Download Lens Aberrations

Lens Aberrations
Two main types of aberrations:
(A) Monochromatic Aberrations
Deviation from the paraxial approximation. The main types of aberrations are:
Spherical aberration, coma, astigmatism.
Spherical aberration
Recall that we used sin θ ≈ θ (first order theory in the paraxial approx.)
Including addition terms in sin θ ≈ θ - θ 3/3! leads to the third-order theory
which can explain the monochromatic aberrations.
Remember that for a single refracting spherical interface in the 1st order approx:
n1 n2 n2 − n1
+ =
= D1
so si
R
If the approximation for the OPL (lo + li) are improved, the 3rd order treatment then
n 1 1
n1 n2 n2 − n1
+ =
+ h2 1
+
2 so so R
so si
R
2
+
n2 1 1
−
2si R si
2
Where h is the distance above the optical axis as shown in the figure.
si(h) < si(0)
Rays striking the surface at a greater distance (marginal rays) are focused
closer to the vertex V than are the paraxial rays and creates spherical
aberration.
Marginal rays are bent too much
and focused in front of paraxial
rays.
Distance between the intersection of
marginal rays and the paraxial
focus, Fi, is known as the L⋅SA
(longitudinal spherical aberration).
Note: SA is positive for convex lens
and negative for a concave lens.
T⋅SA (transverse SA) is the
transverse deviation between the
marginal and paraxial rays on a
screen placed at Fi.
If the screen is moved to the
position ΣLC the image blur will
have its smallest diameter, known as
the “circle of least confusion,”
which is the best place to observe
the image.
Correction of spherical aberration
Rule of thumb: Incident ray will undergo a minimum deviation when αi ≈ αT.
Thus, proper planning of the lens (or lenses) shape can diminish the aberration
αi
α1T
α
T
For an object at ∞, the
round side of lens facing
the object will suffer a
minimum amount of SA.
Note that a planarconvex lens can be
approximated as two
prisms.
α T > α T and the
lower prism results in
a greater deviation.
Fig. 6.16 Spherical Aberration for a planarconvex lens in both orientations.
Similarly if the object and image are to be
nearly equidistant from the lenses (so = si =2f),
an Equi-Convex shaped lens minimizes SA.
f
Marginal rays give smaller
image negative coma
2f
2f
f
Coma
Marginal rays give larger
image positive coma
(comatic aberration) is associated with the fact
that the principle planes are really curved
surfaces resulting in a different MT for both
marginal and central rays.
Since MT = -si/so , the curved nature of the
principal surface will result in different effective
object and image distances, resulting in different
transverse magnifications. The variation in MT
also depends on the location of the object which
can result in a negative (a) or positive coma (b)
and (c), as demonstrated in the left figure.
The imaging of a point at S can result in a
“comet-like” tail, known as a coma flare and
forms a “comatic” circle on the screen Σ
(positive coma in this case). This is often
considered the worst out of all the aberrations,
primarily because of its asymmetric
configuration.
The Coma aberration can be recognized by
observing the change in the image of a far
Distance object, while tilting the lens.
A necessary condition for the absence of coma
is that the system meets the Sine Condition:
sin α o α op
=
sin α i α ip
Astigmatism:
The Meridional Plane contains the
chief ray which passes through the
center of the aperture and the
optical axis.
The Sagittal Plane contains the
chief ray and is perpendicular to
the meridional plane.
Fermat’s principle shows that
planes containing the tilted rays
will give a shorter focal length,
which depends on the (i) power of
the lens and the (ii) angle of
inclination. The result is that there
is both a meridional focus FT and a
sagittal focus FS.
Tilted rays have a
shorter focal length.
Astigmatism: Note that the cross-section of the beam changes from a
circle (1) → ellipse (2) → line (primary image 3)→ ellipse (4) → circle
of least confusion (5) → ellipse (6) → line (secondary image 7) .
7
3
1
2
5
Focal length difference FS-FT depends on
power D of lens and angle of rays.
(B) Chromatic Aberrations
When the ray is composed from several colors having different
refracting index, we can expect that the focal length will depend on the
wavelength.
1
1
1
= (nl − 1)
−
f
R1 R2
f
λ
nl = nl (λ )
Circle of least confusion
Achromatic Doublets
A combination of two lenses, one positive and one negative can minimize
the chromatic aberration.
1 / f = (n − 1) ⋅ (1 / R1 − 1 / R2 )
ρ =1 / R1 − 1 / R2
1 / f = (n − 1) ⋅ ρ
For two lenses in contact: 1 / f = (n1 − 1) ⋅ ρ1 + (n2 − 1) ⋅ ρ 2
For achromatic lens:
Thus:
1 / fr = 1 / fb
(n1r − 1) ⋅ ρ1 + (n2 r − 1) ⋅ ρ 2 = (n1b − 1) ⋅ ρ1 + (n2b − 1) ⋅ ρ 2
n 2 r − n 2b
ρ1
=−
ρ2
n1r − n1b
Abbe number:
V≡
ny −1
nb − nr
ρ1 ρ1 y n2 y − 1
=
=
ρ 2 ρ 2 y n1 y − 1
Dispersive Power = 1 / V
f2y
f1 y
=
(n2 y − 1) ⋅ ρ 2
(n1 y − 1) ⋅ ρ1
=
(n2 y − 1) ⋅ (n1b − n1r )
(n1 y − 1) ⋅ (n2b − n2 r )
f1 y ⋅ V1 + f 2 y ⋅ V2 = 0
Fraunhofer lines
=−
V1
V2
(a)
Graded Index Optical Systems
Flat disk of glass with an index, n(r),
that varies as a function of distance r
from the center is an example of a
GRIN lens.
(b)
Based on the observation that rays
(wave-fronts) slow down in an
optically dense region and speed-up in
less dense regions.
The center of cylindrical lens has n =
nmax along its optical axis. So, along
the optical axis the optical path length
(OPL) is given as (OPL)O = nmaxd
At a height r, (OPL)r ≈ n(r)d.
In order for the rays to converge at a
focus, the planar wavefront must bend
into a spherical wavefront, which
defines surfaces of constant phase.
In order to match the phase at all points
on the wavefront we must require
nmax
n(r)
n(r ) ≅ nmax −
r
2
r
2 fd
(OPL) r + AB = (OPL) O
and
n(r )d + AB = nmax d
also
AF ≈ r 2 + f 2
AB = AF − f
Therefore
n(r )d + r 2 + f 2 − f = nmax d
r2 + f 2 − f
n(r ) = nmax −
d
Assume r << f
Then
(
f 1+ r2 / f 2
n(r ) = nmax −
d
)
1/ 2
−f
r2
≅ nmax −
2 fd
Similar to a multimode-graded index core
optical fiber
The most common device is a GRIN
cylinder a few millimeters in
diameter. They are usually fabricated
using an ionic diffusion process in
which a homogeneous glass is
immersed in a molten salt bath for
many hours.
The focal length is determined by the
index change ∆n < ~0.10. The profile
is usually expressed as
n(r) = nmax(1 – ar2/2)
Rays striking the surface in a plane
of incidence that contains the optical
Used in a
copy machine axis travel in a sinusoidal path have
spatial period T = 2π/a1/2 where a =
a(λ).
Common Applications: Laser printers,
photocopiers, fax machines (i.e. devices
requiring image transfer between surfaces.)
It is possible to create real erect
images by changing the object
distance or length L of the lens.
Radial GRIN lenses are often specified in
terms of their pitch. A 1.0 pitch rod represents
L = T = 2π/a1/2 (full sine wave). A pitch of
0.25 has a length of quarter sine wave T/4.
Note that the block of glass is formed so that n = n(z).
The block can then be grounded and polished into the
shape of a lens. The result is a reduced n for marginal
rays.
It is possible to use this
approach to significantly
reduce the effects of
spherical aberration in
comparison to a regular
lens in (c) to the right.
Superposition of Electromagnetic (E-M) Waves
Any E-M wave satisfies the wave equation:
The equation is linear since any linear
combination will also satisfy this equation:
2
1
∂
ψ
2
∇ψ = 2 2
v ∂t
ψ (r , t ) =
N
i =1
ciψ i (r , t )
This is called the principle of superposition.
Consider the addition of E-M waves possessing the same frequency but having
different phases:
E1 = E01 sin (ωt + α1 ) and
E2 = E02 sin (ωt + α 2 )
Then E = E1 + E2 = E0 sin (ωt + α )
The interference term
where E02 = E012 + E022 + 2 E01 E02 cos(α 2 − α1 )
and
E01 sin α1 + E02 sin α 2
tan α =
E01 cos α1 + E02 cos α 2
Definition: phase difference, δ=
α2 - α1.
E0=min when =(2n+1)!, E0=max when =2n!
The composite wave is harmonic with the same frequency, but the amplitude
and phase are different.
for ε1 = ε2 ,,δ = α2 - α1 = (2π/λ)(x1 - x2)=k0n(x1-x2)
Let α = -(kx + ε)
x1 and x2 are the distances from the sources of the two waves to the point of
observation and λ is index dependent, and so λ = λo/n . Then, we can write
δ = k∆x =
2π
λo
n(x1 − x2 ) = k o Λ = 2π
where Λ = n(x1 − x2 ); Optical
Note that
Λ
λo
=
x1 − x2
λ
Λ
λo
Path Difference (OPD) = Λ
= (# waves in medium)
If ε1 , ε2 = const. the two E-M waves are said to be coherent.
Suppose that we have a superposition of two waves that travel a small
difference in distance (∆x):
E1 = E01 sin[ωt − k ( x + ∆x)] and
and
let
E 2 = E02 sin (ωt − kx )
E01 = E02 ; α 2 − α1 = k∆x
Use sin( A ± B) = sin( A) cos( B) ± sin( B) cos( A) and
kx = k ( x + ∆x / 2) − k∆x / 2
E = E1 + E 2 = 2 E01 cos(k∆x / 2)sin[ωt − k ( x + ∆x / 2)]
and δ = k∆x / 2
Consider two special cases: (1) ∆x = (n + 1/2)λ
λ (out-of-phase)
and (2) ∆x = nλ
λ (in-phase); n = 0, 1, 2, 3, …
Case 1
k∆x 2π 1
(n + 1 / 2)λ = (n + 1 / 2)π
=
λ 2
2
δ = k∆x / 2 = π (n + 1 / 2)
Destructive Interference E = 0, which is a minimum in Intensity.
k∆x 2π 1
Case 2
=
nλ = nπ
δ = k∆x / 2 = nπ
λ 2
2
In both cases n = 0, 1, 2, 3…
Constructive Interference E0 = 2E01, which
is a maximum in Intensity.
For the more general case
in which E01 ≠ E02
Case 2: Constructive
interference δ = 2nπ
Case 1: Partial
destructive interference
δ = 2π(n + 1/2)
Fig. 7.3 Waves out-of-phase by k∆x radians.
Fig. 7.4 The French fighter Rafale uses
active cancellation to confound (frustrate)
radar detection. It sends out a nearly equal
signal that is out-of-phase by λ/2 with the
radar wave that it reflects. Therefore, the
reflected and emitted waves cancel in the
direction of the enemy receiver.
In general, the sum of N such E-M waves is
E=
N
i =1
E0i cos(α i ± ωt ) = E0 cos(α ± ωt )
where E =
2
0
and
N
E +2
2
0i
i =1
tan α =
N
i =1
N
N
j >i i =1
E0i sin α i
E0i E0 j cos(α i − α j )
N
i =1
E0i cos α i
Suppose that we have N random sources (e.g. a light bulb).
Then cos(αi-αj) t = 0
E02 t =NE201 if each atom emits waves of equal E01.
This result is for an incoherent source of emitters.
For a coherent source, we have αi = αj and the sources are in-phase which gives
E02 = E02
t
=
N
i =1
2
E0 i
= N 2 E012 Each atom emits waves of equal E01.
Complex Method for Phasor Additon
E1 = E01 cos(kx ± ωt + ε 1 ) = E01 cos(α1 ωt ) α = −(kx + ε )
~
~
E1 = E01 exp i (α1 ωt ) with E1 = Re E1
The addition of N E-M waves becomes
~
E=
N
j =1
~
Ej =
iα
E0 e =
(
)(
N
j =1
N
j =1
E0 j e
E0 j e
E02 = E0 eiα E0 eiα
)
iα j
iα j
e ± i ωt = E 0 e i α e ± i ωt
The complex amplitude can be expressed as a
vector in the complex plane, and is known as a
phasor. The resultant complex amplitude is the
sum of all constituent phasors.
*
Which can be used to calculate the resulting
irradiance from the complex amplitudes of the
constituent waves.
Imaginary
Axis
Phasor Addition
E1
Real Axis
Consider the sum of two E-M waves: E = E1 + E2
E1 = E01 sin (ωt + α1 ) and
E2 = E02 sin (ωt + α 2 )
Then E = E1 + E2 = E0 sin (ωt + α )
where E02 = E012 + E022 + 2 E01 E02 cos(α 2 − α1 )
and
E01 sin α1 + E02 sin α 2
tan α =
E01 cos α1 + E02 cos α 2
From the law of
cosines we can
easily calculate E02
and further analysis
of the geometry
gives tan α.
E1=5sinωt
E2=10sin(ωt+45º)
E3=sin (ωt-15º)
E4= 10sin(ωt+120º)
E5=8sin(ωt+180º)
Consider the addition
of these five E-M
waves using the
phasor addition below.
In electrical engineering, these
phasors can also be written with
the following notation:
5∠0°, 10∠45°, 1∠-15°,
10∠120°, and 8∠180°
E1 = E01 sin( kx − ωt )
E2 = E02 sin( kx − ωt + α )
Fig. 7.7 The phasor sum
of E1, E2, E3, E4 and E5.
The summation of two sinusoidal functions of the same
frequency using phasor additon. Here E1 is taken as the
reference phasor, and since E2 leads E1 (i.e. its peak occurs at
an earlier location) the angle α is positive. Thus ϕ is positive
and the resultant E also leads to E1.
E1 = E01 sin(kx − ωt − α )
E2 = E02 sin( kx − ωt − 2α )
E3 = E03 sin(kx − ωt − 3α )
E4 = E04 sin( kx − ωt − 4α )
E01 = E02 = E03 = E04
Standing Waves
Consider reflection of E-M waves of a mirror
EI = E0 sin (kx + ωt ) and
E R = E0 sin (kx − ωt )
E = EI + ER = E0 [sin (kx + ωt ) + sin (kx − ωt )]
= 2 E0 sin kx cos ωt = (2 E0 cos ωt )sin kx
Description of standing wave:
Standing wave: Time-varying
amplitude with sinusoidal spatial
variation (see previous slide).
Nodes: x = 0, λ/2, λ, 3λ/2, 2λ...
Anti-Nodes: x = λ/4, 3λ/4, 5λ/4….
Addition of E-M waves of different frequencies
Beats
(1) Equal amplitudes
E1 = E01 cos(k1 x − ω1t ) and
k1 > k 2 , ω1 > ω2 , v ph =
ω
k
E2 = E01 cos(k 2 x − ω2t )
=
ω1
k1
=
ω2
k2
Then E = E01 [cos(k1 x − ω1t ) + cos(k 2 x − ω2t )]
and
α +β
with cos α + cos β = 2 cos
2
cos
Then E = 2 E01 cos(k m x − ωmt ) cos(k x − ω t )
where ω = (ω1 + ω2 ) / 2
and
α −β
2
and ωm = (ω1 − ω2 ) / 2
k = (k1 + k 2 ) / 2; k m = (k1 − k 2 ) / 2
The resultant wave is a traveling wave of frequency and wave number:
Then E = E0 ( x, t ) cos(k x − ω t )
(traveling wave)
with
ω, k
E0 ( x, t ) = 2 E01 cos(k m x − ωmt )
(modulated or time varying amplitude)
Note that the Irradiance is given by the following
I ∝ E02 = 4 E012 cos 2 (k m x − ωmt ) = 2 E012 [1 + cos(2k m x − 2ωmt )]
2ωm = ω1 − ω2
(beat frequency)
ω1
ω2
Detector
with fast
response
ω B = 2ωm = ω1 − ω2
(beat frequency)
(2) Different Amplitudes
Beats can also be observed through the superposition of E-M waves
possessing different amplitudes, as well as different frequencies. The
phasor method can be used to help illustrate the formation of beats.
Heterodyne Principle
Heterodyning is a method for transferring a broadcast
signal from its carrier to a fixed local intermediate
frequency in the receiver so that most of the receiver does
not have to be retuned when you change channels. The
interference of any two waves will produce a beat
frequency, and this technique provides for the tuning of a
radio by forcing it to produce a specific beat frequency
called the "intermediate frequency" or IF.
Group Velocity
The carrier wave exhibits a high frequency
The phase velocity is given by v ph = ω / k
The group velocity is given by v g =
ωm
ωc = ω = (ω1 + ω2 ) / 2
ω1 − ω2
∆ω
=
=
km
k1 − k 2
∆k
This is the rate at which the modulation envelope or energy of the
wave advances or propagates. For a general dispersion ω = ω (k)
vg =
dω
dk
and this speed is usually less than the speed of light c.
ω
Using ω = vk and v = c/n
dω d
dv
d
=
=v+k
vg =
(vk ) = v + k
(c / n)
dk
dk
dk dk
c dn c
c dn
k dn
=v−k 2
= −k 2
= v 1−
n dk n
n dk
n dk
For normal dispersion, dn/dk > 0 and therefore vg < v.
Polarization of Light
Linear Polarization: Begin by defining individual components:
E x ( z , t ) = iˆE0 x cos(kz − ωt ) and
E = Ex + E y
(
E y ( z , t ) = ˆjE0 y cos(kz − ωt + ε )
If ε = 2nπ , n = 0, ±1, ±2, ±3, ...
)
E = iˆE0 x + ˆjE0 y cos(kz − ωt ) " in − phase" components
E = Ex + E y
(
If ε = (2n + 1)π , n = 0, ±1, ±2, ±3, ...
)
E = iˆE0 x − ˆjE0 y cos(kz − ωt ) "180° out − of − phase" components
It is therefore possible to define any polarization orientation with a
constant vector in the x-y plane for the case of linear polarization.
For linear polarization, the state of polarization is often referred to
as a P - state. This is the symbol for a script P.
Circular polarization:
E x ( z , t ) = iˆE0 x cos(kz − ωt ) and
E = Ex + E y
[
E y ( z , t ) = ˆjE0 y cos(kz − ωt + ε )
If ε = −π / 2 + 2nπ , n = 0, ±1, ±2, ±3, ...
]
E = E0 iˆ cos(kz − ωt ) + ˆj sin (kz − ωt )
with E ⋅ E = E02
E = E0 = const.
E0
ω
The electric field vector clearly rotates clockwise while looking back at the
source from the direction of propagation. The frequency of rotation is ω with
a period of T = 2π/ω. This is the case of “right-circularly polarized light”. It is
often expressed as an R – state. This is a script R.
Right circular polarization
(R – state)
E0
ω
Left circular polarization, i.e.
an L– state.
If ε = π / 2 + 2nπ , n = 0, ±1, ±2, ±3, ...
[
]
E = E0 iˆ cos(kz − ωt ) − ˆj sin (kz − ωt )
E = E0 = const.
The electric field vector clearly rotates counter-clockwise while looking back at
the source from the direction of propagation. The frequency of rotation is ω with a
period of T = 2π/ω. This is the case of “left-circularly polarized light”. It is often
expressed as an L– state. This is a script L.
ER + EL = Eε
It is easy to understand that for general parameters E0x, E0y, and ε,
we have elliptical polarization.
E x ( z , t ) = iˆE0 x cos(kz − ωt ) and E y ( z , t ) = ˆjE0 y cos(kz − ωt + ε )
E = Ex + E y
Actual polarizers. Irradiance is
independent of the rotation angle θ
for the conversion of natural light
(unpolarized) to linear polarization.
In the figure below, only the
component E01cosθ is transmitted
I(θ) = I(0)cos2θ, which is known as
Malus’s Law.
If Iu = I (natural or unpolarized light),
then I(0) = Iu <cos2θ>t = Iu/2.
Dichroism –selective absorption
of one of two orthogonal E
components.
1. Absorption of E-field in the ydirection causes e’s to flow.
2. Re-radiation of waves that
cancel incident waves polarized in
the y-direction.
This results in transmission of
waves with E-fields perpendicular
to the wires (i.e., along the xdirection).
Polaroid sheet (H-Sheet), most commonly used linear polarizer.
Contains a molecular analogue of the wire grid.
1. Sheet of clear polyvinyl alcohol is heated and stretched.
2. Then it is dipped in an ink solution rich in Iodine.
3. Iodine is incorporated into straight long-chain polymeric molecules
allowing electron conduction along the chain, simulating a metal wire.
HN-50 is the designation of a hypothetical, ideal H-sheet that transmits 50%
of the incident natural light while absorbing the other 50%. In practice,
about 4% of the light will be reflected at each surface leaving a maximum
transmittance of 92% for linearly polarized light incident on the sheet.
Thus, HN-46 would transmit 46% of incident natural light, and might be the
optimal polarizer. In general, for HN-x, the irradiance of polarized light
transmitted would be I=Io(x/50), where Io is the irradiance for the ideal case.
In practice, it is possible to purchase HN-38, HN-32, and HN-22 in large
quantities for reasonable prices, each differing in the amount of iodine
present.
Tourmaline
Crystal System: Hexagonal (trigonal)
Habit: As well-formed, elongate, trigonal prisms, with
smaller, second order prism faces on the corners. Prism
faces are often striated parallel to direction of elongation
(c-axis). The rounded triangular cross-sectional shape of
tourmaline crystals is diagnostic; no other gem mineral has
such a shape.
Hardness: 7-7.5
Cleavage: none
High birefringence (Two differenct indices of
refraction)
Strong Dichroism
Any transparent gem having a mean R.I. of 1.63 and a
birefringence of 0.015-0.020 is tourmaline.
Tourmaline is widespread in metamorphic, igneous and
sedimentary rocks. Gem Elbaite is, however, nearly
restricted to pegmatites. Literally thousands of tourmalinebearing pegmatites are known; only a few hundred
apparently contain gem quality material in mineable
quantities.
Found in Brazil, Sri Lanka, U.S., Southern California
Tourmaline Boron Silicates)
Chemical
Formula
Specific
Gravity
XY Al B Si OH
(X
Y
Na or Ca)
Hardness
Mg, Li, Al or Fe
Refractive
Index
-
1) There is a specific direction within
the crystal known as the principal or
optic axis.
2) The E-field component of an
incident wave that is perpendicular to
the optic axis is strongly absorbed.
3) The thicker the crystal the more
complete will be the absorption.
4) A plate cut from a tourmaline
crystal parallel to its principle axis
and several mm thick will serve as a
linear polarizer.
5) Absorption depends on λ.
6) Advantages over H-sheet
polarizers with regard to maximum
irradiance permitted and can be used
with high power lasers.