grating, and the frame/ring/cross-grating rotate around the X axis, the Y axis and the Z axis respectively.
The shortcoming of the above apparatus 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 blockage. An alternative apparatus shown in Figure A2 is convenience to perform the experiments Figure A2. The alternative 3-axis rotation apparatus With those apparatuses, the orientation-dependence of the patterns of the cross-grating can be studied thoroughly and conveniently, and novel phenomena are shown.

A2. Coordinate System
To study the orientation-dependence of the patterns, we introduce the coordinate system and the original orientation of the 3-axis-rotation apparatus and grating ( Figure A1a).
Coordinate System: The rotation axis of the frame defines the Z axis; the rotation angles around the Z axis are either "Clockwise " (denoted as "CW ") or "Counterclockwise " (denoted as "CCW ").
The frame can be rotated CW and CCW around Z axis between 0 0 ≤ ≤360 0 . The rotation axis of the ring defines the Y axis, the rotation angles are either "CW " or "CCW ". The ring can be rotated CW and CCW around Y axis between 0 0 ≤ ≤360 0 . The normal vector of the cross-grating defines the X axis. The rotation angles are either "CW " or "CCW ". The cross-grating can be rotated CW and CCW around X axis between 0 0 ≤ ≤360 0 .
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 always. The laser beam points to the same direction always.

A3. Original orientation of the grating
At the original orientation, the X, Y and Z axes form the Cartesian coordinate system (Figure 3a), denoted it as the original coordinate system. The horizontal slits, denoted as S1, of the cross-grating are along the Y axis and creates the vertical pattern P1, while the vertical slits, denoted as S2, of the crossgrating are along the Z axis and create the horizontal pattern P2 (Figure 3b and 3c). The laser source is on the X axis, i.e., the light beam is normally incident on the plane of the grating, referred it as the original orientation. A4. Direction of Rotation: to define the direction of the CW and CCW rotation, we introduce the right-hand rule that states 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 4). Figure A4. Right-hand rule for determining direction of rotation

A5. 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, the effects of the orientations on the patterns need to be considered when the following rotations are performed: (A) Starting from the original orientation, then, rotating the grating around one axis only: (A1) Rotate the grating around the X axis: CW and CCW , respectively (A2) Rotate the grating/ring around the Y axis: CW and CCW , respectively (A3) Rotate the grating/ring/frame around the Z axis: CW and CCW , respectively (B) Starting from the original orientation, then, rotating the grating around 2 axes, respectively:  When rotate the cross-grating, the ring and the frame, we always rotate CCW and CW respectively. For rotating around one axis, rotating CW and CCW creates the mirror-symmetry patterns.