GEOMETRICAL OPTICS Paraxial approximation LINEAR OPERATIONS ON RAYS

GEOMETRICAL OPTICS Paraxial approximation: LINEAR OPERATIONS ON RAYS y a’ OPTICAL SYSTEM a y’ a’ A B C D = . y a y’

PROPAGATION Y+La a 1 L 0 1 STRAIGHT INTERFACE y a 1 0 n y a 2 POSITIVE LENS 1 0 - 1 1 f CURVED INTERFACE 1 n 1 - n 2 Rn 2 n 1 0 n 1 n 2 R

STRAIGHT INTERFACE 1 0 0 n 1 0 1 - n Rn 1 n n CURVED INTERFACE POSITIVE LENS 1 0 - 1 1 f 1 0 1 - n Rn 1 n 1 R

Y’ y L y’ = y + La n

IMPORTANT EXAMPLES Glass plate Thick lens Find the focal distance Find the magnification, transverse and longitudinal Find the image location – real of virtual The “equivalent” thin lens

f 1. 5 1 0 1 2 3

Neg lens f Concave mirror Neg lens

Neg lens f Concave mirror Si = -3/5 0 ai Angular magnification 2. 5 Matrix to image: Magnification: 2/5 1 0 5/2 2 y/5 y 2/5 = 1 Neg lens = a 0 5/2 y 0 1 si 1 0 1 1. 5 0 0 1 1 1 0 1 a 1. 5+2. 5 si 1+si = 2. 5 1 si 1. 5 1 1 a

Significance of Matrix to image: 2/5 1 0 5/2 y’ = (2/5) y a’= y + (5/2) a Inclination from the ray coming from the lens

Neg lens f Concave mirror new s 0 = 18/5 Si = -3/5 0 ai Angular magnification 2. 5 Matrix to image: Magnification: Neg lens 2/5 1 0 5/2 2 y/5 y 2/5 = 1 = a 0 5/2 y 0 1 si 1 0 1 1. 5 0 0 1 1 1 0 1 a 1. 5+2. 5 si 1+si = 2. 5 1 si 1. 5 1 1 a

Neg lens Si. S=i =18/13 -3/5 f Matrix to image: Magnification: Concave mirror S 0 = 18/5 Angular magnification -13/5 -5/13 0 -1 -13/5 -5 y/13 -y -5/13 0 = -1 -13/5 Neg lens 0 ai a y 0 = 0 1 18/5 0 -1 1 0 1 a 1 si 1 0 1 18/5 -13/5 si 1 -si = -13/5 -1 si 18/5 1 1 a

Neg lens Si S =i 18/13 = -21/34 f Concave mirror S 0 = 21/13 0 ai Angular magnification 34/13 Matrix to image: Magnification: 13/34 0 1 34/13 13 y/34 -y Neg lens a 13/34 0 = 1 34/13 = 1 si 1 0 1 21/13 0 0 1 1 0 1 a 1 21/13+34/13 si 1+si = 34/13 1 y 0 si 21/13 1 1 a

TOTAL FROM OBJECT TO FINAL IMAGE Neg lens f Matrix from object to image: Magnification: Neg lens Concave mirror 13/34 0 1 34/13 13/34 x (-5/13) x 2/5 -5/13 0 -1 -13/5 2/5 1 0 5/2

Meter stick – longitudinal image by a Positive lens f = 40 cm 60 cm 160 cm Meter stick Hint: in this particular case, make your life easier by choosing as unit f 60 -si/2 Si = 120 -1/2 160 -3 si -3 Si = 53. 33 0 ai = = 1 si 1 0 1 1/40 1 a 1 si 1 0 1 160 0 0 1 -1/40 1 a 60 0

Meter stick, image by a negative lens (3 -9) f = -40 cm 60 cm 160 cm Meter stick A B Hint: in this particular case, make your life easier by choosing as unit f 60+5 si/2 Si = -24 -1/2 160 -5 si -3 Si = -32 0 ai = = 1 si 1 0 1 1/40 1 a 1 si 1 0 1 160 0 0 1 1/40 1 a 60 0