OPTI 45/55 Midterm Due: Februr, Grdute Students do ll problems. Undergrdute students choose three problems.. Google Erth is improving the resolution of its globl mps with dt from the SPOT5 stellite. The stellite orbits t n ltitude 8 km bove the Erth s surfce. Assume its cmer contins / CCD with sensor size of 6.4 4.8 mm nd piel dimensions of 48 56. Ech piel on the sensor corresponds to.5 m ptch on the ground. You cn ssume thin lens sstem. Answer the following: ht re the dimensions of the piels? The piels re 6.4 mm / 48 b 4.8 mm / 56, which is equivlent to. μm squre piels. ht mgnifiction is needed? The.5 m ptch on the ground corresponds to the. μm on the sensor, so the mgnifiction is given b /, where is the imge height nd is the object height. In this cse, the mgnifiction. -6 /.5.4-6 ht is the imge distnce for the stellite cmer? The mgnifiction is lso equl to the rtio of the imge distnce so the object distnce, so the imge distnce is given b.4-6 (8, m). m. ht is the focl length of the stellite cmer? The Gussin imging equtions tells us tht Effectivel, the object is t infinit. f.m.. 8 f
If we wnt to mke the diffrction spot dimeter mtch the piel size, wht is the size of the entrnce pupil? (Assume λ.5 μm) The diffrction limited spot size is given b Spot Size.44λ f D.μm Solving for D gives 4 mm. Technicll, the working F/# should be used here, but the difference between the F/# nd orking F/# is miniscule here since m is so smll.. Suppose we mesured n unknown thick lens sstem with the reciprocl mgnifiction technique. Using the technique, we determined tht the distnce from the rer principl plne to the imge plne ws mm. e lso determined tht the distnce from the front principl plne to the object plne ws -5 mm. Answer the following questions: ht is the mgnifiction of the lens? The object nd imge distnces re -5 mm nd mm respectivel. The mgnifiction is the rtio of the imge to object distnce or -4. ht is the focl length of the lens? The Gussin imging equtions tells us tht f mm. 5 f How fr do we need to move the lens to hve reciprocl mgnifiction for the sme object nd imge plnes?
Here, there re multiple routes to get the nswer, but bsicll use one of the equtions on pge 5 of the notes nd solve for the distnce d the lens needs to be moved. In terms of the object distnce l : d l d 5 m ( 4) 75mm here re the nodl points locted? Since the lens is in ir, the nodl points coincide with the principl plnes.. The wvefront coefficients for n opticl sstem with n eit pupil dimeter is mm nd the reference sphere rdius is mm re 4 μm, 6.6 μm,.4 μm. Do the following: rite n epression for the wvefront error. 4 ( h,, ψ).. 66h. 4h cos ( ψ) rite n epression for the trnsverse r error. (.( ).h ) (. ( ). h. 68h ) Sketch the trnsverse r error in the cse where h (i.e. on-is). ( ). ( ).
Sme for direction. Sketch the trnsverse r error in the cse where h (i.e. full field) long the nd directions. ( ).. ( ).. 68. 4. In clss, we sid for non-rottionll smmetric sstems, the Seidel berrtions generlize to terms on is. Suppose we hve wvefront (, θ) θ cos Rewrite this wvefront in terms of Zernike polnomils such tht (, θ) Z (, θ) Z (, θ) Z (, θ) Z (, θ) Z (, θ ) ht re the coefficients,,,, in terms of nd?
There re two ws to solve for this problem. First, equte the epressions nd replce cos θ [ cos θ]. cosθ Z (, θ) Z (, θ) Z (, θ) Z (, θ) Z (, θ) Substitute the definitions of ech of the Zernike terms into the right hnd side.. cos θ cos θ ( ) 6 cos θ 8( ) cos θ Collect like terms ( ) ( 8 ) cosθ 6 cosθ 8 θ cosθ cos Compring both sides gives series of simultneous equtions ( ) ( 8 ) 6 8 From inspection, this gives 6
Alterntivel, we cn use the orthogonl properties of the Zernike polnomils to clculte ech of the coefficients vi nm m (, θ) Z (, θ) ddθ n So, 8 ( cos ) ( cos ) 6 ( cos θ)( ) ( cos ) θ ddθ 4 ( cos θ)( ) cosθddθ θ cosθddθ ddθ θ cosθddθ 6 For wht vlues of θ is (,θ) mimum when nd >? For nd >, the peks occurs when cos θ. This occurs when θ, ± Note, the following trig reltionship m be useful: cos θ [ cos θ]. Also, the first few Zernike polnomils re given b