Optics for Engineers Chapter 3

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1 Optics for Engineers Chapter 3 Charles A. DiMarzio Northeastern University July 2012

2 Compound Lens and Ray Definitions Correct Ray Ray Definition Vertex Planes Translation Matrix Optics Ray Refraction July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 1

3 Ray Definitions Ray Information Straight Line Two Dimensions Slope and Intercept Mathematical Formulation Column Vector Two Elements Intercept on Top Reference to Local z Angle on Bottom x V = α Some Books Differ Arbitrary Operation A B M = C D V end = M start:end V start Subscript for Vertex Number July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 2

4 Translation From One Surface to the Next Move Away from Source z 1 to z 2 V 2 = T 12 V 1 Angle Stays Constant α 2 = α 1 Height Changes x 2 = x 1 + α 1 z 12 Matrix Form x2 α 2 = 1 z x1 α 1 T 12 = 1 z July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 3

5 Refraction at a Surface 1 Matrix Form V 1 = R 1V 1 Height Does Not Change x 1 = 1 x α 1 x 1 α 1 = 1 0 x1?? α 1 Angle Changes Ch. 2 θ = γ + α θ = γ β = γ + α tan α = p tan β = p s + δ s δ tan γ = p r δ July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 4

6 Refraction at a Surface 2 Height Does Not Change x x1 =?? α 1 α 1 Angle See Prev. Page nθ = θ n γ + α = γ + α, n x r + nα = n x r + n α. R = 1 0 n r n α = n n r x + n α. R = 1 0 July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 5 P n

7 Cascading Matrices V 1 = T 01 V 0 V 1 = R 1 V 1 V 2 = T 12 V 1 etc. V end = M 0:end V 0 M 0:end = T end 1:end... T 12 R 1 T 01 Multiply from Right to Left as Light Moves from Left to Right. July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 6

8 The Simple Lens 1 First Surface V 1 = R 1V 1 Translation V 2 = T 12 V 1 Second Surface V 2 = R 2V 2 Result V 2 = LV 1 L = R 2 T 12 R 1 July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 7

9 The Simple Lens 2 From Previous Page L = R 2 T 12 R z L = 0 1 P P 1 1 n 1 1 n2 = 1 Strange but Useful Grouping 1 0 L = + z 12 P1 n 1 1 P t 2 n 1 2 P 1 P 2 2 P 2 n 1 2 P t = P 1 + P 2 Initial: n 1 = n, Final: 2 = n, Lens: 1 = n l 1 0 L = + z 12 P1 n n l P t n P 1 P 2 P 2 n n l implict in P 1 and P 2, and thus P t July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 8

10 The Thin Lens 1 The Simple Lens Previous Page 1 0 L = + z 12 P1 n n l P t n P 1 P 2 P 2 n Geometric Thickness, z 12 /n l, Multiples Second Term Set z 12 0 L = 1 0 P n Thin Lens P = P t = P 1 + P 2 Correction Term Vanishes July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 9

11 The Thin Lens 2 Thin Lens in terms of Focal Lengths L = = P n 1 f n = 1 0 n f n Front Focal Length: f = F F L, Back: f = BF L Special but Common Case: Thin Lens in Air L = 1 0 P 1 = f 1 Thin Lens in Air f = f = 1 P F F L = BF L Always True if = n July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 10

12 General Problems and the ABCD Matrix General Equation xend m11 m 12 xstart V end = M start:end V start α end = m 21 m 22 α start Determinant Condition Not Completely Obvious det M = n det M = m 11 m 22 m 12 m 21 Abbe Sine Invariant or Helmholz or Lagrange Invariant x dα = nxdα July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 11

13 Abbe Sine Invariant Equation x dα = nxdα Alternative Derivations Geometric Optics Energy Conservation C. 12 Lens Example Height Decreases by s /s Angle Increases by s/s Example: IR Detector Diameter D = 100µm Collection Cone F OV 1/2 = 30 Telescope Front Lens Diameter D = 20cm Max. Field of View F OV 1/2 = m m = July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 12

14 Principal Planes Concept 1 Thin Lens L = 1 0 P n Simple Equation Easy Visualization High School Optics Good First Try Arbitrary Lens Vertex to Vertex m11 m 12 M V V = m 21 m 22 Possible Simplification M HH = T V H M V V T HV M HH = L Will It Work? July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 13

15 Principal Planes Concept 2 Convert a Hard Problem to a Simple one M HH = T V H M V V T HV M HH = L Useful if a Solution Can Be Found Very Useful if h and h Are Not Too Large July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 14

16 Finding the Principal Planes 1 0 P L = M HH = T V H M V V T HV h m11 m P n = 12 1 h 0 1 m 21 m n = m11 + m 21 h m 11 h + m 12 + m 21 hh + m 22 h m 21 m 21 h + m 22 m 11 + m 21 h = 1 m 11 h + m 12 + m 21 hh + m 22 h = 0 m 21 = P m 21 h + m 22 = n Three Unknowns: Solution if Determinant Condition Satisfied h = n m 22 m 21 Determinant Condition? Yes! P = m 21 h = 1 m 11 m 21 No Assumptions Were Made About M: This Always Works. July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 15

17 Principal Planes Principal Planes are Conjugates of Each Other m 12 = 0 xh 1 0 xh = α H P Unit Magnification Between Them x H = x H n α H Note: Principal Planes May Not Be Accessible July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 16

18 Imaging We Know The Answer Matrix from Object to Image M SS = M H S M HH M SH = T s M HH T s Conjugate Planes x =? x + 0 α M SS = m11 0 m 21 m 22 July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 17

19 Imaging Equation for Compound Lens M SS = 1 s P n 1 s 0 1 = 1 s P s ss P + s n P sp + n Conjugate Plane Rule: m 12 = 0 s ss P + s n = 0 n s + n s = P Measure s and s from h and h respectively. July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 18

20 Compound Lens Matrix Results Magnifications mm α = /n m = 1 s P = 1 s n s + n s = ns s Imaging Matrix m α = s n s + n s + n = s s M SS = m 0 P n m 1 July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 19

21 Thick Lens Thick Lens Equation 1 0 L = P t n + z 12 n l P1 n P 1 P 2 P n 2 Power: P = m 21 P = P 1 + P 2 z 12 n l P 1 P 2 f = n P f = n P Principal Planes h = n n l P 2 P z 12 h = n n l P 1 P z 12 July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 20

22 Thick Lens in Air: The Thirds Rule for Principal Planes Principal Planes and Focal Length f = f = 1 P h = 1 n l P 2 P z 12 h = 1 n l P 1 P z 12 Principal Plane Spacing z HH = z 12 + h + h = z 12 1 P 2 + P 1 n l P P P 1 + P 2 z HH = z 12 + h + h z n l Glass n l 1.5 z HH = z 12 3 July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 21

23 Special Cases h = n n l P 2 P z 12 h = n n l P 1 P z 12 h,h Negative if P,P 1,P 2 Have Same Signs Often True h = 0 if P 2 = 0 Convex Plano or Concave Plano h = 0 if P 1 = 0 Plano Convex or Plano Concave h = h if P 2 = P 1 Biconvex or Biconcave in Air and = n July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 22

24 Example: Biconvex Lens in Air P 1 + P 2 = 10diopters, or f = 10cm Solid=Vertices, Dashed=Principal Planes, Dash Dot=Focal Planes 5 4 z 12, Lens Thickness Locations July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 23

25 Bending the Lens P 1 + P 2 = 10diopters, or f = 10cm Solid=Vertices, Dashed=Principal Planes, Dash Dot=Focal Planes Note Meniscus Lenses in Germanium r 1, Radius of Curvature Inf Locations r 1, Radius of Curvature Inf Locations A. Glass B. Germanium July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 24

26 Example: Compound Lens Matrix Two Thin Lenses M V1,V 2 = L V 2,V 2 T V 1,V 2 L V 1,V 1 Thin Lenses z M V1,V 2 = f f 1 1 M V1,V 2 = 1 z 12 f z 1 12 f 1 + z 12 2 f 1 f 1 2 f 1 z 12 1 f 2 July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 25

27 Compound Lens Results Focal Length Powers add for small separation 1 f = 1 f f 1 z 12 f 1 f 2 Principal Planes h = h = z 12 f 2 1 f 2 + z 12 f 1 f 2 1 f 1 = z 12 f 1 1 f 2 + z 12 f 1 f 2 1 f 1 = z 12 f 1 z 12 f 1 f 2 z 12 f 2 z 12 f 1 f 2 h 0 and h 0 if z 12 0 July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 26

28 Special Case: Afocal z 12 = f 1 + f 2 1 f = f 1 f 1 f 2 + f 2 f 1 f 2 z 12 f 1 f 2 = 0. m 21 = 0, 1 f = 0, or f Afocal h h Principal Planes are not Very Useful Here. July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 27

29 Example: 2X Magnifier 1 We Know How to Do This Object at Front Focus of First Lens Intermediate Image at Infinity Final Image at Back Focus of Second Lens But Let s Use Matrix Optics for the Exercise July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 28

30 Example: 2X Magnifier 2 Lens Vendor Data: Glass=BK7 n = at λ = 633nm Parameter Label Value First Lens Focal Length f mm First Lens Front Radius LA1509 Reversed r 1 Infinite First Lens Thickness z v1,v1 3.6 mm First Lens Back Radius r mm First Lens Back Focal Length f 1 + h mm Lens Spacing z v1,v2 20 mm Second Lens Focal Length f mm Second Lens Front Radius LA1708 r mm Second Lens Thickness z v2,v2 2.8 mm Second Lens Back Radius r 2 Infinite Second Lens Back Focal Length f 2 + h mm July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 29

31 Example: 2X Magnifier Thin Lens Approximation 1 f = 1 100mm mm 20mm 100mm 200mm f = 71.43mm h = h = 20mm 100mm 20mm 100mm 200mm = 7.14mm 20mm 200mm 20mm 100mm 200mm = 14.28mm m s s = 2 s = 2s, and 1 f = 1 s + 1 s = 1 s + 1 2s s = 3f/2 = 107.1mm s = 3f = 214.3mm July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 30

32 Lens Thickness Effects Start with Equations for Thin Lenses Use Principal Planes in Place of Vertices M H1,H 2 = L H 2,H 2 T H 1,H 2 L H 1,H 1 Same Equation as Thin Lens but Different Meaning z M H1,H 2 = f f 1 1 f 1 from H 1 and f 1 from H 1 f 2 from H 2 and f 2 from H 2 z 12 from H 1 to H 2 July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 31

33 2X Magnifier, Revisited 1 Principal Planes h = 20mm 100mm 20mm 100mm 200mm = 7.14mm H 1 to H h = 20mm 200mm 20mm 100mm 200mm = 14.28mm H 2 to H Spacing See Next Page 0.713mm Object and Image Distances s = 3f 2 = 107.1mm s = 3f = 214.3mm July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 32

34 2X Magnifier, Revisited 2 July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 33

35 A Suggestion: Global Coordinates Notation: zh1 First Letter: z The Remaining Characters: Plane Name eg. H1 Need to Set One Plane as z = 0 Example from the Magnifier z = 0 at First Vertex zh1 = h 1 zh = zh1 h = h 1 h Text Error: Not z H = zh1 h = h 1 h etc. July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 34

36 Telescopes 1 Afocal Condition 1 f = 1 f f 1 z 12 f 1 f 2 = 0 if z 12 = f 1 + f 2 Vertex Matrix M V1,V 2 = 1 f 1+f 2 f 1 f 1 + f 2 1 f 2 + f 1+f 2 f 1 f 2 1 f 1 1 f 1+f 2 f 2 = f 2 f1 f 1 + f 2 0 f 1 f2 Imaging Matrix 1 s M SS = 0 1 f 2 f1 f 1 + f 2 1 s 0 f f2 =? 0?? July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 35

37 Telescopes 2 M SS = f 2 f1 s f 2 f1 + f 1 + f 2 s f 1 f2 0 f 1 f2 =? 0?? s f 2 f 1 + f 1 + f 2 s f 1 f 2 = 0 m = f 2 /f 1, Afocal ms + f 1 + f 2 + s /m = 0 s = m 2 s f 1 m 1 + m M SS = s m 2 s m m s July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 36

38 Astronomical Telescope Magnification: Image is smaller 1 m = f 2 f1 But a Lot Closer: m z = m 2 Angular Magnification is Large m α = 1/m July 2012 c C. DiMarzio Based on Optics for Engineers, CRC Press slides3 37

Optics for Engineers Chapter 3

Optics for Engineers Chapter 3 Charles A. DiMarzio Northeastern University Jan. 2014 Chapter Overview Thin Lens 1 s + 1 s = 1 f Thick Lens What are s, s, f? Is this equation still valid? Thin Lens Ch.