ECE 5616Curtis

Paraxial into real surfaces

• Curvature, Radius Power• lens and mirrors

• lens maker equation• mirror and lens in contact• Principle planes

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Real SurfacesRefractive via Fermat’s Principle

Calculate optical path length.

Set equal to our definition of OPL for thin lens

Paraxial power of curved refractive surface.

ECE 5616Curtis

Real Surfacesrefractive via Snell’s Law

Paraxial Snell’s Law

Replace refraction with ray angles

Paraxial approximations to angle, obeying sign conv.

Cancel y, rearrange.

ECE 5616Curtis

2 lens separated by d

⎥⎦

⎤⎢⎣

⎡+−−+−

−1//1//1

/1

12211

2

fdfffdfdfd

F1 F2

d

yiui

= youo

T4R3T2R1T0[ [] ] M = R3T2R1 system matrix

M =

212121

111ff

dfff

Meff

−+==−

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Lens Maker’s EquationNot that anyone would make a lens with it !

This is why it is not good to make lens with

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Real Surfacesreflective

Derive power of mirror by placing object at center of curvature.Since ray strikes surface normally, it must return to the samepoint.

Remember that by our sign convention, R

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Lenses in ContactExample: Magnin Mirror

= a symmetric concave lens with curvatures c1 + a convex mirror of curvature c2

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If real lenses have significant thickness, why have we wasted all this time with thin lens?

All of our paraxial lens equations work with distances measured from the principal planes. However, for high NA optics the planes are actually spherical surfaces.

It turns out that a thick lens (or system of lens) can be replaced with 2 surfaces called principle planes. This assumes axially symmetric optics.

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Cardinal Points

• Guass showed that these quantities completely characterize a thick lens system.

– Front and back focal points/planes– Principle points/planes– First and second nodal points

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Principle Surfaces and Focal points

In the paraxial limit, all rays exiting a compound lens will appear to have encountered a single plane P’ with power. Ditto for reverse travellingrays, but in general P is not at P’.

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Understanding Principle PlanesOverlay previous two drawings and reverse the direction of the ray in the bottom picture (if this bothers you, hold on, it will make sense in a moment). Note that incident rays 1 and 2 appear to converge towards a virtual object at P. Note also that exiting rays 1 and 2 emanate from a virtual image at P’. And finally note that the object and image height are equal.

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Principle Planes: What are they good for ?

Since the principal planes are the conjugates of unit magnification, we can replace any compound lens element with just P and P’. Rays that hit P must be imaged with unit transverse magnification and unit angular magnification to P’. In other words, they “teleport” from P to P’. Ditto on the reverse trip. All the element power F is applied on the second plane (P’ if going forward, P if going backwards).

ECE 5616Curtis

Principle Planes

Warren Smith, “Modern Optical Engineering”, SPIE

P is roughly 1/3 glass thicknessRule of thumb

Note: focal points for negative are reversed

ECE 5616Curtis

Nodal PointsA ray crossing the axis at the first nodal point emerges from the second nodal points parallel to itself. Obviously, for lenses with the same index in object and image space, the nodal points are at the intersection of the principal planes and the axis. This is not true in general (e.g. the eye).

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Ray Tracing Thick Lenses

Same rules as before just rays are extended to principle planes and then they skip the distance in between the principle planes

P1 P2

F F

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Vertex and BFL/FFL

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Summary of a 2 Surface System

N - Conjugate Matrix

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Summary of a 2 Surface System

Remember ϕ1 = (n-1)/R1

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Lens Example

bflfflF

R1 = 40mmR2 =20mmn =1.5d = 5mm

ϕ1= (n-1)/R1 = 0.0125ϕ2= 0.025

1/F = ϕ1 + ϕ2 - d ϕ1ϕ2F = 27.8mm compared to 26.67mm for d=0

bfl = F(1-d ϕ1 ) = 26.1mmffl = F(d ϕ2 -1) = -24.3mm

H2= 27.8-26.1 = 1.7mmH1=27.8-24.3=3.5mm

ECE 5616Curtis

We know how to solve problems

• Now we can calculate F, bfl, ffl from lens parameters

• With bfl, ffl and F you can find location of principle planes (h1 and h2 on figure)

• With measuring distances from principle planes

• All other thin lens equations still work for image position magnification etc

WOW !!!!

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Compound lens

We will do lots of problems next lecture to see how this works…

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Design Problem 1If you are making a system with multiple elements that have to be hard mounted (i.e. no adjustments), how can you use the result of HW problem 1) to help align the system (assuming that you can shim the mounting plate of the plate). Can use you plates to change the location of the focus? Are their issues with using a plate in a converging beam? Is there anyway to minimize these issues? The problem of not having mechanical adjustments is typical in optical engineering as the system is being converted from a research testbed into the final product.

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Design Problem 1

Solution 1: Tip/tilt plate as fine position adjustment of a nearly collimated beam. Δθ = 1° = 17mrad, t=1mm, n=1.5. The beam displacement will be ~5 microns. Can slim in both directions toget X and Y offset.

Solution 2: Put 2 plates tilted at fixed angles in rotating barrels –spin barrels to reach and XY. Similar to Risley prism set.

Comment: can change focus position but with consequences.

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Design Problem 1

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Design Problem 1

VERY LIMITED NA, much worse if plate is tilted

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Design Problem 2

You are given an elliptical beam that has a 2:1 aspect ratio. Use prism(s) to circularize the beam. What are the tolerances on angles, index, etc to get a circular beam to within 5% of perfectly circular?

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Nichia Laser Diode Spec

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Nichia Laser Diode Spec

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Nichia Laser Diode Spec

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Nichia Multimode LD Spec

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Design Problem 2

x

2x

SolutionsSingle prism2 prismsToric/astigmatic lens

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Sample Spec for prism in this use

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Homework #2

Available at the website under homework

http://ecee.colorado.edu/~ecen4616http://ecee.colorado.edu/~ecen5616

Due in 2 weeks

W. Smith “Modern Optical Engineering”

Chapter 4 (again) andChapter 13 (will help for next week)Reading Chapter 17 – cover camera lenses we discussed last time.