Optical path calculation in Pentaprism

[John]

Anyone can help me in calculating the total light
travelling distance in a 25mm X 25mm Pentaprism?

I found out that the total light travelling distance in the
prism multiply by a factor of 1.33 is not right thru
experiment. I already considered the two reflection
surfaces in the pentaprism.

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[Jon Spangler]

paraphrased from MIL-HDBK-141 (out of print), pg 13-38:

for an entrance face height of A, the optical path length of the axial
ray is:
t = 3.4142 * A
t = 3.4142 * 25 mm = 85.36 mm

The air equivalent or apparent thickness, as always, is:
t / n = 3.4142 * A / 1.517 (for BK 7 at the d line)

hope this helps...

regards,
Jon

These are my opinions, not those of my employer.
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[Patrick O'Donnell]

Here goes:

Think of the pentaprism as a cube plus a triangle. The triangle's base is the length of the cube diagonal and each face is at an angle of 67.5 degrees from
the cube flats. The triangle reflects the image by 135 deg at each triangles face, thus emitting the rays with a 270 degree turn. So, what is the OPL?

OPL is the index of refraction times

the path through the front face of the cube (25 mm)
+
the path from the back edge of the cube to the triangle wall (12.5 cot(67.5) = 5.178mm)
+
the path between mirrors, the tricky part. The path taken by a ray entering the center of the face at perfectly normal incidence closes a triangle which
starts at the center of the cube, hits the first mirror, the next mirror and then returns to the center of the cube. Thus the distance between mirrors is
just sqrt(2) * the distance from the center of the cube to the first mirror.

or sqrt(2) * (12.5mm / 2 + 5.178) = 1.414 * 11.428 = 16.162 mm

+
the path from the second triangle wall to the cube border(12.5 cot(67.5) = 5.178mm)
+
the path through the exit face of the cube (25 mm)

In summary:

OPL = index of refraction * (12.5 + 5.178 + 16.162 + 5.178 + 12.5)

Whew!

Regards,

Pat

[Patrick O'Donnell]

Patrick O'Donnell wrote:

> Here goes:
>
> Think of the pentaprism as a cube plus a triangle. The triangle's base is the length of the cube diagonal and each face is at an angle of 67.5 degrees from
> the cube flats. The triangle reflects the image by 135 deg at each triangles face, thus emitting the rays with a 270 degree turn. So, what is the OPL?
>
> OPL is the index of refraction times
>
> the path through the front face of the cube (25 mm)
> +
> the path from the back edge of the cube to the triangle wall (12.5 cot(67.5) = 5.178mm)
> +
> the path between mirrors, the tricky part. The path taken by a ray entering the center of the face at perfectly normal incidence closes a triangle which
> starts at the center of the cube, hits the first mirror, the next mirror and then returns to the center of the cube. Thus the distance between mirrors is
> just sqrt(2) * the distance from the center of the cube to the first mirror.
>
> or sqrt(2) * (12.5mm / 2 + 5.178) = 1.414 * 11.428 = 16.162 mm
>
> +
> the path from the second triangle wall to the cube border(12.5 cot(67.5) = 5.178mm)
> +
> the path through the exit face of the cube (25 mm)
>
> In summary:
>

> OPL = index of refraction * (25 + 5.178 + 16.162 + 5.178 + 25)