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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.
θ θ’θ’ R Right Side Rear-View Mirror L1L1 L2L2 Driver In a previous lecture, the question was asked: What are convex (negative) mirrors used for? One very common use is for the right-side rear-view mirror on an automobile. W W The engineering design task is: Given the size constraints of the mirrors, W, and the geometry of the car layout, L 1, L 2 ; Find R such that θ = θ’. (i.e., make the driver’s field of view equal on both sides.)
Note that, for a convex mirror, l > l’, but also M < 1 (i.e., things appear both nearer and smaller). Whether objects actually appear closer or farther depends on if the user depends mostly on size or binocular vision (or focus) cues for distance. Or, in other words: Other Issues with Convex Mirrors
Shape Factor and Aberration “Bending Lenses” The shape of a lens is defined by the shape factor, X: The following lenses all have the same focal length, but different shape factors: X=0X=1 X>1X=-1X<-1 c1c1 c2c2 For the lens above with shape factor, X=-1, the Zemax LDE is:
LDE for shape factor X=1 The “Solve” on surface 2, Radius, is: This dialog box is accessed by right- clicking on the Radius box in the LDE. The Solve chosen, “Element Power” causes the radius to be automatically calculated so as to maintain a lens power of 0.05mm -1 or a focal length of 20 mm. The first surface of the lens can then be made variable without changing the power of the lens.
The lens’s aberrations will change as its shape factor changes, since this changes the real angles of incidence of the rays impinging on it. We can monitor the aberrations using the operands available in the Merit Function Editor (MFE): Click on the ‘Help’ menu item in the MFE to bring up the operand list: Shape Factor and Aberration
The list goes on for several pages. Clicking on an operand brings up the definition and instructions for use.
For example, clicking on ‘COMA’ in the above list gives this information box: The terms “Surf” and “Wave” refer to the columns in the MFE (Merit Function Editor) which will be labeled as such when the COMA operand is entered.
Other operands for third-order aberrations of interest are: SPHA: Spherical Aberration FCUR: Field Curvature ASTI: Astigmatism Install these four operands in the MFE, setting the “Surf” value to “0”, so that the aberration sum for both surfaces is returned: (Here, the “Hide Column” feature under the “View” menu has been used to shrink unused columns in the editor.)
We set Zemax up with two field points at ∞, one on axis and one at 5 degrees (since only spherical aberration exists on axis – the others require an off-axis field): Likewise, since these are all monochromatic aberrations, we use only one wavelength:
We can then update the MFE and see what aberrations Zemax reports:
Using our Merit Function which reports 3 rd order aberrations, we can write a Zemax macro (see: “Zemax Programming Language”, chapter 25 in the Manual). Here is a macro in the “ZPL” language that loops our system through 50 different Shape Factors and prints out the aberrations: !ThirdOrd.zpl ! !Print out 3rd order aberrations vs. shape factor !Run on Shape.zmx !Editors LDE and MFE must be open for macro to work correctly ! !Number of points to print: N = 50 ! !Surfaces in Shape.zmx frontsurf = 3 backsurf = 4 !Rows in MFE: Spha = 1 Coma = 2 Asti = 3 Fcur = 4 ! !Get starting curvatures (should already be at one extreme) c1 = CURV(frontsurf) c2 = CURV(backsurf) ! Inc = -(c2+c1)/N ! !Loop through shape factors: FOR S1, c1, -c2, Inc !Set curvature of frontsurf: SetSurfaceProperty frontsurf, 2, S1 UPDATE all !Get back surface curvature: S2 = CURV(backsurf) !Calculate the shape value: Shape = (S1+S2)/(S1-S2) !Extract the aberrations from the MFE: SA = OPER(Spha,10) CO = OPER(Coma,10) AS = OPER(Asti,10) FC = OPER(Fcur,10) ! PRINT Shape, " ",SA," ",CO," ",AS," ",FC NEXT Note that there are only 20 active statements in this macro – 20 statements are only comments.
ZPL is much like early “Basic”, but contains a number of functions that access Zemax editor values – such as “OPER”, which accesses values in the MFE – and keywords – like “UPDATE” – which drive Zemax operations. Macros are run from the “Macros” menu from the top of the Zemax window, and they must be in the Macro Directory. Not sure where the Macro Directory is, or you would like to change it? Look under “File/Preferences/Folders”. The output of our “ThirdOrd” macro is a text window: Where the columns are determined by our PRINT statement : PRINT Shape, " ",SA," ",CO," ",AS," ",FC To be Shape factor, Spherical Aberration, Coma, Astigmatism, and Field Curvature.
You can save this text to a file and use it in a plotting program (or you can deal with the primitive plot capabilities of ZPL): Aberration in Waves Shape Factor
Expanded View: As can be seen from the close-up, both S.A. and Coma are minimized at a Shape Factor of ~0.7
Adding weights to the spherical aberration and Coma lines of our MFA: Plus making the radius of the first lens surface variable and optimizing, we get this lens (R1=11.7825, R2=-79.00985, Shape Factor = 0.7405): This is known as a “Best Form” Singlet.
Example of Aberration Correction We’ve seen that Spherical Aberration (S.A.) cannot be eliminated with a singlet lens – but that, for a given power, the shape of the lens controls to a great degree how much S.A. the singlet creates. Aberration in Waves Running our macro on a negative singlet, we also note that a negative lens also has a ‘best form’, but the S.A. is negative whereas the S.A. for a positive lens is positive. Let’s try adding a negative lens to the best form positive lens so as to maintain positive power, but with the form(s) adjusted to (hopefully) elminate S.A.
Spherical Aberration Corrected Doublet Let’s start with a pair of symmetrical lenses: A positive lens with K = 0.02 (f=50 mm) A negative lens with K = -0.01 (f = -100mm) The result will be a combination with f ~ 100 mm. (We can scale the lens after optimization to get an exact focal length.) Close up view:
Here is the LDE for the starting system: And here are the curvature solves on the second surface of each element:
Here is the 3 rd order aberrations plot. Note that the negative lens (surfaces 4&5) has aberrations of the opposite sign from the positive lens (surfaces 2&3), and that some aberration correction is already taking place, even though we haven’t optimized it. A second off-axis field has been added to allow Zemax to calculate the off-axis aberrations. No chromatic aberrations are reported, as only one wavelength has been defined.
Here are the TRA plot and the Wavefront Map for the on- axis field:
Here is the Merit Function Editor (MFE) Notes: Only lines 1 & 2 (spherical and coma) are weighted. We are not trying to do everything at once. Lines 10 & 16 give the shape factors of lens 1 and 2 respectively. Many calculations are possible (with some awkwardness) in the MFE. The operand “DMFS” (Default Merit Function Start) at the end makes sure that, if I add the default merit function (“Design – Sequential Merit Function”), it won’t overwrite my own operands.
Optimize the doublet shapes by opening the “Local Optimization” dialog box (“Tools-Design-Local Optimization”, or just the “Opt” tab at the top of the Zemax window). Before OptimizationAfter Optimization Lens shapes before optimization: Lens shapes after optimization:
The MFE: Spherical and Coma are now essentially zero: As also reported by the Seidel Plot:
While the TRA plot still shows some S.A., note the scale on the diagram --- 0.1 µm. The wavelength for this system was set at 0.55 µm. A report of 1/5 wave TRA is completely meaningless, as the ray trace assumptions are not valid at that scale. This is verified by the MTF plot, which shows no difference between the system (on-axis) and a perfect lens:
Alternate Combination of Powers Formula (How to add negative lenses and still have positive power) By following a ray entering the system of two lenses parallel to the axis, we were able to find the standard combination of powers formula by eliminating all references to ray variables: Assume K 1, K 2 are thin lenses: We would like to develop an equivalent combination of powers expression that is given in terms of the parallel input ray, as this will be useful in designing systems for aberration correction. K1K1 K2K2 f u 1 =0 u’ 1 =u 2 u’ 2 h1h1 h2h2 d1d1 n1n1 n’ 1 = n 2 n’ 2
Refract through first element: Since: Refract through second element: Substituting for K1K1 K2K2 f u 1 =0 u’ 1 =u 2 u’ 2 h1h1 h2h2 d1d1 We could replace the two lenses with their equivalent lens, then add a third lens and do the same analysis getting: And, by induction, for n lenses:
This combination of powers formula gives us some insight into optical design principles. For example, in the Cooke Triplet design, the negative element is positioned such that it subtracts less power from the combination than it’s intrinsic power: h1h1 h2h2 Remember that the ray heights referred to are those of a ray traced into the system parallel to the axis – the ray that is used to define the power of a combination of lenses. This allows the negative lens to cancel some of the aberrations of the positive lenses, without having too great an effect on the total power.
Returning to our corrected doublet: Suppose we select a range of wavelengths covering the visible and see how the system behaves:
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:
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.
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.
Dispersion and Achromats Zemax’s “Glass Map” (‘Analysis – Glass and Gradient Index – Glass Map’):
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)
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
From the image equation, we have: Dispersion 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 whos ratios sum to zero: where the K’s are implicitly calculated at the d wavelength.
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.