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1. Optomechanical Setup
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"Setup" here means the optomechanical skeleton.
It comprises three main items: object, lens, and image.
They are lined up on the optical axis as shown in figure 1.

For the layout, we have to answer several questions:
a) Where is the object positioned (point  O ; object distance  g )?
b) How big is the object (point  O ; object size  G )?
c) Where is the lens (principal point  H ) positioned (layout center)?
d) Where is the image positioned (point  O' ; image distance  b )?
e) How big is the image (point  O' ; image size  B )?

(Here and in the subsequent text, I choose small letters for distances and capital letters for Sizes.
But the Letters O, H, F and O', H', F' indicate points on the optical axis.)

From the above questions a) thru e), normally:
b) object size and e) image size are given per design task;
and questions a), c) and d), i.e. positions and distances, are to be answered by the skilled engineer.


    fig.1: skeleton (11 kByte)

1.1 Distances
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In figure 1 there are two important rays.
They enable you to geometrically deduce the image distance  b  .
And they are called the principal ray and the axial ray.
Both are shown originating from the very highest point of the object.

The principal ray intersects the lens at its center, that is, at the principal point   H .
If we resort to the paraxial approximation (i.e. small field angles)  and to thin lenses (i.e. principal plane distance  0 ), then the lens does not have any influence on the principal ray; the latter stays straight and unrefracted.

The axial ray is refracted by the ideal lens in such a way that it intersects the optical axis in the focal point  F' .

("Brothers" of the principal ray originate from lower points of the object and hit the lens center; none of them is influenced by the lens.
Brothers of the axial ray originate from lower object points and approach the lens in parallel to the optical axis; they are all refracted to meet in   F' .)

If you know focal length  f  and object distance  g , you can now geometrically construct the image distance  b  by using one principal ray and one axial ray.
Though I admit: Drawing and constructing isn't the most precise solution.

But you can easily derive the famous "lens equation":
1/g   +   1/b   =   1/f
...and transform it to...
b  =   1   /   (1/f   -   1/g)       (1)
All setups that fulfill equation (1) will deliver optimum image definition.


1.2 Sizes and Scale
--------------------
From paragraph 1.1 we see that for any object distance  g
(as long as it is greater than  f )
we will get an associated ("conjugated") image distance   b
for a crisp and sharp image.
And we know: the longer we choose  g , the smaller the image will get.
... How small?

Please look at figure 1 again. Consider the principal ray and the upper half  G/2  of the object size  G . The principal ray is and remains straight; that's why  G/2  will increase linearly with increasing object distance  g :
      G/2 = k * g     -->     G = 2 * k * g
And the principal ray even remains straight when penetrating the lens; so, the same linear relation holds for the image:
      B/2 = k * b     -->     B = 2 * k * b
We'd like to know, to which scale  m  the object is imaged. So we divide image size by object size:
        m = B / G = b / g          (2)

Some people like to use plus and minus signs to indicate positive and negative direction on the optical axis. If we do so, the scale  m   gets a negative value. Which also indicates that the image is upside down.


Having derived image distance  b  from object distance  g  and from focal length  f  per equation (1), we can now calculate the scale  m  per equation (2).


1.3 Glass sheets
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Looking through a lens, you'll immediately notice that you look through a lens. Of course. But this is not as self-evident if you look through a flat glass sheet.
If the sheet is clean, clear and non-reflecting then you might ... bounce your head against the show-window.
So, is a plane glass sheet free of influence on optics?


No, there is an influence. From physics lessons at school you probably remember that a slanted ray, penetrating a plane glass sheet, is shifted prallel. The amount of shift depends on glass thickness  t , on glass refractive index  n , and, sad to say, on slant angle. I'm sure you can easily derive the amount of parallel shift by using the law of refraction (Snellius and Descartes, about 1600).
More important to our application is: A glass sheet between a lens and its focus shifts the focus by some increment  db  away from the lens (caused by the aforementioned parallel shift).
Resorting to paraxial approximation
(that is, rays nearly parallel to the optical axis;
angles  a  small enough that   sin a = a   ),
you can calculate the increment  db  of the image distance:
      db = t * (n-1) / n             (3)
Since this paraxial approximation is not very exact, we can even increase the coarseness and assume that glass had a refractive index of simply 1.5:
      db = t / 3                    (4)
Simple and handy.

But now we must carefuly keep in mind the distinction between optical and mechanical length:
a) If the optical properties of the setup (e.g.focussing) shall be maintained and a glass sheet is inserted, then the "mechanical" length is to be increased.
b) If you look through the same mechanical length with and without a glass sheet, then the "optical" length looks shorter with the glass sheet.

This lengthening / shortening is often mixed up. To correct for such a failure, you often have to increase mechanical path lengths afterwards, until they no longer fit into their housings.


Subject of my last consideration on plane glass sheets is "correction".
The increment  db  does not only depend on glass thickness and refractive index, but also on the slant angle and (via refractive index) on the wavelength of light.
Lens systems are more or less thoroughly corrected in such a manner that rays from any point of the exit pupil find their way to their image point ... even at arbitrary (visible) wavelengths.
Simply inserting a glass sheet will destroy this correction. In a demanding application, even 0.5mm glass may be too much. Images will get blurred by rainbow-colored edges.
So as a rule, define all glass sheets (thickness, material, position) on the object side and on the image side at the time of lens specification.



1.4 Principal Plane Distance     (omitted in this free edition)
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In the above calculation, the lens is assumed to be "thin" and hence, to have just one principal plane.
Real lenses with real glass thicknesses are better represented by the assumption of two principal planes.


In this paragraph, you can read more about how the distance between these two principal planes modifies
* object distance
* image distance
* object-to-image distance
* scale

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Last modified Nov.28th, 2002 23:46