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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. There will be a homework assignment posted today, as well as several Zemax files which may be helpful.

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**Right Side Rear-View Mirror**

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. θ’ W θ R L2 L1 W Driver The engineering design task is: Given the size constraints of the mirrors, W, and the geometry of the car layout, L1 , L2 ; Find R such that θ = θ’. (i.e., make the driver’s field of view equal on both sides.)

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**Other Issues with Convex Mirrors**

l l’ C f’ Observer Or, in other words: 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. Automobiles have been using convex right-side rear-view mirrors for ~100 years. The warning has been required for only about the last 20 years or so. What is remarkable, is that this warning usually can’t be read by the driver, but only by the passenger. Exactly what the logic behind this is, is unclear: Is the passenger expected to warn the driver? What if there is no passenger? Are people less aware of this common optical phenomena today than they were 100 years ago? While this is not specifically an engineering issue, it might be prudent to find out why this warning is required before doing any significant work for an automobile company!

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**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<-1 X=-1 X=0 X=1 X>1 c1 c2 For the lens above with shape factor, X=-1, the Zemax LDE is:

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**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.

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**Shape Factor and Aberration**

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:

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**The list goes on for several pages**

The list goes on for several pages. Clicking on an operand brings up the definition and instructions for use.

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**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.

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**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.)

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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:

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**We can then update the MFE and see what aberrations Zemax reports:**

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Using our Merit Function which reports 3rd 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: !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 !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 By “early Basic”, I mean 1980’s! Forget about debuggers, looking at variables changing, step-by-step execution etc, etc. The only debuging tools available are your pencil and paper (pretend to be the computer), and copuous numbers of PRINT statements. Note that there are only 20 active statements in this macro – 20 statements are only comments.

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**The output of our “ThirdOrd” macro is a text window:**

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.

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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

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Expanded View: As can be seen from the close-up, both S.A. and Coma are minimized at a Shape Factor of ~0.7

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**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= , R2= , Shape Factor = ): This is known as a “Best Form” Singlet.

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**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. 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. Aberration in Waves 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.

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**Spherical Aberration Corrected Doublet**

Let’s start with a pair of symmetrical lenses: A positive lens with K = (f=50 mm) A negative lens with K = (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:

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**Here is the LDE for the starting system:**

And here are the curvature solves on the second surface of each element:

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**Here is the 3rd 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.

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**Here are the TRA plot and the Wavefront Map for the on-axis field:**

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**Here is the Merit Function Editor (MFE)**

Any line in the MFE can be used as an optimization control (i.e., added to the merit value) by filling in the “Target” and “Weight” columns. 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.

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**Lens shapes before optimization: Lens shapes after optimization:**

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 Optimization After Optimization Lens shapes before optimization: Lens shapes after optimization:

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**The MFE: Spherical and Coma are now essentially zero:**

As also reported by the Seidel Plot:

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**While the TRA plot still shows some S. A**

While the TRA plot still shows some S.A., note the scale on the diagram 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:

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**(How to add negative lenses and still have positive power)**

Alternate Combination of Powers Formula (How to add negative lenses and still have positive power) K1 K2 f u1 =0 u’1=u2 u’2 h1 h2 d1 Assume K1, K2 are thin lenses: n’2 n1 n’1 = n2 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: We’ve seen that negative lenses have the oppposite 3rd order aberrations of positive lenses, so a key method of correcting these aberrations is to add negative lenses and surfaces to the system. Most lens systems, however, need to have net positive power, so we would like to see how to add significant negative power without destroying the net positive power required. 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.

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**And, by induction, for n lenses:**

Refract through first element: K1 K2 f u1 =0 u’1=u2 u’2 h1 h2 d1 Refract through first element: Since: Refract through second element: Substituting for We could replace the two lenses with their equivalent lens, then add a third lens and do the same analysis getting: Note the input ray must be parallel to the axis in order to recover the net focal length. This is because of the property of lenses (and systems of lenses) that they will focus an object at infinity at the lens’ focal point. And, by induction, for n lenses:

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**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: 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. h1 h2 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.

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**Returning to our corrected doublet:**

Suppose we select a range of wavelengths covering the visible and see how the system behaves:

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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:

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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.

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**Here is the Seidel (3rd order) aberration plot**

Here is the Seidel (3rd 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.

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**Dispersion and Achromats**

Zemax’s “Glass Map” (‘Analysis – Glass and Gradient Index – Glass Map’):

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**Dispersion nF = n(0.486µm) (blue) nd = n(0.588µm) (green)**

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. F (hydrogen) ≈ 486nm d (helium) ≈ 588 nm C (hydrogen)≈656 nm The corresponding indices of refraction (for a given glass type) at these wavelengths are tradionally labeled: nF = n(0.486µm) (blue) nd = n(0.588µm) (green) nC = n((0.656µm) (red)

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Dispersion In the “glass map”, the index of refraction is the index at the d line (nd), 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:

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**Dispersion From the image equation, we have:**

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.

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**Achromats For two thin lenses in contact:**

The lenses chromatic focal shifts cancel each other. The condition that the combination have zero focal shift between the F and C wavelengths is: 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.

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