ECEN 4616/5616Jan 18, 2013

Notes from Jan 16:

The question was asked: “Why does a quadratic form make a perfect lens?”

The answer is that a quadratic sag does not make a perfect lens – a spherical lens is a perfect lens in the paraxial approximation, and in that approximation, it is a quadratic. If the region of interest is extended past the paraxial region, so that the quadratic is no longer a good approximation of the sphere, then the sphere will not be an ideal lens.

Suppose we let a plane wave hit the flat side of a plano-convex lens – that way, we don’t have to make the thin-lens approximation, since the plano side will have no effect on the wave, only the convex side.

Assume the lens has an index of 1.5 and is in air (n=1). Immediately after the lens, the wave will have a fraction, (n-1=1/2), of the lens’s sag imposed on it; i.e., S2 = S/2, at every distance r from the axis.

If the curvature of the lens is c, its sag as a function of r is:

Hence, the sag of the wavefront is:

For S2to be a spherical surface there must exist a new curvaturec2, such that:

for all r. For the expressions to equal for all r, each term must be equal.

Solving for the new curvature term by term, we get:

  • 1st term:
  • 2nd term:
  • 3rd term:

and etc. Hence, no c2 exists allowingS2to be written in the form of a sphere, and S2 is therefore not a sphere – except in the paraxial assumption represented by the first term in the expansion.

Thin Lens Imaging Equation:

Previously (Class Notes: 1/16, pg.4), we derived the thin lens construction equation by letting a plane wave impinge on the lens and noting that the curvature of the wave after exiting the lens was:

(where R1, R2 are the radii of curvature of the 1st and 2nd lens surfaces, using the sign rules).

The plane wave became a spherical wave converging on a point at distance F to the right of the lens. f is defined as the “Focal Length” of the lens and is called the “Power” of the lens. C is positive if f is to the right of the lens, and negative if f is to the left, according to the sign rules.

A thin lens with power C will therefore add curvature C(and sag ) to an incoming wave, where C can be calculated from the surface curvatures and index of the lens.

Let us consider a spherical wave which is diverging from a point that is distance O to the left of the lens (hence, O < 0, according to the sign rules):

At the lens, the wave will have.curvature .

The lens will add to this the curvature , so that the total curvature of the wave is now .

In general, the wave will now be a spherical wave converging to a point I away from the lens. (I could be anywhere – including at ∞ -- but we will temporarily assume that it is a point to the right of the lens.) Hence the curvature of the wave immediately after the lens is also given by .

Equating these two expressions for the wave’s curvature (after the lens) and re-arranging, we get the thin lens imaging equation (also called the Gaussian Lens Formula):

This equation relates the “object point”, O (where the wave came from), the “image point”, I (where the wave is going) to the lens focal length, f.

Paraxial Ray Tracing:

Sign Rules:

  1. The (local) origin is at the intersection of the surface and the z-axis.
  2. All distances are measured fromn the origin: Right and Up are positive
  3. All angles are acute. They are measured from:
  4. From the z-axis to the ray
  5. From the surface normal to the ray
  6. CCW is positive, CW is negative
  7. Indices of refraction are positive for rays going left-to-right; negative for rays going righ-to-left. (Normal is left-to-right.)

Notation Rules:

Unprimed rays, angles and distances are before the ray is refracted by the surface, primed values are after.

Paraxial Assumptions:

  1. Surface sag is ignored as negligible.
  2. angles sines tangents

Positive Values:

  • Angles: u, i, i’
  • Lengths: l’,R,h

Negative Values:

  • Angles: a, u’
  • Lengths: l

Angle of incidence & refraction:

Ray Angles with z-axis:

Snell’s Law (Paraxial):

hence:

Thin Lens Equation:

The thin lens equation can be derived by applying the above surface imaging equation sequentially to both surfaces of the thin lens.

Note that a thin lens is defined as two surfaces with negligible separation (both surfaces intersect z-axis at origin), considered paraxially.

We will use subscripts 1, 2 to indicate which surface we are tracing through. Note that the resultant values (primed) from surface one are the input (unprimed) values for surface two: i.e.,

Using the surface imaging equation twice, we have:

Noting that

We get, after substitution:

Adding these two equations gives:

(Paraxial thin lens equation)

(Notice that this also gives us the relation between the focal length, object and image:

where F is the focal length, O is the distance to the object and I the distance to the image.

Observations about the thin lens equation:

  1. l, l’ are independent of h – all rays from A go to A’ regardless of (paraxial) height on the lens: => A’ is the (paraxial) image of A
  2. We can use this equation without regard to the sign of the values, or whether the image is real or virtual, as long as the sign conventions are followed.
  3. The power of the lens is the sum of powers of each surface. This will be true for any number of (paraxially) close surfaces – which will be used in deriving achromats.

Paraxial Refraction and Transfer ray tracing:

The thin lens equation allows us to trace images through a series of surfaces or thin lenses, but doesn’t allow the path of a specific ray to be traced. Hence, we need a set of equations for this task that don’t involve the object and image distances, l,l’.

From the Paraxial Snell’s law, and making some of the same substitutions as above:

Hence is the Gaussian Refraction Equation for a surface whose power is defined as . Remember that , where R is the center of curvature of the surface, whose magnitude and sign are given by the sign and coordinate conventions, and n, u are the index and angle before (to the left of) the surface, n’, u’ are the quantities after (to the right of) the surface.

The Gaussian Refraction Equation gives you the change in angle as a ray with height h passes through a surface, allowing arbitrary rays to be drawn.

There is one more step necessary to trace a ray through multiple surfaces, and that is a transfer equation that allows you to calculate the change in height of a ray as it travels to the next surface.

Important note on notation:

The distance from a surface to the next surface is considered a property of the first surface: Hence the distance from surface S1 to S2 is labeled d1, not d2. This is a convention not only in paraxial tracing, but in all ray trace programs, as well.

There are two transfer equations, whose derivation is self-evident (provided the sign conventions are carefully followed):

For the imaging equation, the output image distance for surface i, adjusted by the distance between surfaces, becomes the input object distance for surface i+1.

For the refraction equation, the output height modified by the output slope times d gives the input height for the next surface:

Use of paraxial equations for finite height rays:

Notice that the paraxial formulas (refraction, transfer) for Gaussian tracing are all exact formulas, regardless of finite ray height, IF:

  1. “Angle” variables, u, u’are interpreted to be the tangents of the actual ray angles.
  2. “Lenses” and “surfaces” are considered to be planes perpendicular to the z-axis at the local coordinate origin.
  3. “Paraxial Lenses” have no curvature or sag, only power.
  4. All refraction takes place at the intersection of a ray with the surface plane.
  5. The variables u, u’ are modified by the surface according to the Gaussian refraction equation – Snell’s law is ignored.

Under these conditions, the Gaussian trace through a “paraxial” surface simulates the performance of an ideal lens – one which images all rays in exactly the same place as paraxial rays, regardless of the distance from the axis that a ray intersects the surface.

All ray trace programs have a (mis-named) “Paraxial Lens” type which behaves as such an ideal lens. These idealized ‘lenses’ have no thickness or index of refraction, only extent in r and optical power. One simply specifies a focal length (1/K). There is no need to restrict rays to the actual paraxial region.

Useful lens equations from Gaussian ray tracing:

The paraxial ray trace equations are:

  1. The diffraction equation giving the change in angle of a ray through a surface:
  1. And the transfer equation, giving the ray’s height at the next surface):

These equations (and strict adherence to the sign rules) allow one to derive many useful expressions about lenses.

Using Gaussian ray traces to generate lens formulas:

Trace a horizontal, marginal ray through two lenses (why a horizontal ray?):

Refract through first element:

Transfer to second element:

Refract at second element:

Eliminating h1 and using where K is the combination power, we get the:

General formula for combination of two systems.

(Easy to remember & independent of the sign convention)

Why doesn’t n1 enter into this equation?

(n1 is included in the calculation of K1)

Thick Lens Equation:

Let:

Then:

Or, in terms of focal length and radii:

Thick lens formula (Lensmaker’s formula)

Thin Lens Equation:

Just set d=0 in the thick lens equation:

Focal length of lenses not in air:

Suppose that n2’1 (lens is not in air):

Then, the LHS of the lens equations becomes , andK2 will be calculated differently.

Two thin lenses spaced in air:

Let:

Then:

(This also works for thick lenses, if d is measured between principle planes)

Afocal systems:

Suppose that

Then

Hence if the lenses are spaced by the sum of their focal lengths, we get an afocal system – e.g., a telescope.

Zoom lens:

What happens if d = f1?

Then f = f1 and the second lens has no effect on the total power at all.

This is the principle behind zoom lenses: The power contribution of a lens (or lens group) to the total lens is dependent on the relative positions of the lenses. A useful expression for calculating this is the:

Alternate combination of power formula:

If a paraxial ray parallel to the axis is traced through a sequence of thin lenses, at an initial height, h1, the total system power can be solved in terms of the individual lens powers and the ray intercept heights on each lens:

Graphical Ray Tracing Through Thin Lenses

We will assume, without further proof, that the paraxial effect of a thin lens on the curvature of wavefronts is, to first order, unchanged by tilting the lens through a “paraxial” angle.

Given this, there are a number of rays that can be immediately traced through a thin lens, whose power is known, without further calculation:

I)Incident rays parallel to the axis, pass through the focal point.

(The corollary is that incident rays that pass through the focal point emerge parallel to the axis.)

II)Rays through the center of the lens are undeviated.

In the paraxial approximation, the lens is an infinitesimally thin plane-parallel plate at the center.

III)Any parallel bundle of rays will meet at the focal plane at the point where the central (undeviated) ray in the bundle meets that plane.

The “focal plane” is a plane perpendicular to the axis at the focal distance. The central ray of a bundle is often called the “chief ray”.

IV)For object and image distances determined by the thin lens imaging equation, a bundle of rays diverging from any point on the object plane will converge to a point on the image plane at the point where the chief ray from the object point intersects the image plane.

pg. 1