James,
We have been looking at the lines on the double slit experiment. We have been discussing the source at twice the distance from the slits as the screen.
In your last post you challenged me:
Let’s look at some examples using my coordinate system. As usual the points are as follows (not in proportion):

As before, we have a coordinate x,y,z system that is centered on the source (S) at (0,0,0).
Distance between (x1, y1, z1) and (x2, y2, z2) is SQRT ((x1-x2)(x1-x2) + (y1-y2)(y1-y2) + (z1-z2)(z1-z2)).
Visible light (used for the double slit experiment) has the following range of wavelengths:
Violet is 380 nm = 380 x 10 e(-9) meters = 3.8 x 10 e(-7) m.
Red is 760 nm = 760 x 10 e(-9) meters = 7.6 x 10 e(-7)m.
In the examples, all numbers will be in millimeters, unless otherwise stated. The number “d” is the distance between the slits, and is always 1.
As usual, we need the screen and the source at least 1000d from the slits. The gradient from M to E,F must be the same gradient from E,F to the source. This means the spacing in the z direction is the same as the spacing in the x direction.
[size=150]Example 5: Source at 2000d, screen at 1000d[/size]
The points are as follows:
A (2000,-0.5,0)________B (2000,0.5,0)___________L (3000,4.5,0)
E (2000,-0.5,-4)_______F (2000,0.5,-4)__________M (3000,4.5,-6)
The paths differences summary is:
Path Diff L_A_S to L-B-S________________0.004499954
Path Diff M-E-S to M-F-S________________0.004499945
Difference between path differences____8.99945E-09
The difference between the path differences is even smaller than before: 9 x 10 e(-9) mm = 9 x 10 e(-12) m. This is about 100,000 times smaller then the wavelengths of visible light. Hence points L and M would have the same phase relationship – if L is a bright spot then so is M.
Let’s be devil’s advocate here – perhaps the “up and down” changing of the path length differences is happening in between L and M, and I’m just lucky that M is back to the same path difference as L.
To check this out, let’s look at the points in between L and M. The only difference between them is in the z direction: L is at 0 and M is at -6. Let’s try M at -1, -2, -3, -4, and -5 to see if we can see some sort of up and down change.
[size=150]Example 6: M at -1[/size]
A (2000,-0.5,0)___________________B (2000,0.5,0)____________________L (3000,4.5,0)
E (2000,-0.5, -0.666666667)________F (2000,0.5,-0.666666667)__________M (3000,4.5,-1)
Path_Diff_L_A_S_to_L-B-S_______________0.004499954
Path_Diff_M-E-S_to_M-F-S_______________0.004499954
Difference_between_path_differences____2.49656E-10
[size=150]Example 7: M at -2[/size]
A (2000,-0.5,0)___________________B (2000,0.5,0)____L (3000,4.5,0)
E (2000,-0.5, -1.333333333)F (2000,0.5, -1.333333333)M (3000,4.5,-2)
Path_Diff_L_A_S_to_L-B-S 0.004499954
Path_Diff_M-E-S_to_M-F-S 0.004499953
Difference_between_path_differences 9.99535E-10
[size=150]Example 8: M at -3[/size]
A (2000,-0.5,0)____B (2000,0.5,0)L (3000,4.5,0)
E (2000,-0.5, -2)F (2000,0.5, -2)M (3000,4.5,-3)
Path_Diff_L_A_S_to_L-B-S 0.004499954
Path_Diff_M-E-S_to_M-F-S 0.004499952
Difference_between_path_differences 2.24964E-09
[size=150]Example 9:– M at -4[/size]
A (2000,-0.5,0)___________________B (2000,0.5,0)____L (3000,4.5,0)
E (2000,-0.5, -2.666666667)F (2000,0.5, -2.666666667)M (3000,4.5,-4)
Path_Diff_L_A_S_to_L-B-S 0.004499954
Path_Diff_M-E-S_to_M-F-S 0.00449995
Difference_between_path_differences 3.9995E-09
[size=150]Example 10: M at -5[/size]
A (2000,-0.5,0)___________________B (2000,0.5,0)____L (3000,4.5,0)
E (2000,-0.5, -3.333333333)F (2000,0.5, -3.333333333)M (3000,4.5,-5)
Path_Diff_L_A_S_to_L-B-S 0.004499954
Path_Diff_M-E-S_to_M-F-S 0.004499948
Difference_between_path_differences 6.24914E-09
Here is the trend:
Z coord of M _______Difference between path differences (mm)
-1_______________________2.49656E-10
-2_______________________9.99535E-10
-3_______________________2.24964E-09
-4_______________________3.9995E-09
-5_______________________6.24914E-09
-6_______________________8.99945E-09
What does this show? The phase difference gets bigger between z coordinates of -1 to -6, at a slowing rate. All of the phase differences are at least 100,000 times smaller than a wavelength of visible light, so they do not matter. If L is a bright spot then so is M.
In conclusion, moving the source at 2000d from the slits and the screen at 1000d has dramatically shown there is no problem whatsoever - the interference lines on the screen make sense.
You will have to show us your calculations, so we can see the problem you are talking about.
Eugene Morrow