Optical bench http://webphysics.davidson.edu/Applets/optics4/default.html

1 Phys 322 Lecture 17 Chapter 6

More on geometrical optics

Thick lenses Analytical ray tracing Aberrations (optional) Thick lens: terms

First focal point (f.f.l.) Primary principal plane First principal point

Second focal point (b.f.l.) Secondary principal plane Second principal point

Nodal points

If media on both sides has the same n, then:

N1=H1 and N2=H2 Fo Fi H1 H2 N1 N2 - cardinal points Thick lens: principal planes

Principal planes can lay outside the lens: Thick lens: equations Note: in air (n=1)

1 1 1 x x  f 2 1  1 1 n 1d    o i effective  n 1   l l s s f l   o i focal length: f  R1 R2 nl R1R2 

f nl 1dl f nl 1dl Principal planes: h1   h2   nl R2 nl R1

yi si xi f Magnification: M T        yo so f xo Thick lens: example Find the image distance for an object positioned 30 cm from the vertex of a double convex lens having radii 20 cm and 40 cm, a

thickness of 1 cm and nl=1.5 1 1 1   so si f f f

30 cm so si 1  1 1 n 1d   1 1 0.51  1  n 1   l l  0.5   l     f  R1 R2 nl R1R2  20  40 1.5 20 40 cm f  26.8 cm s  30cm  0.22cm  30.22 cm 26.80.51 o h   cm  0.22cm 1 1 1 1  401.5   30.22cm s 26.8cm 26.80.51 i h2   cm  0.44cm 201.5 si  238 cm Compound thick lens

Can use two principal points (planes) and effective focal length f to describe propagation of rays through any compound system Note: any ray passing through the first principal plane will emerge at the same height at the second principal plane 1 1 1 d For 2 lenses (above):    H11H1  fd f2 f f1 f2 f1 f2 H22H2  fd f1 Example: page 246 Analytical ray tracing

note: paraxial approximation Refraction equation: nt1t1  ni1i1  D1 y1

nt1  ni1 Transfer equation: y2  y1  d21t1 D1  R1

Ray tracing of lens system: sequentially apply these equations Matrix treatment: refraction In any point of space need 2 parameters to fully specify ray: distance from axis (y) and inclination angle () in respect to optical axis. Optical element changes these ray parameters. Refraction: note: paraxial approximation nt1t1  ni1i1  D1 yi1 Reminder:

yt1=yi1 yt1  0 ni1i1  yt1  A B    A  By         C D y  C  Dy

Equivalent matrix nt1 t1  1 - D1 ni1i1        presentation:  yt1  0 1  yi1 

 ri1 - input ray rt1  R1ri1  R1 - refraction matrix

 rt1 - output ray Matrix: transfer through space

Transfer:

ni2i2  nt1t1  0 yt1

y yi2  d21 t1  yt1 i2  yt1

Equivalent matrix ni2 i2   1 0nt1t1        presentation:  yi2  d21 nt1 1 yt1 

 rt1 - input ray  T21 - transfer matrix ri2  T21rt1  ri2 - output ray System matrix

y y y i2 i2 i1 yt1

ri1 rt1 ri2 rt2 ri3 rt3 R1 T21 T32 R3

rt1  R1ri1 ri2  T21rt1  T21R1ri1

Thick lens ray transfer: rt2  R2T21R1ri1

System matrix: A  R2T21R1 rt2  A ri1 Can treat any system with single system matrix Thick lens matrix d

nl A  R2T21R1

y y 1 - D  i2 i2 R    yi1 y   t1 0 1   1 0 Reminder: T     A B a b   Aa  Bc Ab  Bd  d n 1       C D c d  Ca  Dc Cb  Dd 

1 - D 2  1 01 - D1  A      0 1 dl nl 10 1  system matrix of thick lens D d D D d  2 l 1 2 l  For thin lens d =0 1 - D1 - D 2 -   l  nl nl  A  1 - 1/ f   dl D1dl  A     1  0 1   nl nl    Thick lens matrix and cardinal points

ni11 a11  V1H1   a12

ni2 a22 1 V2H2   a12

 D d D D d  1 2 l - D - D -  1 2 l  n 1 2 n  a a  A   l l    11 12  d D d    l 1 l  a21 a22   1   nl nl  n n a   i1   t2 in air 1 12 a12   effective focal length fo fi f Matrix treatment: example rI

 rO

rI  TI 2AlT1OrO

nI I   1 0a11 a12  1 0nOO           yI  d I 2 nI 1a21 a22 d1O nO 1 yO 

(Detailed example with thick lenses and numbers: page 250) Mirror matrix note: R<0

1  2n / R M     0 1 



n r  ni     M   yr   yi 

yr  yi

nr  ni  2nyi / R

r  i  2yi / R Aberrations Aberrations - deviations from Gaussian optics. Chromatic aberrations - n depends on wavelength  Monochromatic aberrations - rays deviate from Gaussian optics  3 5  7 Taylor series: sin      ... 3! 5! 7! Paraxial approximation: sin     3 Third order theory: sin   3! Departures from the first order theory observed in the third order leave to the following primary aberrations: • spherical aberrations • coma Philipp Ludwig Seidel “The five Seidel • astigmatism (1821-1896) aberrations” • field curvature • distortion Spherical aberrations

Paraxial approximation: n n n  n 1  2  2 1 so si R

deviation from the first-order theory

2 2 n n n  n  n  1 1  n  1 1   Third order: 1  2  2 1  h2  1     2     s s R 2s  s R  2s  R s  o i  o  o  i  i   L.SA = longitudinal spherical aberrations image of an on-axis object is longitudinally stretched . positive L SA - marginal rays intersect in front of Fparaxial T.SA = Transverse (lateral) spherical aberrations image of an on-axis object is blurred in image plane Spherical aberrations

LC - circle of least confusion, smallest image blur

Spherical aberration depends on object and lens arrangement: Wavefront aberrations

John William Strutt (Lord Rayleigh) 1842-1919

Lord Rayleigh criterion: wavefront aberration of /4 produces noticeably degraded image (light intensity of a point object image drops by ~20%) Zero SA

For points P and P’ SA is zero

Oil-immersion microscope objective

http://micro.magnet.fsu.edu/primer/java/aberrations/spherical/ Hubble telescope

COSTAR - corrective optics space telescope axial replacement module Arecibo observatory

1000 ft radiotelescope (3cm - 3m) Coma (comatic aberration) • Aberration associated with a point even slightly off the optical axis Reason: principal planes are not flat but curved surfaces Focal length is different for off-axis points/rays

Negative coma: meridional rays focus closer to the principal axis Coma (comatic aberration)

Vertical coma

Horizontal coma Astigmatism • Aberration associated with a point considerably off the optical axis Focal length for rays in Sagittal and Meridional planes differ for off-axis points

http://www.microscopyu.com/tutorials/java/aberrations/astigmatism/ Field curvature

Focal plane is curved: Petzval field curvature aberration

Negative lens has field plane that curves away from the image plane: Can use a combination of positive and negative lenses to cancel the effect

http://www.microscopyu.com/tutorials/java/aberrations/curvatureoffield/index.html Distortion

Transverse magnification MT may be a function of off-axis image distance: distortions Positive (pincushion) distortion Negative (barrel) distortion

http://www.microscopyu.com/tutorials/java/aberrations/distortion/index.html Correcting monochromatic aberrations

• Use combinations of lenses with mutually canceling aberration effects • Use apertures • Use aspherical elements

Example: Chromatic aberrations

1  1 1     nl 1    f  R1 R2 

Refraction index n depends on wavelength Chromatic aberrations

A.CA: axial chromatic aberration

1  1 1     nl 1    f  R1 R2 

L.CA: lateral chromatic aberration Achromatic Doublets

Combine positive and negative lenses so that red and blue rays focus at the same point Achromatized for red and blue  1  1 1  1 1 1 d    nl 1    For two thin lenses d apart:    f  R1 R2  f f1 f2 f1 f2   1   n1 1 1  n2 1 2  d n1 1 1 n2 1 2 f 

n1R 11  n2R 12  dn1R 11n2R 12  Achromat:fR=fB  n1B 1 1  n2B 1 2  d n1B 1 1 n2B 1 2 http://www.microscopyu.com/tutorials/java/aberrations/chromatic/index.html  Achromatic Doublets 

n1R 11  n2R 12  dn1R 11n2R 12    n1B 1 1  n2B 1 2  d n1B 1 1 n2B 1 2

Simple case: d = 0

 n  n Focal length in yellow light 1  2B 2R (between red and blue): 2 n1B  n1R 1  nY 1  fY Combine:  n 1 f 1  2Y 2Y  n 1 f 2 1Y 1Y f n  n  n 1 2Y   2B 2R 2Y f1Y n1B  n1R n1Y 1 Achromatic Doublets

f n  n  n 1 2Y   2B 2R 2Y f1Y n1B  n1R n1Y 1

n  n n  n Dispersive powers: 2B 2R 1B 1R n2Y 1 n1Y 1

n2Y 1 n1Y 1 Abbe numbers (dispersive indices, V-numbers): V1  V2  n2B  n2R n1B  n1R f2Y V1   f2YV2  f1YV1  0 f1Y V2

Typical BYR colors: B = 486.1327 nm (F-line of hydrogen) Y = 587.5618 nm (D3 line of helium) R = 656.2816 nm (C-line of hydrogen) Table of V numbers - page 270 http://www.microscopyu.com/tutorials/java/aberrations/chromatic/index.html Achromatic lenses Crown Flint

Flint Achromatic triplet: Cooke triplet focus match for 3 wavelengths (Denis Taylor, 1893)

http://www.microscopyu.com/tutorials/java/aberrations/chromatic/index.html Crown Achromatic doublet: example Design an achromatic doublet with f = 50 cm Use thin lens approximation.

f1 f 1 1 1 Solution: f2V2  f1V1  0 f2    f1  f f f1 f2

f1 f V2  f1V1  0 f1  f

V1 V2 V2 V1 f1  f f2  f V1 V2

Technically: want smaller R, i.e. longest possible f1 and f2 Solution: use two materials with drastically different V Use figure 6.39 (page 271) Achromatic doublet: example

V2= 36.37

V1= 63.46

V Achromatic doublet: example Design an achromatic doublet with f = 50 cm

Solution: V1 V2 V2 V1 f1  f f2  f V1 V2 63.46  36.37 f  0.50  0.2134 m f2  0.3724 m 1 63.46

f f 1  1 1  n =1.51009 (for yellow line!) 1 2   1  nl 1    f  R1 R2  n2 = 1.62004 Negative lens: 1  1 1    R  f n 1 0.2309 m  n2 1    1 2 2 f2  R1   Positive lens: 1  1 1   n1 1      R1  0.2059 m f1  R1  0.2309m 