Inclined, Expanded and Curved Interference Pattern of Double Slit Experiment---Mathematical Description (1)

: Young’s double slit experiments express the mystery of quantum mechanics. To explore the mystery, varieties of the double slit experiments were performed. It has been shown that the following three phenomena emerged simultaneously in one double slit experiment: (1) the interference patterns incline towards the axis that perpendicular to the axis the diaphragm rotating around; (2) the interference patterns curved; (3) the distances between the fringes of the interference patterns expanded. To determine the direction the interference pattern curved towards, we proposed the right-hand rule for the clockwise rotating diaphragm and the left-hand rule for the counterclockwise rotating diaphragms. In this article we derived the mathematical formulars for the first phenomenon that interference patterns incline. We show that the inclination of the interference pattern is determined by two angles, one is the original orientation of the double slit, other one is the rotating angle of the diaphragm, either clockwise or counterclockwise. Declaration: this work potential interesting conflict


Introduction
Young's double slit experiment was first performed in 1801 [1,2], which, 100 years later, led to wave-particle duality. Feynman called the double slit experiment "a phenomenon which is impossible […] to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery [of quantum mechanics]." [3]. Moreover, the nature of photons truly puzzled Einstein. He wrote to M. Besso: "All these 50 years of conscious brooding have brought me no nearer to the answer to the question: What are light quanta?" [4].
In the standard double slit experiments, the characteristics of the interference patterns are described by three factors, wave length, the spacing between two slits and distance between the double slit and screen, as described by equation, !"#$% = & ' . Note that (1) there is only one factor related to the parameter of the diaphragm of the double slit, i.e., the spacing d; (2) the fringes of the interference pattern distribute along a straight line; (3) the equation was derived for a special situation that the light beam is perpendicular to the plane of the diaphragm of the double slit.
We have raised a question whether the orientations of the diaphragm affect the characteristics of the interference pattern? To answer this question, several experiments have been performed [5] [6] [7] [8] [9]. We have shown experimentally that, by rotating the diaphragm of the double slit around one axis, three phenomena have been observed in the same interference pattern simultaneously: (1) the interference patterns incline towards the axis that perpendicular to the axis the diaphragm rotating around; (2) the interference patterns curved; (3) the distances between the fringes of the interference patterns expanded. To determine the direction of the interference patterns curved towards, we propose Left-hand Rule and Right-hand Rule [9].
In this article we propose mathematical formula to describe the inclination of the interference patterns of the double slit experiments.

Inclination of Interference Patterns
First let us review several examples of the inclinations. Figure 1 shows the diaphragm of the double slit.  Figure 2b and 2c show the patterns' inclinations with difference rotation angles and support the Left-hand rule [9] Second, rotating the diaphragm 60 0 and 75 0 respectively around Y-axis clockwise.  Figure 3a shows the interference pattern before the diaphragm rotating. Figure 3b and 3c show the patterns' inclinations with difference rotation angles and support the Right-hand rule [9]. After we derive the mathematical formulas for the inclination in Section 3, we will show that the experiments in Section 2 support formulas in Section 4.

Derivation of Mathematical Formula for Inclination of Interference Patterns
The phenomena take place when the diaphragms rotate, we expect that the formula relates with the rotation angle. Now we study the effects for two configurations. The double slit-AB is in the Y-Z plane of the diaphragm. The source is on X-axis. The angle ∠a is the original angle between the double alit-AB and Z-axis. Z'-axis is the projection of Z-axis on the detector.

Configuration-1:
The "P" represents the interference pattern formed on the detector. The "D" represents the projection of the double slit-AB on the detector. The interference pattern "P" is perpendicular to the projection "D".
The angle ∠b is the angle between "P" and Z'-axis. The angle ∠a' is the angle between "D" and

′ > ′ ′
The line ′ represents the projection of the "D" on Z'-axis. The angle ∠ ′ ′ ′ is the angle between the projection of the "D" and Z'-axis. The ∠ ′ becomes ∠ ′ ′ , denoted as ∠ ". We have The interference pattern is perpendicular to the projection of the double slit. Now the angle between the interference pattern "P" and Z'-axis is The larger the diaphragm's rotation angle, the shorter the line ) ) and the larger ∠ ′ ′ ′. Thus, the interference pattern inclines to Z'-axis when the diaphragm rotates.
Next to derive the mathematical formula.
Since = ′ ′, to find the ∠ ′ ′ ′, we need to find ′ ′. For this aim, let us draw the projections of the line o'm' on the X-Z-plane, denoted as the line o'q". the angle of the "D" rotated counterclockwise is ∠ " ′ ′ or ∠ (Figure 7).
The angle of the pattern relative to the Z-axis is The larger the rotation angle ∠ , the larger the angle ∠ ) ′ ) , namely the interference pattern inclining to Z'-axis. The angle between the "P" and Z'-axis is depending on the original angle ∠ of the double slit-AB and the rotating angle ∠ of the diaphragm.
Next let us study the situation of rotating the diaphragm clockwise, the projection of the double slit, i.e., the line o'q" is the line o'q' as shown in Figure 8. So, we obtain the same Eq. 6.

Configuration-2:
Configurataion-2 is shown in Figure 9 below: The double slit-AB is in Y-Z plane of the diaphragm. The source is on X-axis. The angle ∠a is the original angle between the double alit-AB and Z-axis. Z'-axis is the projection of Z-axis on the screen.
The "P" represents the interference pattern formed on the screen. The "D" represents the projection of the double slit-AB on the screen. The "P" is perpendicular to the "D". The angle ∠b is the angle between "P" and Z'-axis. The angle ∠a' is the angle between "D" and Z'-axis. At the original orientation of the diaphragm, ∠ ′ is equal to the ∠a. We have
To find the mathematical description, we need to find the relation between ∠ ′ ′ ′, ∠ and the diaphragm's rotation angle. Since = ′ ′, to find the ∠ ′ ′ ′, we need to find ′ ′. For this aim, we draw Figure 12 that is in the X-Z plane. The ′ ′′ is the projection of the ′ ′ on X-Z'-plane. The ′ ′ is the projection of ′ ′′ on Z'-axis. The angle of the "D" rotated clockwise is ∠ " ′ ′ or ∠ .
When the diaphragm rotates around Y-axis an angle ∠ clockwise, the line ′ becomes ′ ′′, i.e., Substituting Eq. (7) into Eq. (9), we have Since *+ " ! + = ∠ , Eq. (10) gives The angle of the interference pattern relative to the Z-axis is Next let us rotate the diaphragm counterclockwise, the "D", i.e., o'q", is shown in Figure 13. the projection of the o'q" is the o'q' that is the same as that in Figure 12. Therefore, we will obtain the same mathematical expression, Eq. 12.

Figure 13 Schematics of the diaphragm rotating counterclockwise
Conclusion: The directions of the diaphragms rotate, either the clockwise or counterclockwise, have no effect on the inclination of the interference pattern.

Testing the Mathematical Expressions
We test the mathematical formulars with existing experiments/observations [9], in which the original angle between the double slit and Y-axis is ∠ = 45 ( . Then rotating the diaphragms 60 ( and 75 ( respectively. There are two configurations.   We have done the experiments and observed the following [9].

Configuration
Rotating the diaphragm around Y-axis counterclockwise: (b) 60 0 ; (c) 75 0 (Figure 14). The angles between the interference patterns and Z-axis are shown by green lines.
. We have done the experiments and observed the following [9].

Summary
By rotating the diaphragm of the double slit around one axis, we observe three phenomena simultaneously, namely, the interference patterns curved, expanded and inclined simultaneously [9] (see attached video). To determine the direction of the interference patterns curved towards, we proposed Left-hand Rule and Right-hand Rule. The experiments obey the rules. However, the underlying physics of the Rules is unclear.
In this article we derived the mathematical formulars for calculating the inclination angles of the interference patterns, attribute to the rotations of the diaphragms. We show that the inclination angles depend only on both the original angle of the double slit and the rotation angles of the diaphragm. The directions of the diaphragms rotating, either clockwise or counterclockwise, have no effect on the inclination.
The experimental observations support the formulars.

Appendix:
A-1: Video: the link to the video is on the last page.
Attached Video shows the evolution of the characteristics of the interference patterns of the double slit experiments.

A-2: Left-Hand Rule and Right-Hand Rule to Determine Directions Patterns Curved towards
We have shown that, attribute to the rotations of the diaphragm, either clockwise or counterclockwise, the patterns/interference patterns curved [5] [6] [7] [8].
To determine the direction the patterns curved towards, we propose Left-hand Rule and Right-hand Rule [9].
Left-hand Rule: For the curved pattern created by diaphragm rotating counterclockwise.
To determine the direction of the patterns curved towards, point the left thumb to the source, the index finger is aligned with the direction of the original pattern, and the middle finger will point in the direction of the patterns curved towards, which is attribute to the counterclockwise rotation of the diaphragm. To determine the direction of the patterns curved towards, point the right thumb in the direction of the source, the index finger is aligned with the direction of the original pattern, and the middle finger will point in the direction of the patterns curved towards, which is attribute to the clockwise rotation of the diaphragm.