1. 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.

3. Last lecture, we optimized an air-spaced doublet and achieved zero 3 rd order spherical and coma aberrations, as well as diffraction-limited focusing along the axis: (File is ‘Sacorrection.zmx’, available on the website. ) Before OptimizationAfter Optimization Lens shapes before optimization: Lens shapes after optimization:

4. Returning to our corrected doublet: Suppose we select a range of wavelengths covering the visible and see how the system behaves:

5. Looking at a stretched and zoomed section of the layout, we see that the red, green, and blue wavelengths don’t focus at the same distance:

6. This aberration is also shown in the “Chromatic focal Shift” analysis window (“Analysis – Miscellaneous – Chromatic Focal Shift”), which shows ~ 1.5 mm change in focal length over the visible.

7. Here is the Seidel (3 rd order) aberration plot. Note that “Axial Color” (another term for ‘chromatic focal shift’) is the largest remaining aberration. There are both positive and negative contributions, however – perhaps to correct for axial color we need another degree of freedom.

9. Dispersion and Achromats Zemax’s “Glass Map” (‘Analysis – Glass and Gradient Index – Glass Map’): “Abbe Number” is a measure of dispersion – the change in index of refraction with wavelength. (Note that low Abbe number = high dispersion.) ‘Crown’ glass ‘Flint’ glass Low dispersion High dispersion

10. Dispersion The index of refraction of glasses is traditionally taken at three wavelengths (for use in the visible). These are the Fraunhofer lines for hydrogen and helium ( a nearby line from mercury was used before helium became widely available). These were convenient, because they spanned the visible spectrum and could be duplicated with great precision in any well-equipped laboratory. 1.F (hydrogen) ≈ 486nm 2.d (helium) ≈ 588 nm 3.C (hydrogen)≈656 nm The corresponding indices of refraction (for a given glass type) at these wavelengths are tradionally labeled: 1.n F = n(0.486µm) (blue) 2.n d = n(0.588µm) (green) 3.n C = n((0.656µm) (red) The ‘Abbe Number’ is defined as:

11. Index of Glass Far from any absorption resonances, the index of refraction of a dielectric like glass is a mostly linear interaction between the electric field of the light and the electrons in the outer shells of the atoms. The value of the index generally increases with increasing density of the glass. Some of the additives used to modify the base index of silica are: High index – high dispersion (‘Flint’) glass: Lead Oxide (PbO) (Traditional, but now discontinued.) Titanium Dioxide (TiO2) Zirconium Dioxide (ZrO2) Low index –low dispersion (‘Crown’) glass: Phosphorus Pentoxide (P 2 O 5 ) Zinc Oxide (ZnO) Boron Trioxide (B 2 O 3 ) Barium Oxide (BaO)

12. While the density of the glass is the main effect on index far from an absorption resonance, nearer a resonance, the strength and width of the resonance is more important. Resonances and Anomalous Disersion Near a resonance, however, the index changes rapidly, and reverses direction near the peak of the absorption curve. Anomalous Dispersion The drop in index with decreasing wavelength is called “anomalous dispersion”. Note: n<1 (!)

13. Devices Based on Anomalous Dispersion: “Optical Cloaking Device” What can be done with an material with n<1? One design for an “optical cloaking device” depends on n<1 material causing the phase velocity of light to exceed light allowing the waves to pass around the cloaked area but still meet up with the waves that pass outside the cloak.

14. MetaMaterials ‘Metamaterials’ is the name given to artifically constructed materials with anomalous dispersion – even reaching negative indices in a narrow frequency band. At microwave frequencies, such materials have been constructed by simply stacking up a number of resonant circuits: At optical frequencies, arrays of 3D sub-wavelength holes works: It has been proposed (although not tried yet) to construct an array of posts in the water that create a ‘negative index’ region around some structure (such as an oil platform) so that waves pass around the structure in the negative index region and the structure is ‘cloaked’ from any influence by the waves. Model ‘Wave Cloak’

15. MetaMaterials and Zemax Zemax uses formulas like the Sellmeier formula to calculate index: Where the L’s represent wavelengths (squared) where resonances exist. Zemax allows you to create new glasses (see Ch. 23, “Using Glass Catalogs” in the user’s manual). By carefully placing resonances in an artificial glass, you can create the necessary indices for cloaking.

16. MetaMaterials and Zemax Alternatively, you can define the index of any existing glass to be negative by checking a box on the glass catalog dialog: An interesting student project would be to model one of the several invisibility cloak designs in Zemax.

17. Indices of some common glasses: The sharp upturn in index in the UV indicates resonances nearby. Note that the slope of the index (in the visible) is much less for the crown glasses than for the flints. Hence, our scheme for creating an achromat (color-corrected doublet) will be to combine a positive crown glass lens with a weaker negative flint lens so that the combination still has positive power, but that the focal shifts cancel. F d C Visible

18. In the “glass map”, the index of refraction is the index at the d line (n d ), and the ‘Abbe Number’ (often called the “V-number” by those who aren’t sure how to pronounce “Abbe”) is: The significance of this value follows from the calculation of the chromatic variation of power for a thin lens. Power at d: Power at F: Power at C: Consider: Hence: Dispersion

19. We can also derive the dispersion equation from the image equation: Dispersion Correcton Given that the Object distances are all the same. So, the change in power is proportional to the power divided by the Abbe number. (It can also be shown that change in magnification with wavelength – transverse chromatic aberration – is also proportional to. Hence, chromatic aberration (longitudinal and transverse) will be corrected (paraxially) if we use two lenses whose chromatic power shifts sum to zero: where the K’s are implicitly calculated at the d wavelength.

20. Achromats For two thin lenses in contact: The condition that the combination have zero focal shift between the F and C wavelengths is: The lenses chromatic focal shifts cancel each other. Solving these two equations simultaneously, we get the construction equations for an Achromat: Obviously, this only works if the lenses are made of glasses with significantly different Abbe Numbers.

21. Designing an Achromat Let’s pick two glasses (one crown, one flint) with a large difference in V-number: BK7: n d = 1.5168, V d = 64.1673 SF59: n d = 1.952497, V d = 20.3637 Let K = 0.02 (f = 50mm), and the diameter = 25mm (F/2 lens) Evaluating the construction equations: Note that we have chosen glasses that have near the maximum difference in both index and V d. This will have consequences, as we will see later.

22. Starting Solution for Zemax While the calculated powers (for the given glasses) guarantee a thin-lens that is achromatic, there are still things we want Zemax to optimize: 1.The lens must have some thickness, so the thin lens approximation will not be entirely correct. 2.As we have seen, there should be enough degrees of freedom in the shapes of the elements to correct 3 rd order Spherical and Coma. We still need a specific design to enter into Zemax, however, so we will pick the simplest solution and make the positive lens symmetrical (c 1 =-c 2 ≡c): Hence: Since we intend to cement the two elements together, the first surface of the negative element has to have the same curvature as the second surface of the positive element, namely -0.0283 mm -1. For the negative element, therefore: Hence:

23. Putting these values into Zemax’s LDE: Notice that we have specified a separation of 0.01 mm between the two elements. This is not strictly necessary, but does allow us to easily change our minds about cementing the elements together later. If we explicitly put the lenses together, the LDE would look like this:

24. The element thicknesses in the LDE were arbitrarily chosen to give the lens sufficient edge thickness: Zemax reports the focal length as 51 mm: Quantities on the status bar are changed by accessing the File-Preferences-Status Bar dialog box. And the ‘Chromatic Focal Shift’ analysis window shows that we didn’t get the F and C lines to focus at the same plane: These differences are due to the fact that our analysis assumed the thin lens approximation, but the actual lens has finite thickness.

25. To adjust for the thick lens effect, we open two “sliders” and manually adjust the first and last surfaces of the doublet until we are able to meet both the focal length and achromatic requirements. (Sliders available from ‘Tools-Design-Slider’) Notice that Zemax reports the focal shift range as 31 µm, but the “Diffraction Limited Range” is only ~14 µm. What’s up with that?

26. F d C Visible We picked two glasses with a wide difference in V-numbers. The high dispersion (low V-number) of SF59 is probably due to the nearness of an atomic resonance in the UV, while the low dispersion (high V-number) of BK7 indicated a relatively farther away resonance. As this chart shows, indices near a resonance are significantly more non-linear than indices far away.

27. Here are the specific indices of refraction for the selected glasses:

28. The construction equations for the achromat allow us to match the focal shift at two wavelengths, F (0.486 µm) and C (0.656 µm), but other wavelengths will have different power shifts if the index curves are of significantly different shapes. Here is a plot of the residual power mismatch after matching the F and C shifts:

29. Apparently, we should have checked the dispersion curves to get a more compatible pair. This means, in practice, that the V-numbers have to be closer together. According to the construction equations, however, that also means that the individual elements are going to have stronger powers and hence more spherical aberration: Zemax has an analysis window that allows you to compare up to 4 glasses’ dispersion curves: (Analysis-Glass and Gradient Index- Dispersion Diagram)

30. An easier solution, however (and one consistent with our design philosophy of not reinventing things) is to look through the lens catalogs in Zemax and see what other designers have used: A search on focal length and diameter in the Edmund catalog gives us 11 lenses. Highlighting one and selecting the “Prescription” tab at the bottom reveals the glass selections (as well as lots of other things):

31. SF10 BAF10 A look at the dispersion diagram for the Edmund lens shows a much closer match in the dispersion diagrams than we chose.

32. There is something we can do about the “Diffraction Limited Range”, however. If we look at the ray diagram near the focal point, it is obvious that it diverges at a rate dependent on the system aperture and focal length: a b B Given we define an “Acceptable Blur Diameter”, B, then the Depth-of-Focus depends on the F/# of the lens: a: DOF for F/2 b: DOF for F/5 Zemax always uses the diameter of the Airy Pattern as the Blur value, so when the F/# is increased, the ray slopes become less AND the Blur value increase.

33. In the case of our attempt at an achromat design, increasing the F/# to 4 (by reducing the entrance pupil diameter to 12.5mm) allow the inescapable chromatic shift to fall (just) within the diffraction limited range: