A6525 Fall 2015 Solutions to Problem Set #2. This is the case of a single plano-convex lens. The specifications are:

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1 A655 Fall 05 Solutions to Problem Set # Problem : This is the case o a single plano-convex lens. The speciications are: Focal length ~ 5 cm Diameter D = 0 cm Index o reraction n =. Size o aperture stop s = 9 cm Thickness o lens t = cm a) From the lensmaker s equation = ( n ) r r with r, r = ( n ) Thus, r = -. 5 = 6.5 cm b) Do a system plot and spot diagram with ray trace program. BFD = 3.66 cm Figure -: Let is system plot o the p-c lens. On the right is a spot diagram. Full scale is cm. File is: pc with conic surace ( = 5).ray c) rms (on-axis) = 0.56 cm, rms ( deg. o-axis) = d) Vary eccentricity to minimize the rms spot size. Eccentricity =.0 rms (on-axis) = 0.0 cm Best case - see Figure - below. rms ( deg. o-axis) = 0.09 cm A PS-

2 PC with conic surace Sm_RMS EC_03 Figure -: Optimization plot o smallest rms vs. eccentricity o the curved surace o the plano-convex lens. Dierent symbols represent dierent distances o-axis (rom 0 to.0 degree). The best eccentricity is.. e) Show spot diagrams or the eccentricity optimized case. Figure -3: Spot diagram or p-c lens with e =.. The source is one degree o-axis. The on-axis case gives a perect image. The ull scale is 0.5 cm. A PS-

3 ) Do the same calculations or a convex plano-lens. Figure -4: Let is system plot o the c-p lens. On the right is a spot diagram. Full scale is 0.5 cm. File is: cp with conic surace ( = 5).ray rms (on-axis) = cm, rms ( deg. o-axis) = Vary eccentricity to minimize the rms spot size. Eccentricity =.09 rms (on-axis) = cm Best case - see Figure -5 below. rms ( deg. o-axis) = cm CP with conic surace 0.0 Sm_RMS EC_0 Figure -5: Optimization plot o smallest rms vs. eccentricity o the curved surace o A PS-3

4 the convex-plano lens. Dierent symbols represent dierent distances o-axis (rom 0 to.0 degree). The best eccentricity is.. Show spot diagrams or the eccentricity optimized case. Figure -6: Spot diagram or c-p lens with e =.09. The source is one degree o-axis. The onaxis case gives a near perect image. The ull scale is 0.05 cm. g) Which case is better? Clearly the C-P lens is better both with and w/o optimization. Table a summarizes the cases below. Table a: Summary o P-C/C-P comparison Case Eccentricity Rms spot size θ = 0 θ = P-C P-C C-P C-P A PS-4

5 Problem : A summary o results is given in Table a: Table a: Ray-trace results or double lens system First Lens Second Lens rms spot size Case r e r e r e r e θ = 0 θ = a Cases:. As designed rom two-lens, thin lens, and lensmakers equations.. Did a hand search starting with both eccentricities equal to Bent lenses to reduce spot size 4. Same as case 3 but optimized by varying eccentricity o suraces o second lens a. Did a grid search with variation window Details o the Solution: For a two lens system separated by a distance d, the object a distance o rom the irst (letmost) vertex and the image a distance i rom the rightmost vertex, we have i o d( + o) o o + ( d)( + o) + d d = = d d Where the irst limit is or the object at minus ininity and the second limit has the two ocal lengths set equal. We can solve the above equation or the ocal length, = i + d / + ( i + d / ) id The principal plane will be located at the stop because o the symmetry o the system so the ocal length,, o the doublet will be i+d/. Putting this into the above equation yields the ocal length o the individual lenses in terms o the total ocal length and lens separation. = + ( d / ) d Since d = cm and we want = 5 cm, = 9.04 cm. For a plano-convex lens the lensmakers equation simpliies r = ( n ) or the curved surace and so or n =., we have r = 3.9 cm. A PS-5

6 Case : I we used the back ocal distance, i = 5 cm instead o we get = 3.03 cm and r = 34.4 cm. As per the instructions in the problem, we will us these latter numbers. The results o a trace with 000 rays are given in Table a and spot diagrams or on-axis and one degree o-axis are shown below. The back ocal distance is 3.39 cm. Figure -: A diagram o the starting coniguration o the doublet system. Figure -: Spot diagrams or Case (lenses are plano-convex with spherical suraces). Let: On-axis, right: one degree o-axis. Full scale is 0.5 cm. A PS-6

7 Case : Using the variation window, we now search or a better spot size by varying the eccentricity o suraces and 4. The results are given in Table - (Case ). Note that there is a partial degeneracy between the two eccentricities. Spot diagrams are given in Figure -3. The spherical aberration is basically gone and residual coma is evident. Figure -3: Spot diagrams or Case (lenses are plano-convex with conic suraces). Let: On-axis, right: one degree o-axis. Full scale is 0. cm. Case 3: Now we try optimizing by bending the lenses (starting rom Case ). Initially the search was done with bending to two lenses (surace & together and surace 4 and 5 together) rom -0.4 to 0.4 (cm - ) about the nominal (Case ) values. The search space was then narrowed. The spot diagrams are shown in Figure -4 and the results rms spot sizes are list in Table -. A system plot is shown in Figure -5. Figure -4: Spot diagrams or Case 3 (lenses are bent with spherical suraces). Let: Onaxis, right: one degree o-axis. Full scale is 0.05 cm. A PS-7

8 Figure -5: Schematic o optical system ater bending lenses and to minimize the rms spot size (Case 4). Case 4: We now vary the eccentricity o the suraces o the second lens. This basically can be used to get rid o spherical aberration. Figure -6 shows the resulting spot diagrams. There is still residual coma in the system. There is also signiicant ield curvature and we have not worried about chromatic aberrations, so this system still has a ways to go. Note that the rms spot size or degree o-axis is about 0.9 arcminutes. Not great but much better than the original doublet. Figure -6: Spot diagrams or Case 4 (lenses are bent with conic suraces). The source is one degree o-axis. The on-axis case gives a near perect image. Full scale is 0.05 cm. A PS-8

9 0.8 Optimized doublet (bending & ecc <> 0) 0.6 EC_ EC_04 Figure -7: Optimization plot o smallest rms vs. eccentricity suraces 4 and 5 or Case 4. The smallest rms is e = 0.7 or surace 4 and e = 0.0 or surace 5. This optimization is done or an on-axis source. There is not much dierence or a source one degree oaxis. Ray-trace iles used: File pc with conic surace ( = 5).ray cp with conic surace ( = 5).ray Description Plano-convex lens with eccentricity optimization Same or convex-plano lens doublet (pc-cp, conics) ( = 5).ray Case, set ecc = 0 to get case doublet - optimized ( = 5).ray Case 4, set ecc = 0 to get case 3 A PS-9