Hecht by YHLEE;100510; 5-1

Chapter 5 Geometrical Optics-Paraxial Theory 5.1 Introduction An imaging system Rays diverge from S and converge to P → P is a perfect image of S

An optical system in terms of wave: The object space → An optical system → The image space ↑ ↑ Collection Reshape of wavefront. of wavefront. The principle of reversibility → S is a perfect image of P : Conjugate points • A real imaging system:

A point object → An optical system → Blur spot (Still an image) ↑ ↑ Finite extent. Diffraction Partial collection. The best possible image is called diffraction limited λ ⇒ 0 → Rectilinear propagation of the light : Geometrical optics ↑ No diffraction

Hecht by YHLEE;100510; 5-2 5.2 Lenses A refracting device that reshapes the wavefront. A. Aspherical Surfaces Aspherics : Lenses and mirrors with non-planar or non-spherical surfaces (Difficult to make) Hyperboloidal and ellipsoidal refracting surfaces → Perfectly parallel beam

B. Refraction at Spherical Surfaces Spherical surfaces are easier to grind than aspheric surfaces [p 153] A spherical surface can have aberrations (Imaging errors), but they can be corrected to make the system diffraction limited. Refraction at a spherical surface

Optical Path Length of a ray S → A → P : OPL n l n lo i= +1 2

( ) ( )22 2 coso o ol R s R R s R= + + − + ϕ

( ) ( )22 2 cosi i il R s R R s R= + − + − ϕ

Fermat’s principle

d OPL

db gϕ

= 0 → nl

nl R

n sl

n slo i

i

i

o

o

1 2 2 11+ = −

FHG

IKJ (1)

Note (1) This eq. is exact (2) Different ϕ → Different A. → Different is , Different position P Different , o il l . (NOT imaging)

Hecht by YHLEE;100510; 5-3 • Paraxial rays : Rays nearly parallel to the optical axis

Small ϕ and h → cos 1ϕ ≈ , sinϕ ϕ≈ Then (1) becomes

ns

ns

n nRo i

1 2 2 1+ =−

: Independent of A (Perfect image at P )

This is called First-order optics, Paraxial optics, or Gaussian optics → A perfect image is possible with spherical surfaces Deviations from the paraxial approximation (Ideal system) → Aberrations • The first and second focal lengths The first focal length or the object focal length

ns

ns

n nRo i

1 2 2 1+ =−

↑ ↑ = fo , =∞ The second focal length or the image focal length

ns

ns

n nRo i

1 2 2 1+ =−

↑ ↑ =∞ , = f i • A virtual image : Rays diverge from it in the image space A virtual object : Rays converge toward it in the object space

Note s Ri < <0 0, Note s Ro < <0 0,

(Sign convection Table 5.1)

Hecht by YHLEE;100510; 5-4 C. Thin Lenses Lens : It consists of two or more refracting surfaces. It has at least one curved surface. Simple lens : One element lens Compound lens : Many element lens Thin lens Thick lens Convex, converging, or positive lens Concave, diverging, or negative lens

• Special lenses Gradient-index lens (GRIN), Holographic lens Thin-Lens Equations

At the first refracting surface : ns

ns

n nR

m

o

l

i

l m

1 1 1+ =

− , si1 0< , virtual image

At the second surface : 2 2 2

l m m l

o i

n n n ns s R

−+ = , R2 0< , s s do i2 1= − +

Add two eqs. : ns

ns

n nR R

n ds d s

m

o

m

il m

l

i i1 2 1 2 1 1

1 1+ = − −

FHG

IKJ + −

b g b g

A thin lens, d→0, in the air, nm = 1

1 1 1 1 1

1 2s sn

R Ro il+ = − −FHG

IKJb g : Thin-lens equation or Lensmaker’s formula

1 1 1s s fo i

+ = : Gaussian lens formula

The focal lengths : 1 1 1 1 1

1 2f fn

R Ri ol= = − −FHG

IKJb g

Hecht by YHLEE;100510; 5-5 • Parallel rays at different incident angles → Different focal points on σ centered on C

For paraxial rays, σ becomes a plane near the optical axis → Focal plane In a lens, parallel paraxial rays focus onto a second focal plane, or back focal plane.

Similarly the first focal plane or the front focal plane is in the object space.

Finite Imagery Imging by a lens

Three rays are used (1) A ray through the optical center (2) A ray trough the first focal point (3) A ray parallel to the optical axis

Hecht by YHLEE;100510; 5-6 A thin lens can be replaced by a plane

Sign convention y yo i> <0 0,

Gaussian lens eq.

From the triangles AOFi and P P Fi2 1 → yy

fs f

o

i i=

−b g (1)

From the triangles S S O2 1 and P P O2 1 → yy

ss

o

i

o

i= (2)

From (1) and (2) → 1 1 1s s fo i

+ =

Newtonian lens eq.

From the triangles BOFo and S S Fo2 1 → yy

fs f

i

o o=

−b g (3)

From (1) and (3) → x x fo i =2

xo > 0 : The object is to the left of Fo xi > 0 : The image is to the right of Fi • The image by a thin lens

Hecht by YHLEE;100510; 5-7 The ray parallel to the optical axis determines the height of the real image

Nonlinear transformation along the axis The object from ∞ to 2F → The image from F to 2F. Object moves along the axis → Axial change > Height change • Transverse magnification : /T i oM y y≡

Longitudinal magnification : 2/L i o TM dx dx M≡ = − Thin-Lens Combinations

[1] Ignore L2 for a moment [2] Rays 1 and 3 to find ′P1 [3] Line from ′P1 to O2 [4] Put lens L2 on O2 No change for ray 4 But refraction of 3 by L2 [5] The image by 3 and 4

Hecht by YHLEE;100510; 5-8 Analytic calculations

Analytic calculations

For lens L1 : 1 1 1

1 1 1s s fo i+ =

For lens L2 : 1 1 1

2 2 2s s fo i+ = , where s s di o1 2+ =

Combine the three equations

sf d f s f s fd f s f s fi

o o

o o2

2 2 1 1 1 1

2 1 1 1 1=

− −

− − −

//b gb g

so1 → ∞ , ( )( )

2 12

1 2i

f d fs

d f f−

=− +

: back focal length

si2 → ∞ , sf d f

d f fo11 2

1 2⇒

−

− +b gb g : front focal length

When d=0, f f l b f lf f

f f. . . . .= =

+1 2

1 2

→ 1 1 1

1 2f f f= + : Effective Focal Length

The total transverse magnification

1 2T T TM M M=

Hecht by YHLEE;100510; 5-9 5.3 Stops A. Aperture and Field Stops Aperture stop : It limits the amount of light reaching the image (The rim of a lens) Field stop : It limits the size of the image (The edge of a film) It determines the angular extent of the object, Field of View.

B. Entrance and Exit Pupils Entrance pupil : The image of A.S. formed toward the object. Exit pupil : The image of A.S. formed toward the image.

The cone of EnP → The amount of light entering the system. The cone of ExP → The amount of light leaving the system.

Hecht by YHLEE;100510; 5-10 Chief ray : A ray from an off-axis object point crossing the center of A.S It passes through the centers of EnP and ExP. Marginal ray : A ray from the axial object point touching the edge of A.S

Vignetting : Cone of rays becomes narrower for off axis object points. Image fade out from on-axis to off-axis.

C. Relative Aperture and f-Number The irradiance at the image ~ /D f2 2

f-number is defined as f fD

/#≡ : speed of camera lens

f/1.4 at 1/500 sec = f/2 at 1/250 sec :The same amount of light at the film

Hecht by YHLEE;100510; 5-11 5.4 Mirrors For high reflection : Polished substrate + Al coating + Protective coating. For ultrahigh reflection : Multilayered dielectric coatings. A. Planar Mirrors Same distance from mirror to object and to image

Inversion via reflection

Twice tilt of the reflected beam B. Aspherical Mirrors

C. Spherical Mirrors The same lens maker’s formula

1 1 1 2

o is s f R+ = = − : f > 0 for concave mirrors, f < 0 for convex mirrors

Hecht by YHLEE;100510; 5-12 5.5 Prisms Applications : Beam splitters, Polarizing devices, Interferometers, Measuring n( )ω . Main characteristics : Dispersion. Change in the image orientation. Change in the beam propagation. 5.6 Fiber Optics 5.7 Optical Systems A. Eyes (1) A single centered lens : Vertebrates. A real image on retina (2) A multifaceted compound lens : Insects. No real image on retina Electrical synthesis in the nervous system (3) A simple lensless hole Structure of the Human Eye

The eye is almost spherical (24mm long, 22mm across) Cornea : The strongest convex element nc = 1376. (water nw = 133. ). Crystalline lens : A layered fibrous mass Cornea + Crystalline lens (Double-lens system) Fish moves the lens itself. Shell fish contract or expand the whole eye. Birds of prey change the cornea curvature. Iris : Aperture stop (Eye color, 2mm ~ 8mm) Aqueous humor : Thick gel Retina : electrochemical reactions. Two kinds of photoreceptor cells in retina Rods : High speed, operation at dim light, black and white, sharp image, Cones : Low-speed, operation at bright light, color image. Operation wavelength → 390 nm ~ 780 nm (310 nm ~ 1050 nm, for some people) Macula : Center of the retina. Twice as many cones as rods. Fovea centralis : 0.3 mm in dia. at the center of macula. Rod-free region. (The image of full moon ~0.2mm) The perception of an image

→ A continuous analysis of the time-varying retinal image by the eye-brain system For detailed view of image

The continuous movement of eye ball → Continuous shift of the image across the photoreceptors. No fading out of image.

Hecht by YHLEE;100510; 5-13 Accommodation (The fine focusing) The lens → Ligaments → Ciliary muscles.

Near point : The closest point the eye can focus. 7cm for teens 12 cm for young adults 28~40 cm for middle-aged 100 cm for 60 years of age Far point : The most distant point the eye can focus. Infinity for the normal eye. B. Eyeglasses

Diopter = 1f

: f in m

1 1 1

1 2f f f= + → 1 2= +a a a

The normal eye = 58.6a : 19=a from the crystalline lens, 43=a from the cornea) The second focal point not on the retina due to cornea, lens or eye ball length → Farsightedness, nearsightedness, astigmatism ↑ Common cause

Hecht by YHLEE;100510; 5-14 Nearsightedness Parallel rays are focused in front of the retina. The far point distance is not infinity.

The focal length of the correction lens = The far point distance → Moving the far point to infinity.

Example An eye with a far point of 2m

1 1 1f s so i= + → f m= −2 (a=-0.5D)

↑ ↑ =∞ , -2, Bring objects from infinity to within 2m

Hecht by YHLEE;100510; 5-15 Farsightedness Parallel rays are focused behind the retina. The near point distance is longer than the normal (25cm).

Example An eye with the near point of 125cm

1 1 1f s so i= + → f = 31cm (a=+3.2D)

↑ ↑ 25cm -125cm Astigmatism Due to an uneven curvature of the cornea.

Hecht by YHLEE;100510; 5-16 C. The Magnifying Glass D. Eyepieces or Ocular E. The Compound Microscope F. The Camera The pinhole camera

Well defined, undistorted image over a wide angular field and a large range of distances ↑ ↑ Great depth of focus Great depth of field [Fig. 5.101] [p. 217] G. The Telescope 5.8 Wavefront Shaping A. Adaptive Optics B. Phase Conjugation