OPTI 201R Homework 9 Solutions

Transcription

1 OPTI 20R Homework 9 Solutions. Unknown thick lens system measured with reciprocal magnification technique. At the first position where the object is sharply focused on the image plane m = 2. The lens needs to be shifted by d = 25 mm to bring the object into focus at the second position. The system is shown in Fig.. Position : Fig. z = z + f Equ. z = m z Equ. 2 so, z = ( m ) f Equ. 3 Position 2: z 2 = z 2 + f Equ. 4 z 2 = m 2 z 2 Equ. 5 so, z 2 = ( m 2 ) f Equ. 6 In addition: z z + z PP = z 2 z 2 + z PP which is: z z = (m )z = z 2 z 2 = (m 2 )z 2 Equ. 7 Hence: (m m 2 )(m m 2 ) m m 2 f = 0

2 Here the thick lens with finite focal length f operates at finite conjugates with a fixed object to image distance, so m and m 2 have non-zero values. Apparently, m and m 2 are not equal, for the above equations to hold, must be satisfied. m m 2 = 0 Equ. 8 Finally: z d = z 2 Equ. 9 which is: so: ( m ) f d = ( m 2 ) f f( m m 2 ) = d substituting in Equ. 0: f = d = m d m m m 2 Equ. 0 a) From Equ. 8, magnification in the second position m 2 = m = 0.5. b) From Equ. 0, focal length f = m d m 2 = 2 25 mm 4 = 50 3 mm = mm c) From Equ. 3, z = ( m ) f = d +m = 25 mm. From Equ. 2, z = m z = m d +m = 50 mm. d) The thick lens is used in air, with the index of refractions of both side of the lens being the same. Therefore, the nodal points coincides with the principle points of the thick lens, which is sketched in Fig.. e) Object image separation z oi = z z + z PP = 50 mm + 25 mm + 0 mm = 85 mm. 2

3 2. A reverse telephoto lens with two thin lenses with powers Φ = mm and Φ 2 = mm, as shown in Fig. 2 a) Gaussian reduction Fig. 2 Φ = Φ + Φ 2 Φ Φ 2 t Equ. d = Φ t Equ. 2 Φ BFD = d + f R = d + f = Φ t + = Φ t Φ Φ Φ Therefore: Φ t = ΦBFD, substitute this relation into Equ. : Φ = Φ + Φ 2 Φ Φ 2 t = Φ + Φ 2 Φ 2 ( ΦBFD) = Φ + ΦΦ 2 BFD Hence: Φ( Φ 2 BFD) = Φ The total power of the zoom lens is given by: Equ. 3 b) Φ = Φ Φ 2 BFD Gaussian reduction Φ = Φ + Φ 2 Φ Φ 2 t Equ. Therefore: t = Φ +Φ 2 Φ Φ Φ 2 3

4 c) BFD (mm) Φ (mm ) t (mm) z = t BFD (mm) z 2 = BFD (mm) f R (mm) d) Plotted in MATLAB: 4

5 Use ynu raytracing and not Gaussian reduction to get the results. Credit will not be given for Gaussian reduction. Obviously, your answers should be identical between the two techniques, so you can use Gaussian reduction to check your answers. Turn in a copy of your spreadsheets and clearly show how you calculated the various values from the elements of the spreadsheet. 3. The parameters of a Hastings triplet are shown in the figure below. (a) Use ynu raytracing to find the location of all six cardinal points, as well as the BFD and FFD Include your raytracing worksheets and show all work. To find the rear focal point, set the height of this ray to zero and solve for tt tt = 0 tt 4 = mmmm = BBBBBB The rear focal length is determined by the intersection of the incident and emerging ray. From the raytrace From BBBBBB = ff RR + dd uu 4 = yy ff ff RR = = mmmm RR Page

6 dd = = mmmm By symmetry, the locations of the front cardinal points mirror the rear cardinal points, so dd = mmmm. The nodal points are located at the principal planes. (b) Answer the following: What is the front focal length of the system? ff FF = mmmm What is the rear focal length of the system? ff RR = mmmm What is the total power of the system? φφ = = mmmm For an object located 25 mm to the left of the front principal plane, where is the image formed and what is its magnification? zz 25 = zz = mmmm mm = = 2.63 Page 2

7 4. A Gregorian telescope has two reflective mirrors. The first mirror has a radius of curvature of R = 500 mm. The second mirror lies to the left of the first mirror such that t = 50 mm. The radius of curvature of the second mirror is R2 = 25 mm. (a) Sketch the telescope and label the vertices, V and V2. (b) Use ynu raytracing to trace a ray with y 50 mm and u 0. Where is the rear principal plane located with respect to the vertex V2? = = To find the rear focal point, set the height of this ray to zero and solve for tt tt = 0 tt 2 = mmmm = BBBBBB The rear focal length is determined by the intersection of the incident and emerging ray. From the raytrace From BBBBBB = ff RR + dd uu 2 = yy ff ff RR = 50 = mmmm RR 0.52 dd = = mmmm Page 3

8 (c) Where is the rear focal point located with respect to vertex V2? tt 2 = mmmm = BBBBBB 5. A system consists of two thin lenses separated by a distance of 70 mm. The focal length of the first lens is 3.75 mm and the focal length of the second lens is 0 mm. An object is located 4 mm to the left of the first lens. The object has a height of 0.5 mm. Using ynu raytracing, trace the following to rays through the system. (a) The a-ray has a height y a0 = 0 at the object and a slope u a = 0 25 incident on the first surface. See below.. (b) The b-ray has a height y b 0 = 0. 5 mm at the object and a slope u b = incident on the first surface. (c) Calculate the optical invariant at each surface to double check you work. Skip this since we didn t discuss (d) What is the slope u a2 of a-ray leaving the second lens? The slope is zero. (e) Where does the b-ray cross the optical axis after the second lens? The ray crosses.667 mm to the right of the second lens. (f) What is the ratio u u b2 b? This is known as the angular magnification and is equal to 6 in this case. Page 4

9 (g) Sketch the system and the locations of the two rays as they propagate through the system. Page 5