Geometrical optics: rays and images - Texas A&M...
Transcript of Geometrical optics: rays and images - Texas A&M...
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Geometrical optics: rays and images
A ray is an imaginary line along the direction of propagation of the light wave: this line is perpendicular to the wave front
If description in terms of rays is enough,it is geometrical optics.
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Reflection and refraction; index of refraction; Snell’s law.Image of an object in an optically more dense medium
sin sin
incident water refracted air
water air
water water air air
n n n n
n nn nθ θ
= =
>=
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Image by a plane mirror
incident reflectedθ θ=
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Sign rules for objects and images
0; ' 0 's s s s> < = −
plane mirror
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'ymy
=
Lateral magnification
Caution: lateral magnification can be less than 1, (m<1), and even negative!
m is negative for inverted images
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Image by a spherical mirror
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2
tan tan tan'
'
h h hs s R
h h hs s R
φ α θ β φ θα β φ
α β φδ δ δ
α β φ
= + = ++ =
= = =− − −
= = =
1 1 2+ =s s' R
Image by a spherical mirror;paraxial approximation, focal point and focal length
paraxial rays; paraxial approximation
1 1 2 ' / 2'
s Rs R
+ = =∞
Rf =2
focal point and focal length
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Image by a spherical mirror; spherical aberration; focal point and focal length
paraxial approximation
1 1 2 / 2'
f Rs s R+ = =
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Concave mirror: focal point
1 1 1's
+ =∞ f
Rf =2
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Concave mirror: image of an extended object
' 'y smy s
= = −
Attention: always find two most convenient rays to find an image of a point
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Concave mirror: image of an extended objectExample 34.1
' 'y smy s
= = −
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Concave mirror when s<f; plane mirror as a limiting case of a concave mirror (f→∞)
' ' 1y smy s
= = − =1 1 2 1 1 0 | ' | | |
' 's s f
s s R s s+ = ⇒ + = = <
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Convex mirror: image of an extended object
1 1 2 0, ' 0' | |
s ss s R+ = ⇒ > <
−
' 'y smy s
= = −
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Convex mirror: virtual focal point
1 1 2 0, ' 0' | |
s ss s R+ = ⇒ > <
−
' 'y smy s
= = −
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Convex mirror: virtual focal point; f<0
1 1 2 0, ' 0' | |
s ss s R+ = ⇒ > <
−
' 'y smy s
= = −
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Convex mirror: virtual focal point; f<0example 34.3
1 1 1' | |s s f
+ =−
' ' 1.76 0.023475
' 0.0234 160 3.8
y smy s
y my cm cm
−= = − = − =
= = × =
1.81 1 1 1 1' | | 1.8 75' 1.76
f cm
s f ss
= −
= − = − −−
= −
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Principle rays for concave and convex mirrors
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Principle rays for concave and convex mirrors
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Spherical refracting image
1 1 2' | | '
a b b an n n ns s R s s R
−+ = − ⇒ + =
' ' '//
a b
b a
y n s s nmy n s s n
= = − = −
Caution: spherical mirror and spherical refractions act very differently
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Plane refractive image as a limit of spherical refracting image
1 1 2' | | '
1 1: 0 0' '
a b b a
a b
n n n ns s R s s R
n nRs s s s
−+ = − ⇒ + =
→∞ + = + =
' ' 1a
b
y n smy n s
= = − =
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Caution: don’t confuse lateral magnification, m=1, with other directions.
Plane refractive image as a limit of spherical refracting image
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Plane refractive image as a limit of spherical refracting imageExample 34.7
1 1.330 0' ' 2.00
' 1.5
a bn ns s s
s m
+ = + =
= −
Note that although image is inside water, one has to use n of the air: nb =n(air)=1 !
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Spherical refracting image;Example 34.5
'1 1.52 1.52 1 ' 11.3
8 ' 2' 1.00 11.3 0.93
1.52 8.0
a b b a
a
b
n n n ns s R
s cmcm s cm
n s cmmn s cm
−+ =
−+ = =
+×
= − = − = −×
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Spherical refracting image;Example 34.6
'1.33 1.52 1.52 1.33 ' 21.38 ' 2
' 1.33 ( 21.3) 2.331.52 8.0
a b b a
a
b
n n n ns s R
s cmcm s cm
n s cmmn s cm
−+ =
−+ = = −
+× −
= − = − = +×
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Lenses; converging lenses
1 1 1' | |s s f
+ =
f1 =f2
Thicker in the middle
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Lenses; converging lenses
Caution: left hand remains left
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Lenses; diverging lenses
Thinner in the middle
1 1 1' | |s s f
+ =−
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Lens maker’s equation for thin lens
1 2 1 2
1 1 1 1 1 1( 1) ( 1)| | | |
n nf R R f R R
⎛ ⎞ ⎛ ⎞= − − ⇒ = − +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
Caution: sign of a radius, R1 or R2, can be negative!
R2 <0R1 >0
Double convex
1 1 1' | |s s f
+ =
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Lens maker’s equation for thin lens
1 2 1 2
1 1 1 1 1 1( 1) ( 1)| | | |
n nf R R f R R
⎛ ⎞ ⎛ ⎞= − − ⇒ = − − +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
Caution: sign of a radius, R1 or R2, can be negative!
1 1 1' | |s s f
+ =−
R1 <0R2 >0
Double concave
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Lenses: principal rays
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Lenses: principal rays
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Two lenses: image of an image
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Eye as an optical system
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Eye as an optical system: farsighted/nearsighted eyes
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Eye as an optical system;correction of nearsighted eye
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Eye as an optical system;correction of farsighted eye
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Eye as an optical system;correction of farsighted eye
Example 34.13 the near point is 100cm (i.e., much further than 25cm). What contact lenses are required? Caution: contact lenses, rather than glasses.Image is on incoming side of the source, i.e., negative!
1 1 1' | |
25 ' 10033 1/ 0.33 3.0
s s fs cm s cmf cm diopters
+ =
= = −= = +
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Example 34.14 the far point is 50cm (i.e., very short). What glasses are required? Caution: glasses, rather than contact lenses.Image is on incoming side of the source, i.e., negative!
1 1 1' | |
' (50 2) 4848 1/ 0.48 2.1
s s fs s cm cmf cm diopters
+ =−
= ∞ = − − = −= − − = −
Eye as an optical system;correction of nearsighted eye