This is a really basic tutorial. However, it does assume understanding of the basic ideas in my Single-lens Ray-Trace Tutorial
Mirrors are created by setting the
After the mirror, light travels to the left so
THICKNESS is a negative number
The signs for the radius of curvature are the same independent of the direction of travel
If the light is reflected back through the same optical elements, these surfaces must be repeated
# Execute this cell first %matplotlib inline import numpy as np import matplotlib.pyplot as plt def efl_thick_mirror(n, R1, R2, d): """The effective focal length of a mirror with the second surface R2 reflecting""" return -n * R1**2 * R2 /2 / (d - d *n + n *R1) / (d - d* n + n *(R1 - R2) + R2)
Let's model a mirror with a radius of curvature of 300mm. The paraxial approximation says that the focal length $f$ should be $R/2$
R=-300 #mm, negative because center of curvature is to the left of the surface f = R/2 print("The paraxial focal length of the mirror is %.1f mm" % f)
The paraxial focal length of the mirror is -150.0 mm
To simulate this in OSLO, we will place the object infinitely far away from the mirror. We will make the object 10mm high and the entering beam radius also 10mm. The mirror is 3 inches in diameter of 75mm.
We can construct the system in OSLO with just three surfaces: OBJ, AST, IMS. The object, the aperture stop (which of course is the mirror) and the image surface. Everything is in air so it is super simple.
First, notice that in the glass column, we have selected
REFL_HATCH to cause the rays to be reflected at this surface. (Either
REFL_HATCH will reflect the light, it is just that the latter option indicates the opaque side of the mirror with hatching.)
Second, we get the expected effective focal length of -150mm in the upper right hand corner.
Now when we turn on the drawing, we get the disappointing ray trace shown below
To improve this drawing, force OSLO to autofocus by clicking the
THICKNESS -> Autofocus - paraxial focus button in the
IMS column. Now we see
Since I would like the rays to start farther to the left (at say 160mm from the mirror), one way is to create a new surface that has a specific aperture but does nothing else.
Insert a new row. In the
APERTURE RADIUS column of the new row select
APERTURE STOP (A). Next, in the
SPECIAL column of this new row, select
SURFACE CONTROL -> GENERAL to open a dialog box. Then in the
Surface appearance in lens drawings select
Drawn like shown below.
Then the Surface Data Table should look like
and the Autodraw window will look like
This OSLO lens file can be downloaded as concave-mirror.len
Let's assume that we have the same mirror, but now it is covered with 2mm of BK7 glass. What changes?
Well, now we have two more surfaces. The first is the air-glass surface for light entering the glass mirror and the second is for the glass-air surface for light leaving the mirror. So, we need to add rows to the lens data spreadsheet before and after the mirror surface.
The first air-glass surface is straightforward. The radius of curvature is -300mm, the thickness of the glass is 2mm, the aperture is the same as the mirror, and the glass is BK7.
The reflecting surface now must be changed so that its
THICKNESS will place the next surface at the correct location. After the reflection, light travels to the left and distances are now negative. Therefore the thickness to the next surface is -2mm. (Since the reflected rays are assumed to propagate in the same medium as before, the rays will still be in the BK7 glass.)
The glass-air surface has the same radius of curvature and the same aperture radius as previous two surfaces. Leave the thickness as 0.
Finally, force OSLO to autofocus by clicking the
THICKNESS -> Autofocus - paraxial focus button in the IMS column again.
At this point the Surface Data Table should look like
And the ray trace looks like
We see that the glass has modified the effective focal length from -150mm to -149.1mm.
This OSLO lens file can be downloaded as glass-mirror.len
Finally we can calculate the expected EFL using the paraxial approximation
n = 1.5168 #BK7 refractive index for the He line 587.6nm R1= -300 R2= R1 d = 2 f =efl_thick_mirror(n,R1,R2,d) print("The paraxial focal length of the mirror is %.1f mm" % f)
The paraxial focal length of the mirror is 149.1 mm