1

Paraxial raytracing: Refraction and Translation

2

Sign Conventions: • angles

• radii

• distances

u

CC

r

y

t

3

u (-)

Sign Conventions

Angles

CCW from axis (+)

CW from axis (-)

u (+)

4

Sign Conventions

Radii

CC to left of surface: r (-)

CC

r (-)

CC

r (+)

CC to right of surface: r (+)

5

Sign Conventions

Distance

Above axis: y (+)

y (-)

Below axis: y (-)

Above axis: y (+)

y (+)

6

Sign Conventions

Distance

Left to right: t (+)

t (+)

7

n'n

θ

1. Refraction at an interface

θ’

8

α

θ’

n'n

u’

αu’

θ’ = (u’ – α)

u

θα

θ = (u – α)

n sin(θ) = n’ sin(θ’) n sin(u - α) = n’ sin(u’ - α) For small angles: sin(θ) ~ θ n (u - α) = n’ (u’ - α) nu - nα = n’u’ – n’α nu + n’α - nα = n’u’

1. Refraction at an interface

u

n’u’ = nu + α (n’ – n)

(note: α is neg as drawn)

9

α

1. Refraction at an interface

n’u’ = nu + α (n’ – n)

ry

n'n

10

α

1. Refraction at an interface

n’u’ = nu + α (n’ – n)

ry

n’u’ = nu - (n’ – n) y r

For small α: α = - y r n'

n

n’u’ = nu - y (n’ – n) r

(n’ – n) r

= optical power[ ]

sin(α) = -y/r

α = sin-1(-y/r)

15

n’ > n

r (+)

(n’ – n) r

= optical powern’ < n

r (-)

n’ < n

r (+)

n’ > n

r (-)

(+)(n’- n) > 0 r > 0

(+)(n’- n) < 0 r < 0

(-)(n’- n) < 0 r > 0

(-)(n’- n) > 0 r < 0

16

u’

u

1. Refraction at an interface

u

n’u’ = nu - y (n’ – n) r

nu n’u’

y

(n’ – n) r

17

2. Translation between two surfaces

n'

18

2. Translation between two surfaces

n'

u’y`

y

angle u’ is constant; height y y’

u’

t

19

2. Translation between two surfaces

n'

u’

y

y’ = y + t • u’

t

Since we’re carrying around index-angle product n’u’:

y’

=

y + t • u’ = y + n’u’

tn’

y`

y’ = y + n’u’ tn’

20

y’

=

y + t • u’ = y + n’u’

tn’

2. Translation

1. Refraction n’u’ = nu - y (n’ – n) r

n'

u’y`

Δy

y y

t

u

n

r

21

u = 0

n = 1 n = 1.4

t = 7 t = 2

n = 1.5

y = 4

r = 10 r = -6 r = -20

n = 1

Find: back focal length (BFL); distance from last surface to F’