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ECEN 4616/5616 Optoelectronic Design Class website with past lectures, various files, and assignments: http://ecee.colorado.edu/ecen4616/Spring2014/ (The first assignment will be posted here on 1/22) To view video recordings of past lectures, go to: http://cuengineeringonline.colorado.edu and select “course login” from the upper right corner of the page. Lecture #27: 3/17/14

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Strehl Ratio When optimizing an optical design, it is rarely useful to put a lot of work into getting the geometric spot size much smaller than the diffraction spot size (the Airy Pattern, for circularly symmetric systems). When the geometric spot size is similar to the size of the the Airy Pattern, however, the actual MTF is determined by a complicated interference interaction between the incident waves at the image plane. A very useful measure of how an optical system will perform is known as the Strehl Ratio. This is the ratio of the intensity of the peak of the systems Point Spread Function compared to the intensity at the peak of the Airy Pattern of a perfect system. (The measure is named for Karl Strehl, a German mathematician and astronomer, who proposed it in 1895.) In Zemax, the Strehl Ratio is returned by the Merit Operand STRH, which can be used both in the Merit Function Editor and in the Universal Plot feature under the ‘Analysis’ menu. It is also returned by the Huygens PSF window (also in the ‘Analysis’ menu):

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Strehl Ratio The Strehl Ratio is a measure of how well a system actually performs at imaging. It is often described as the “Simplest, most meaningful way of expressing the effect of wavefront aberrations on image quality”. As an example, how does one decide which of these two aberrations will produce the ‘best’ image? Going by the geometric spot statistics (RMS radius and Geometric radius), we would probably choose the right hand one.

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Strehl Ratio The wavefront maps are not much help: How do you compare two different shapes of distortion? The Strehl Ratio gives us a simple way to decide which is likely best:

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Strehl Ratio The Strehl Ratio can vary between 0 and 1, with 1 being a perfect system, and 0 meaning “no discernable PSF”. Approximate Rules for Strehl Ratios: Strehl Ratio = 0.8 Most people can’t see any improvement beyond this point. (A criteria due to Lord Rayleigh, once again!) Strehl Ratio = 0.9 A good system – not worth much effort to improve. Strehl Ratio = 0.95 A very good system – stop optimizing and build! The Strehl Ratio is also an approximate measure of how much of the PSF energy is within the central peak: Strehl Ratio = 1.0 Strehl Ratio = 0.5

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A comparison between MTF’s and Strehl Ratios: No Aberration: Spherical Aberration: Strehl Ratio = 1.0Strehl Ratio = 0.72 Strehl Ratio

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Why the Strehl Ratio is important for optimizing optical designs: What are the problems with using the Strehl Ratio? The Strehl Ratio is computationally expensive (hence slow) to calculate. This is, however, less of an issue each year (as long as Moore’s Law continues). Optimizing the strehl ratio is a useful way of refining optical systems, since it reduces the complexities of 2D wavefront errors (and MTF values) into a single number. Since the merit function requires the merit of a trial system to be a single number, the Strehl ratio is an easy way to get there. Other combinations of aberration constraints may over-restrict the optimizer’s options compared to their importance in the final system. The more flexibility the optimizer has to traverse the design space, the more likely it is to find a good solution.

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Strehl Ratio It is relatively simple to measure the Strehl ratio, if at least part of the object is a point source. Hence, for astronomical imaging, the Strehl ratio can be used to detect a “Lucky Imaging” situation. Light from Objective Gated (switchable) Detector at image “Lucky Image” Detector ControlElectronics Beam Splitter A rough design for an amateur “Lucky Imaging” telescope system: The lucky image detector can be as simple as an x-y adjustable pinhole which is centered on a guide star, with a single detector behind. When the intensity passed through the pinhole exceeds a set amount, the detector is set to integrate – when the power level falls, the detector is turned off. This would remove both tip-tilt errors and de-focus errors, at the expense of only integrating for a fraction of the observing time.

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GRIN Lenses A GRadient INdex lens is one in which the index of refraction changes with position. The lens in the Human eye is an example of a natural GRIN lens. A normal lens, but with the index a function of Z, can correct for more aberrations than a lens with a fixed index. This material is called Gradium, and is available from Light Path Technologies, Inc. The GRADIUM surface type is included in the Zemax program, and several stock lenses from Light Path are available in the Zemax lens catalog. This finds uses where there is not room for a more complex lens, but a higher degree of aberration correction is required. A simple plano- convex lens: With a Z-axis index gradient: Can be diffraction limited:

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GRIN Lenses The more usual type of GRIN lens is a cylindrical rod of glass with an index that varies in a radial direction with distance from the axis of the cylinder. You can see how this could focus light conceptually by considering a discrete approximation – a cylinder with layered indices: (θ i is the angle inside the central layer, w.r.t. the z-axis.) Φ0Φ0 n0n0 Assuming that n0 > n1 > n2 > n3, we can see that the input ray will have decreasing angles with the axis as it passes each successive interface, until it finally undergoes Total Internal Reflection and heads back to the axis, since, and θ i = 90 – θ 0. Assuming that n k is the index of the last layer (which could be air, or a cladding layer), then the ray will escape the rod, if. ΘiΘi

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GRIN Lenses Since sin(θ) = cos( ϕ ) ≡ Z 0, the direction cosine of the ray in the layer – and this also holds true for all other layers (i.e., sin(θ n ) = Z n ), we can re-write Snell’s law for GRIN media in terms of Z-direction cosines as: θ ϕ Z n1n1 n2n2 Where Z is the direction cosine w.r.t. the Z-axis (and the projection of a unit- length ray onto the Z-axis).

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GRIN Lenses GRINs as Lenses: Under what conditions will a rod with radially dependent index act as a lens? When rays starting from an object point re-collect at an image point, as in the figure below: This will happen if rays from a launch point follow a periodic path whose period remains constant over a range of launch angles. Paraxial Model: By solving the GRIN Snell’s Law, for a function, n(r), which causes a ray to follow a sinusoidal path: we can find a profile that will achieve periodic imaging.

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The text, Mouroulis & Macdonald, does this derivation on pages 168 – 172. In the paraxial approximation, a quadratic profile does the job: GRIN Lenses The index profile: will cause rays entering one face of the rod to follow sinusoidal paths of the form: where the amplitude, r 0 is related to the angle of incidence at the axis of the rod by

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GRIN Lenses GRIN lenses are characterized by their Pitch Length, P. The pitch length is the number of cycles that light will make in the given length of the GRIN rod. The above GRIN has a pitch length of one. For example: a GRIN lens with a pitch length of ¼ will focus parallel incoming light to a point on the back of the rod: (Pitch slightly < ¼)

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GRIN Lenses Our text also develops paraxial ray-tracing equations for GRIN lenses: Paraxial transfer equation: Paraxial refraction equation: The paraxial equations are useful for calculating the Numerical Aperture of GRIN lenses, for example, by calculating the output ray angles for different aperture heights. Refer to the text for examples of this.

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GRIN Lenses Finite Ray Trace Model: For tracing real rays, Zemax approximates a grin lens as a sequence of thin lenses. Consider a thin slice of a GRIN lens: Because the index of refraction is less further from the axis, the optical path length is less there and hence the wavefront edges advance more than the center as it traverses the slice. This is, to first order, identical to the effect that a thin lens (shown as dashed lines) would have on the same wavefront. The thin lens has a constant index of refraction, but the OPD is reduced away from the center, since the thickness is also reduced further from the axis.

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GRIN Lenses Since the change of index in the GRIN slice corresponds to a change of thickness in the thin lens, the quadratic GRIN profile is equivalent to the quadratic surface approximation to a spherical lens. Hence, real GRIN profiles often diverge from a pure quadratic profile, to allow non- paraxial rays to focus at a point independent of initial ray angle. Zemax lists nearly a dozen different GRIN profiles that are used in making stock GRIN lenses. When Zemax traces rays through a GRIN lens, it divides the lens into a number of thin slices and treats each as a thin lens. Rays are traced using Snell’s Law and the local slope (derivative) of the index, dn/dr, at the ray’s intersection with the surface. (dn/dr corresponds to the slope of a regular len’s surface, and hence to the angle of incidence.)

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GRIN Lenses There is a parameter in the Lens Data Editor, delta-t, which controls the thickness of the slices. By default, Zemax divides a GRIN into 20 slices/pitch. The only way to know if this is enough is to change the delta-t parameter and see if anything changes noticeably. For almost all purposes, the default works well.

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GRIN Lenses Non-Paraxial GRIN lenses: The quadratic profile was calculated using paraxial assumptions. Often it is desired to use GRIN lenses at higher apertures, hence there are many modifications to the quadratic profile. Zemax lists 10 types of gradient “surface”, which are variations on the index profile formula. A typical one (for the GRADIENT2 surface) is: Other surface formulas allow index change in the Z direction, and, of course, the ends of the GRIN rod can also be curved like a normal lens, a modification which Zemax handles easily. It is easier to design GRIN lenses than to get them built – usually it is better to stay with commercially available GRINs, unless your budget is very large. Custom GRIN lenses usually come in (total) lengths of many meters. They can be cut to any desired pitch length. The reason for so many GRIN surfaces is to model the many stock lenses and types of GRINs available.

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GRIN Lenses Systems that use GRIN lenses: GRIN lenses are useful in a number of optical systems. Many of these are due to the fact that a GRIN rod (with P=1) will act as an image relay: Each integer of pitch acts as a one-to-one, upright image relay:

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GRIN Lenses Because of their inherent image relay ability, GRIN lenses are regularly used in endoscope relays:

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GRIN Lenses The scanner bar on a common photocopy machine is typically a linear array of GRIN lenses arranged to produce a one-to-one, upright, image relay, with images from adjacent lenses overlapping in registration. This works much better than trying to relay each pixel, as that would result in very small lenses with poor resolution. Modeling this phenomena accurately can be somewhat of a challenge.

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