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Mirrors

In contrast to lenses and refracting surfaces, mirrors are reflecting optical devices.They have the advantage of working in a much broader frequency range, since theyin general do not suffer any dispersion.

Planar mirrors (Fig. 5.40):

i.e. the object and its image are equidistant from the mirror surface. In fact, bysetting f =, and Si negative on the right of the mirror, the mirror formula is

identical to the Gaussian Lens formula. The transverse magnification is:

Therefore, the image formed by a plane mirror is life-sized, virtual and up-right.

The mirror image is inverted, i.e., left hand is imaged as right hand.

Plane mirrors are frequently used to redirect a beam of light.

iStoidenticalis oSVPAVAS

1 o

i

o

i

T S

S

Y

Y

M

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Spherical Mirrors

Spherical mirrors (Fig. 5.50):

According to the law of reflection,qi = qr

In SCA, using Law of Sines, we have:

In CAP, we have:

Therefore,

i

SC

SCA

SA

qsinsin

r

CP

PCA

PA

qsinsin

SCAPCASCAPCA o sinsin180

PA

CP

SA

SC (47)

Furthermore, SC = So -|R| and CP = |R| - Si

Considering |R| = -R (C on the left of vertex)

Thus, SC = So + R, CP =(Si + R)

In the paraxial region, SA So, PA Si , and Eq. (47) becomes:

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Spherical Mirrors

Eq. (48) is equally applicable to concave (R < 0) and convex (R > 0) mirrors.

We can find that object and image focal length are equal:

Eq. (48) can be simplified as:

(48) Mirror formulaRSSS

RS

S

RS

ioi

i

o

o211

or

2

Rff oi

fSSio

111 (50)

For a concave mirror f> 0 (R0), as shown in Fig.

5.51. For a convex mirror, the formed image is behind the mirror, reduced and virtual.

Fig 5.51

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Spherical Mirrors

Summary of imaging properties:

(1) any parallel off-axis bundle of rays will be focused to a point on the focal plane

(2) a finite planar object perpendicular to the optical axis will be imaged in a planeperpendicular to the optical axis

(3) the transverse magnification is:MT= - Si/So

(4) the sign convention is summarized in Table 5.4, please note that Si is takenpositive when it is on the left ofV(rather than right, as it is for lenses)

(5) Table 5.5 summarizes the image properties for both concave and convexspherical mirrors. DRAW THE RAY DIAGRAM FOR EACH CASE!!

(6) the ray diagram for both concave and convex mirrors are shown in Fig. 5.53

Fig 5.53

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Aspherical Mirrors

Paraboloidal mirror (Fig. 5.45): formed by rotating a parabola along its axis.

E.g.:y2 = 2px, focal point: F(p/2, 0) A concave paraboloidal mirror can focus abundle of // rays in its focal point F, even under non-paraxial conditions.

Conversely, a point source on the focal point will generate an emission ofparallel rays (plane waves).

y

xF

Paraboloidal mirror are used in variety of applications, such as flash lights, automobileheadlight reflectors and giant radio telescope antennas.

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Aspherical Mirrors

Fig 5.47

Ellipsoid mirror (Fig. 5.47): formed by rotating an ellipse along its axis:

with focal points F(c, 0),

Its two foci are perfect conjugate points, i.e., any ray that goes through one focalpoint has to go through the other focal point after reflection. Therefore, a pointsource at one focal point will form a perfect image at the other focal point.

12

2

2

2

b

y

a

x

22 bac

y

x

F1 F2

Magnetic imaging

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Optical Systems

Camera (Fig. 5.102): a single lens reflex (SLR) camera is composed of a lens,used to form an image on the film, an iris diaphragm, to control the amount oflight reaching the film and quality of image and a shutter, to control theexposure time.

Iris diaphragm: (1) controls the amount of light reaching the film (small f/#, morelight) (2) controls the depth of field (DOF).

DOF: a range of object distance centered at the object plane, which still gives clearand sharp image at the film plane. The smaller the aperture, the larger thedepth of field, since the image blurring induced by non-paraxial rays is reducedfor smaller aperture.

o

o

iT

o

L dxx

fdxM

x

fM

2

22

2

2

(1) For a givenf(or zoom), the smaller the xo, the smaller the dxo (at fixed dxi).Therefore, smallerxo, smaller the DOF, the bigger thexo, the bigger the DOF

(2) Similarly, for givenxo, the bigger thef, the smaller the DOF

Here, dxo is the DOF anddxi reflects the quality ofthe cameras sensor.

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Human eye

Human eye (Fig. 5.80): the principle of human eye is very similar to that of a camera,can be viewed as a positive lens + light-sensitive surface.

Cornea has an index of refraction of 1.376; aqueous humor: n=1.336; iris (2 8 mm ID);crystalline lens (~22, 000 fibrous layers, at center n = 1.406 and at the edge n =

1.386, focal length can be changed by shape change), vitreous humor chamber (n =1.337) with black inner shell choroid (dark layer to absorb the strayed light), on topof it covered with retina (light receptor cells)

Two kinds of photoreceptor cells: rods and cones.

Cones: work in bright light, detailed and colored view, lack of sensitivity in low light level

Rods: work well in dark condition, has higher sensitivity, fast response time, no color sense

Blind spot: the area of exit of the optic nerve, containing no receptors

Yellow spot (macula): 2.5-3.0 mm in diameter contains twice as many cones than rods, atthe center it shows the fovea centralis (more densely packed cones) for the sharpestand most detailed image.

Check this out: http://photography.bhinsights.com/content/photographic-eye.html

http://photography.bhinsights.com/content/photographic-eye.htmlhttp://photography.bhinsights.com/content/photographic-eye.htmlhttp://photography.bhinsights.com/content/photographic-eye.htmlhttp://photography.bhinsights.com/content/photographic-eye.html

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Eyeglasses

Normal eyes : The fine focusing of the human eye is performed by adjusting thecrystalline lens with the cilliary muscles (a muscle disc supporting the lens). Thus, bychanging the focal length of the lens, the image distance is kept constant. For normaleyes, the far point is at infinity and the near point, the nearest point that the eye canfocus on, is about 25 cm or 10 inches.

Myopia (Nearsightedness) (Fig. 5.83): the power of the lens is too large for the axiallength of the eye: the parallel rays are brought to focus in front of the retina.Therefore, the far point is closer than infinity, and all the points beyond the far pointappear blurred. To correct this condition, a negative lens is introduced to diverge a bitthe rays, as shown in Fig. 5.83.

Hyperopia (Farsightedness) (Fig. 5.85): the second focal point of a relaxed eye liesbehind the retina, usually due to the shortening of the axial length of the eye. As aresult, its near point moves further away from the eyes: one cannot see the nearbyobjects clearly. In this case, a positive corrective lens is introduced to help imagingthe nearby objects on the retina.

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Eyeglasses

Dioptric powerD : reciprocal of the focal length, 1D = 1m-1

Usage example: for thin lenses in contact, we have D = D1 + D2

Astigmatism: the focusing power of an eye is not the same along different directions,due to an uneven curvature of the cornea. Needs cylindrical lens for correction.

Example (Fig. 5.84): Suppose an eye that suffers myopia has a far point of 2 m, whatis the focal length for a correcting contact lens?

If the virtual image of an object located at infinity is formed at 2 m by a negative lens, the

eye will see the object clearly. We have:

fD

1

distancepointfarthem2i.e.2

11111

f

SSfio

The above equation is for a contact lens with focal length fc .

Repeat the problem for eyeglasses placed 16mm in front of the eye

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Eyeglasses

Contact and spectacle lens : usually the eye glasses (with focal length fl)arepositioned at the first focal point of the cornea (d~16 mm in front of the eye), to

avoid extra magnification of the image over the one formed by the unaided eye.

We will derive the relation between fl andfc.

The b.f.l. of the eye (fe) plus spectacle: (72)

le

le

ffd

fdflfb

...

The combined focal length for eye plus contact lens: (73)ec fff

111

Set the inverse of b.f.l. equal to 1/f above and we have:

dfflc

11

dD

DD

l

lc

1

or (74)

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Eyepieces

Case (3): set So = f, so the virtual image is at infinity (L = ). We have:

(79)

This mode is the most pleasing to the eye and is widely used in eyepieces.

fd

DdMP oo

Eyepieces (ocular) (Fig. 5.93, 94, 95) : it is basically a magnifier which is used toview the image of the object formed by a preceding lens system. It provides a virtualimage of the intermediate image, most often located at infinity, so it can becomfortably viewed by a normal, relaxed eye.

Its magnifying power:

The design of an eyepiece is very complex in order to reduce a variety of aberrations andmaintain superior image quality. A few commonly used eyepieces are shown in Fig.5.93, 5.94, 5.95.

f

dDdMP oo

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Compound Microscope

Compound microscope (Fig. 5.99) : It is made of an objective (lens) and an eyepiece.The objective is first forming a real, inverted, magnified image right in the field stop.Then, the eyepiece magnifies further the intermediate image.

The magnifying power:MP = MToMAe

MTo : transverse magnification of the objective, MTo =- xi / fo . Thexi is set as 160 mm, calledtube length L = xi = 160 mm.

MAe : the angular magnification of the eyepiece.MAe= 254/fe , 254 mm = 10 in. is a

standard near point distance for a normal eye.

eo ffMP

254160

Therefore, (81), with both focal lengths in mm.

Numerical aperture (NA) : defined as: NA = ni sin qMAX (82)

Where ni is the refractive index of the immersing medium (air, water, oil, etc.) adjacent tothe objective lens, and qMAX is the half angle of the maximum cone of light picked upby the entrance pupil (objective), i.e. half angle subtended from the objective lens to

the object.

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Telescope

Numerical aperture (NA) : it determines (1) the brightness of the image (2) moreimportantly, the resolving power of a microscope, i.e., the minimum transversedistance between two object points that can be resolved in the image.

Telescope (Fig. 5.106, 5.107): enlarge the retinal image of a distant object.Similar to the microscope, the objective first forms an inverted real image, whichis further magnified by the eyepiece. Since the object is in effect at infinity, itsimage is formed at the second focus of the objective. Usually, the eyepiece islocated with its first focal point overlapping on the second focal point of theobjective. The telescope in this configuration forms an afocal system, i.e. a systemwithout a focal point, since a parallel beam in results in a parallel beam out.

The angular magnification:

e

o

u

a

f

fMP

tan

tan(83)

The image formed by the telescope is an inverted, virtual image. When the imageorientation is important, an additional system is added, such as the double Porro

prism in the binoculars.

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Thick Lenses

Two approximations (basis for the first approximation):

(1) Thin lens: d

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Thick Lenses

Nodal points : The ray going through the opticalcenter O will emerge parallel to the incidentdirection. The extension of both the incomingand the outgoing rays will cross the optical

axis in two points, the nodal points (N1 andN2). When the lens has the same medium on

both its sides, these two point coincide withtheH1 andH2 points.

Two focal points, two principal points and twonodal points constitute the cardinal points ofthe system. If we know the position of these

points, than the image position, size andorientation can be uniquely determined.

N2

ON1

fSS io

111Gaussian formula still holds for a thick lens: (6.1)

The effective focal length is given by (math, next class):

(6.2)

2121

1111

1

RRn

dn

RR

n

f l

ll

l

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Thick Lenses

Principal plane position : the position of the principal planes is given by V1H1 = h1and V2H2 = h2which are positive when the plane lies to the right of their respective

vertices.

Newtonian lens formula and transverse magnification also hold:

2

1

1

Rn

dnfh

l

ll

1

2

1

Rn

dnfh

l

ll (6.4)

(6.3)

2fxx

io

o

i

o

i

T x

f

f

x

Y

Y

M

(6.5) (6.6)

Properties of principal planes: (1) they are conjugate planes (2)MT = 1, so they

are also called unit planes. (3) any ray directed towards a point on the firstprincipal plane will emerge from the lens as if it is originated from thecorresponding point (the same height) on the second principal plane.

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Thin Lens vs. Thick Lens

Thin lens :

Reference point: optical center

dl0 dl 0

Thick lens :

Reference points:H1 & H2

o

i

o

i

o

iT

x

f

f

x

S

S

Y

YM

2fxx io

fSS io

111

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Ray Diagrams

Fo

F i

F i Fo

Fo

FiH1 H2

Fi FoH1 H2

Thin Lens Thick lens

Positive Lens Positive Lens

Negative Lens Negative Lens

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Example

Find the image distance (P245, Fig.6.4): An thick double-convex lens with R1= 20 cm andR2 = -40 cm, thickness d = 1 cm, index of refraction of 1.5. An

object is positioned 30 cm from the lens. Find the image position.

The system principal plane positions (referenced to each vertex):

H1is to the right ofV1, andH2is to the left ofV2. Finally, So = 30+ 0.22, we

have

cm8.261/cm)40)(20(5.1

115.1

40

1

20

1

15.1

111

1

1

2121

fRRn

dn

RRnf l

ll

l

cm22.0

)40(5.1

115.18.261

2

1

Rn

dnfh

l

ll

cm44.0)20(5.1

15.08.261

1

2

Rn

dnfh

l

ll

2frommeasuredcm,238and

8.26

11

2.30

1HS

Si

i

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Compound Thick Lens

Compound thick lens (Fig.6.5): If we know the focal lengths, principalpoint positions and their separation, we can calculate the effective focallength, principal points for the compound length:

2112

1221

where111

HHdff

d

fff (6.8)

The principal plane position of the system:

2

111

f

fdHH

1

222f

fdHH (6.9) (6.10)

Thus, by knowing the effective focal length and the locations of the principalplanes, we can represent the above two think lenses as a single effective thicklens. For an optical system consisting of more than two thick lenses, we canapply the above procedure successively to work out the final focal length andlocations of principal planes. Please look at the example on P246.

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Matrix Review

A matrix is a square or a rectangular array of numbers or functions (elements)that obey certain rules. Its elements are labeled by two subscripts. An elementat ith row andjthcolumn is denoted as: aij.

A matrix with m rows and n column is:

mnmm

n

n

aaa

aaa

aaa

A

11

22221

11211

Equality: A = B, if and only ifaij = bij, for all values of i and j.Addition: A + B = C, if and only ifaij + bij = cij , for all values of i and j.

Commutation: A + B = B + A

Association: (A + B) + C = A + (B + C)

Scalar Multiplication: C = A =(A), cij =aij , alsoA= A.

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Matrix Review

Matrix Multiplication:

A B = C, if and only if , the element cij is

formed as a scalar product ofith row ofA withjth column ofB.

k

kjikij bac

Association: (AB)C = A(BC)

Distributive law: A(B+C) = AB+AC , however multiplication of matrices is usually not

commutative, i.e.,ABBA , in general.

Example: 2x2 square matrix

2221

1211

2221

1211 andGivenbbbbB

aaaaA

we have:

22222121

12121111

baba

babaBA

2221

1211

aa

aaA

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Matrix Review

2222122121221121

2212121121121111

2221

1211

2221

1211

babababa

babababa

bb

bb

aa

aaAB

222121

212111

2

1

2221

1211

2

1havewe,For

gagagaga

gg

aaaaAG

ggG

Determinant of A: detA = |A| = a11a22 - a12a21

Unit matrix: AEAAEE

and,10

01