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**ECEN 4616/5616 Optoelectronic Design**

Class website with past lectures, various files, and assignments: (The first assignment will be posted here on 1/22) To view video recordings of past lectures, go to: and select “course login” from the upper right corner of the page.

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**Picking up our Achromat Design**

We had to reduce the aperture to 12.5 mm (F/4) in order to match the inescapable chromatic focal shift range (due to choice of glasses) with the diffraction limited Depth of Focus: The Depth of Focus is determined by the distance where the geometric blur reaches a predetermined “Acceptable Blur Size”, which for Zemax is always the Airy Pattern diameter. b DOF (DOF exaggerated by 700X)

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**Optimization of the Achromat**

Let’s let Zemax optimize the achromat and see how good it becomes. Here is the LDE for the system. Notice that we’ve made all three surfaces variable, the thickness of each lens, and the distance to the image plane. We will need to constrain these values in the merit function to keep them reasonable. Surface #1 Surface #4 t3 t1 t4 Surfaces #2&3 Surface #5

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We want to monitor the spherical aberration and constrain the chromatic focal shift and focal length, so we start with the operands: SPHA This monitors 3rd order spherical aberration in the doublet. AXCL “Axial Color” – this is the same as chromatic focal shift. Notice that we are only checking the shift between Wave #1 (F) and Wave #3 (C), as those are the only wavelengths at which focal shift can be nulled. EFFL “Effective Focal Length” – we will use this to constrain the lens to be 50 mm fl. DMFS “Default Merit Function Start” – this is where the default merit function will be inserted. If this operand weren’t here, the default merit function would overwrite the other operands I entered. I haven’t put any weights in yet – I’m waiting to see how they compare with the Default Merit Function (next).

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**How do we know which wavelengths are #1 and #3?**

That’s the way they are entered into the wavelength dialog box:

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**Selecting “Design-Sequential Merit Function” on the MFE menu…**

…brings up the Default Merit Function dialog box: The ‘Help’ button at the lower right brings up an explanation of the choices. We have left everything default except that Glass thickness limits have been selected: Min glass thickness = 2 Max glass thickness = 8 Min glass edge = 1

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We click “OK” and the Default Merit Function appears in the Merit Function Editor (MFE), below our manually entered operands:

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We put weights in for the operands we entered and adjust them to get a reasonable contribution to the total merit value: The goal is to have the operands contribute to the total merit value, but not either be swamped by or dominate the Default Merit Function value.

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**Optimizing does reduce the merit value somewhat:**

Spherical aberration is only reduced by ~10%: And other analyses are also improved by a similar small amount – the lens is still a long way from diffraction limited:

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**Let’s try giving the system another degree of freedom: **

The two lenses are separated by 3mm and all surfaces are now variable: Additionally, the lens Semi-diameters are reduced to 8mm, since the aperture has been stopped down to 6.25 mm radius. The system now looks like this: Notice that the lens power has increased (fl reduced) due to the spacing between lenses – in accordance with the combination of powers formula:

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**This time, optimization produces a dramatic improvement:**

And 3rd order spherical nearly disappears: And the system is now diffraction limited:

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**Cemented doublet Air Spaced doublet**

Adding the extra degrees of freedom by spacing the elements apart proved to be the key to getting a good lens. Nothing except a better selection of glasses, however, is capable of reducing the chromatic focal shift.

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**Stock Achromats How do the commercially available achromats compare?**

Under “Tools-Catalogs-Lens Catalogs” (or just the “Len” tab) we bring up the following dialog box: A constrained search brings up the following F/2 achromats: Highlighting one of them and clicking on the “Insert” button puts the lens into the LDE. Note that we have constrained the search (in the Edmund Optics catalog – one of many) to focal lengths between 48 and 52 mm and diameters from 20 to 25 mm. Also we specify that we’re only interested in doublets with spherical surfaces.

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Starting with a blank system (use the “New” tab to clear out everything), the inserted lens is placed in the LDE: The catalog number is inserted into the ‘Comment’ field. After using ‘Tools-Design-Quick Focus’ to fill in the distance to the image plane ( mm), we can look at the Layout window:

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**Use the same analyses that we used on our designed achromat:**

Surprisingly, it’s not as good as our attempt at a cemented doublet! Apparently, it is a good idea to check stock lenses for suitability. This lens might be OK in a visual application, but wouldn’t work very well at, say, inserting light into a fiber.

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Making the surface curvatures and BFL variable, and using the same merit function we used on our achromat design, optimization results in ~20% improvement in the Merit Value:

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**Both Spherical and Axial Color are reduced by 50%:**

At any rate, our cemented achromat design appears not to have been so bad after all. As we found with our monochromatic doublet, spherical cannot be completely removed with the degrees of freedom afforded by a cemented doublet. For applications requiring better lenses, it would seem wise to choose air-spaced achromats instead. Why the manufacturer didn’t do this is puzzling. Perhaps this doublet was designed for the discontinued glasses, BAF10 and SF10 and not updated for the “equivalent” substitutions, N-BAF10 and N-SF10.

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Glass Substitution Finding the best glasses for a design used to be a long and arduous task, with much done by trial and error. Fortunately, Zemax has recently introduced a “Glass Substitution” feature into their ‘Global’ and ‘Hammer’ optimizers (not the standard optimizer): The ‘S’ next to a glass name means that there is a “Substitution Solve” set on the glass. It’s not necessary to use the glass catalogs that come with Zemax – Chapter 23 of the manual “Using Glass Catalogs” describes how to create new catalogs that can contain glasses chosen from any of the other catalogs, or specified by the user. If you are designing a custom optical lens for a company, it’s a good idea to contact the shop that will be making them and ask what are their preferred glasses (and how much each costs!). You can then make up a catalog specifically for this shop and thereby save time and money in the fabrication. High volume applications are now almost all made in China, so it would be wise to design using Chinese Glasses rather than depend on nothing going wrong with a substitution. Right-clicking on a glass name in the LDE brings up a “Glass Solve” dialog box. Choosing “Substitute” from the pull-down menu and specifying a Glass Catalog to use allows Zemax to try different glasses.

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Running the ‘Hammer’ Optimizer for a short while (on our original achromat design) results in Zemax picking a pair of glass types that result in a far superior achromat: ‘Proper’ glass selection is apparently a very important part of a good design. Fortunately, it is now much easier than it used to be.

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Equivalent V-Number How might we involve more than two lenses in a color-correction scheme? (Or, alternatively, how can compound lenses combine to correct color aberrations?) One could make every pair of lenses achromatic, but this might cause inconvenient constraints on a design. Suppose we design a doublet with a non-zero change in chromatic focal power: From the chromatic power shift formula: we can define: Therefore, we can say that the effective V-number of our doublet is: Adding even more elements to a corrected system makes automated glass selection even more important. Hence we can produce a compound lens with an effective V-number that wouldn’t be available from a single glass. This can prove useful in correcting optical instruments with widely separated elements such as microscope and telescopes – the eyepiece can be designed to correct the chromatic aberrations of the objective, for instance, allowing more freedom to design the objective’s performance.

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**Depth of Field (DOF) and Resolution**

We’ve seen that the geometric DOF is determined by the convergence angle (N.A.) of the optics and the definition of an “Acceptable Blur Size” (b): b DOF Zemax simply assumes that the acceptable geometric blur size is the diameter of the Airy Pattern: But, for a digital detector whose pixels are larger than this, a diffraction-limited lens can actually be a liability.

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The Depth of Field (DOF) and depth of focus (dof) are strictly a function of the geometry of the optical system and the “acceptable blur diameter”, b. Given that b and B are related by the magnification, m, the DOF and dof are related by the magnification squared, m2 : And, in the approximation where sin tan, we have:

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Hyperfocal Distance The hyperfocal distance is defined as the distance to focus a camera such that objects at infinity will still be “acceptably” in focus (i.e., the geometric PSF will not be larger than the acceptable blur diameter). The following drawing illustrates an optical system focused at the hyperfocal distance: l’ dZ b f’ Objects at focus at F, with the maximum blur diameter, b. The lens is focused at the conjugate of l’ = F+dZ/2, and the minimum in focus distance is the conjugate of l’+dZ/2. Hence, the hyperfocal distance (point of focus) is:

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**Solving the image equation for the nearest distance in focus:**

In Summary, if the lens is focused at the hyperfocal distance L, then everything from ∞ to the near distance, N, will be acceptably in focus. Hyperfocal distance Near focus distance (Note that Wikipedia gets this formula wrong.)

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**The Nyquist Frequency Limit So, what is meant by “correctly sample”?**

Given a spatial frequency, NF (in cycles/mm); The Nyquist theorem says that the minimum spatial rate you can use to correctly sample this image is: where is the spacing of the samples, and is the sampling frequency. Since , the Nyquist theorem says that the minimum sampling frequency we can use to ‘correctly’ sample a given spatial frequency is twice that frequency. If, for example, we have a digital detector with pixel spacing p, the highest spatial frequency we can correctly sample has a period of 2p. In other words, the minimum sampling rate for a spatial frequency is twice per cycle. So, what is meant by “correctly sample”?

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**Undersampling results in Aliasing**

Sampling Points Spatial Frequency Aliased Frequency This is an example of an undersampled spatial frequency: The sample points are spaced a little more than once per cycle. Note that the given samples are indistinguishable from sample points obtained from a spatial frequency (in blue) with a much longer period. There is no data that can determine that the higher frequency is even present. When the image is viewed, what you will see is the lower (aliased) frequency, which is not really in the image at all.

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**Digital detectors and resolution**

It’s clear from the preceeding analysis, that the blur size to use in calculating the depth of field of a digital camera should be the pixel size. However, having a lens which has a resolution (at the detector plane) greater than , where is the pixel spacing will result in aliasing in the image. For example, a detector with a pixel spacing of 10µm would have a Nyquest frequency limit of 50 cycles/mm. The diffraction limit for a F/2.8 lens (at 0.55 µm wavelength) is 650 cycles/mm. Therefore, it is not only unnecessary to use a diffraction-limited lens on such a detector, it will cause problems with the image. On many color detectors, the pixels have color filters in alternating rows of RGRGRG, and BGBGBBG. If the PSF from a point falls only on one pixel, the color information about that point is, not only lost, but aliased into a (possibly) false color.

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This is the reason that many camera lenses seen in the literature fail to reach the diffraction limit – not only is there no need, too much resolution (beyond the detector limit) can result in aliasing with digital detectors. (This wasn’t a problem with film.) Typical camera lens from a famous designer: MTF plotted to the diffraction limit: MTF plotted to the Nyquist limit (10µm pixels):

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