Optomechanical Systems Engineering
Lesson 2 – Optical Fundamentals
Keith J. Kasunic, Ph.D.
Technical Director
Optical Systems Group, LLC
2.1 Optical Fundamentals
2.1.1 Imaging
• A white card pointed at a scene does not produce an image
image.
• Lenses and curved mirrors modify wavefronts to create an image.
• A diverging wavefront, for example,
can be converted to a converging
Converging
wavefront, creating a point-to-point
reproduction (i.e., an image) of the
object.
j
• The lens index and radii determine
the type and degree of modification.
(a)
Image
Object
Lens
Copyright © 2012 by Keith J. Kasunic
2-2
2.1 Optical Fundamentals
v1 = c/n1
Wavefront
W
f
t
Propagation
> n1
0.2 to 30 microns m)
m = 10-6 meters
= 39.4 inches
25 um ~ 0.001” (1 mil)
v2 = c/n2

Snell’s Law: n1sinI1 = n2sinI2
Copyright © 2012 by Keith J. Kasunic
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2.1 Optical Fundamentals
2.1.1 Imaging
Surface Normal
i
Snell’s Law
for curved
surfaces
Paraxial Rays
Marginal Rays
n> 1
n= 1
Wavefronts and Rays
Converging
Image relies on the same
optical path length OPL =
∑ niLi for all rays
OPL = n1L1 + n2L2 + n3L3
As rays ttravell farther
A
f th in
i
air, they move through a
thinner part of the lens
Copyright © 2012 by Keith J. Kasunic
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2.1 Optical Fundamentals
How Does a Lens Change Wavefront Curvature?
Lens Maker’s Equation
(Thin Lens)
1
1
1
 (n  1)   
f
 R1 R2 
R1 = radius of1st surface of lens
R2 = radius of 2nd surface of lens
• The wavefront exits from the thinner edges of the
lens first, and moves faster than in the lens
• The shape of the transmitted wavefront depends
on the shape of the surface
• Spherical surfaces create spherical wavefronts
g or diverge
g
which converge
• Imperfections in the surface will create errors in
the wavefront
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2.1 Optical Fundamentals
Lens Types
Positive Lenses
Negative Lenses
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2.1 Optical Fundamentals
Where is the Image Located?
“Imaging Equation”
(a)
(a)
so = ∞
si = f
Co = 0
1 1 1
 
f so si
(b)
so = si = 2f
Co = 1/2f
(c)
so = f
si = ∞
f = lens focal length
so = object distance
si = image distance
Co = 1/f
Copyright © 2012 by Keith J. Kasunic
2-7
2.1 Optical Fundamentals
Imaging
• For a given focal length, the
object
bj t di
distance
t
so determines
d t
i
where the image is located (si).
• For a distant object, the lens
has enough power to bring the
collimated wavefronts to focus
at the focal length f.
• But the lens doesn’t have
enough power to first
straighten
g
out a diverging
g g
wavefront, and then bring it to
focus at the focal length.
1 1 1
 
f so si
The sum of the wavefront
curvatures (1/so + 1/si) is a
constant ((= 1/f,, lens power)
p
)
Ci = 1/f
Co = 0
Co = 1/so
• So real images move away
Co = 1/f
from the lens as the object
moves in closer.
Copyright © 2012 by Keith J. Kasunic
Ci = 1/si
Ci = 0
2-8
2.1 Optical Fundamentals
Summary of Imaging
• The curved surfaces of a lens changes
the curvature of the incident wavefront.
Less
power
• How much a lens modifies the
wavefront depends on the lens refractive
power  = 1/f = C1 + C2
• From a wavefront perspective, the sum
off the
th wavefront
f t curvature
t
incident
i id t on
and exiting from the lens is a constant.
• In most cases,, imperfections
p
in the
index or surface will create errors in the
wavefront.
• Controlling these imperfections is a key
task of the optomechanical engineer.
Copyright © 2012 by Keith J. Kasunic
2-9
2.1 Optical Fundamentals
2.1.2 Field of View (FOV)
• Not all p
points on the object
j
are “on-axis”.
• Instead, almost all points are “off-axis”, and these points determine the
field-of-view (FOV) of the system.
• The concept
p of a ray
y–
i.e., a perpendicular to the
wavefront – is used to
determine the FOV.
f
• Parallel rays represent planar
wavefronts.
• Converging or diverging rays
represent focusing or defocusing
wavefronts.
Copyright © 2012 by Keith J. Kasunic
2-10
2.1 Optical Fundamentals
2.1.3 Relative Aperture
• Image
g “brightness”
g
is
proportional to the diameter D
of the lens collecting light.
• The collected power is ~ D2,
so image brightness is ~ D2.
• The same power spread out
over a larger screen area
reduces
d
th
the iimage b
brightness.
i ht
• Image area is ~ f2, so image
brightness is ~ 1/f2.
• Putting these two concepts
together, we find that image
brightness is ~ (D/f)2.
• The relative aperture (f/#)
depends on the inverse of this,
or f/# = f/D.
Dia. D
Projection screen
or white
hit paper
f
• The longer focal length has
a smaller IFOV, reducing the
g collected by
y
amount of light
each pixel.
Copyright © 2012 by Keith J. Kasunic
2-11
2.2 Optical Fundamentals
Wavefront Error and
Aberrations
Vincent Van Gogh,
g ,
“Starry Night”
Vincent Van Gogh
Gogh,
“Café Terrace at Night”
Copyright © 2012 by Keith J. Kasunic
2-12
2.2 Optical Fundamentals
Wavefront Error and Aberrations
• The image quality may be a convolution of the geometrical image with
th aberration-limited
the
b
ti li it d lens
l
performance
f
(point
( i t spread
d function)
f
ti )
• Each point on the object is blurred by the point spread function
Spherical
Aberration
Lens Point
Spread
Function
(PSF)
Copyright © 2012 by Keith J. Kasunic
2-13
2.2 Optical Fundamentals
Wavefront Error and Aberrations
• Even for a lens with perfectly spherical surfaces, the wavefront may
have errors (aberrations)
• The reasons for aberrations and wavefront error (WFE) include:
1. Lens design (residual aberrations)
2 Fabrication
2.
3. Alignment
4. Environmental Factors: stress and strain, vibration, temperature
changes temperature gradients,
changes,
gradients etc.
etc
Copyright © 2012 by Keith J. Kasunic
vs. RMS
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2.2 Optical Fundamentals
Wavefront Error and Aberrations
• Understanding what lens designers do gives us better insight into the
challenges of opto-mechanical design
• First-order designs based on geometrical optics do not produce good
images…
• The lens designer’s job is reduce
red ce the aberrations fo
found
nd in a 1st-order
order
design to an “acceptable” level, thus improving image quality
First-order design for a MWIR imager
Credit: Warren J. Smith, Modern Optical
Engineering (4th Edition), McGraw-Hill, 2008
Aberration-corrected design
g for
the same imager
Copyright © 2012 by Keith J. Kasunic
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2.2 Optical Fundamentals
2.2.1 Lens Design – Spherical Aberration
• For lenses with spherical
p
surfaces,, it is seen as a difference in focal
length for the paraxial (O) and marginal (R) rays
• It is also seen as a large symmetrical blur of any point on the object
• A result of the nonlinearity of Snell’s law creating changes in optical path
length (or optical path difference, OPD):
• Larger incidence angles bend differently than smaller
Credit: Warren J. Smith, Modern Optical
Engineering (4th Edition), McGraw-Hill, 2008
Copyright © 2012 by Keith J. Kasunic
2-16
2.2 Optical Fundamentals
Lens Design – Coma
• Seen as a cone-shaped
p blur for off-axis p
points
• Coma is a result of upper (A) rays incident on the top of the lens seeing
a different refraction angle i than the lower (B) rays
• Blur size for a minimum-spherical lens depends on both the f/# and offaxis angle up
i
up
Up = off-axis angle
g ((radians))
Credit: Warren J. Smith, Modern Optical
Engineering (4th Edition), McGraw-Hill, 2008
Copyright © 2012 by Keith J. Kasunic
2-17
2.2 Optical Fundamentals
Lens Design – Astigmatism
• Seen as distinct lines as the image plane is moved through focus
• A result of the asymmetry of incidence angles for rays in the tangential
plane, but not in the sagittal
Angular blur size  at
intermediate focus, for
minimum-spherical shape
Credit: Warren J. Smith, Modern Optical
Engineering (4th Edition), McGraw-Hill, 2008
Copyright © 2012 by Keith J. Kasunic
2-18
2.2 Optical Fundamentals
Lens Design – Field Curvature
• Seen as a parabolic “best surface” for the image of off-axis points
• Not an aberration of image blur, but of image location (along z-axis)
Credit: Warren J. Smith, Modern Optical
Engineering (4th Edition), McGraw-Hill, 2008
Copyright © 2012 by Keith J. Kasunic
2-19
2.2 Optical Fundamentals
Lens Design – Distortion
• Seen as a scrunching or squeezing of off-axis points
• Not an aberration of image blur, but of image location (in detector plane)
Ref:
f James C.
C Wyant and K. Creath,
C
““Basic Wavefront
f
Aberration Theory for
f Optical
O
Metrology”,
” in
APPLIED OPTICS AND OPTICAL ENGINEERING, VOL. Xl, Academic Press, 1992.
Copyright © 2012 by Keith J. Kasunic
2-20
2.2 Optical Fundamentals
Lens Design – Axial Chromatic Aberration
• Seen as a colored blur surrounding a central point as focus is changed
• A result of the wavelength dependence of the refractive index n()
• Depends on the index dispersion through the Abbe V-number (V/#)
Longitudinal axial:
f = f/V
Transverse axial:
Credit: Warren J. Smith, Modern Optical
Engineering (4th Edition), McGraw-Hill, 2008
Copyright © 2012 by Keith J. Kasunic
2-21
2.2 Optical Fundamentals
Lens Design – Lateral Chromatic Aberration
• Seen as a color separation in the image plane for off-axis points
• A result of the wavelength dependence of the refractive index n()
19
1.9
Refractive Index, n( )
N-SF6
1.8
1.7
1.6
N-BK7
1.5
1.4
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Wavelength,  (um)
Credit: Warren J. Smith, Modern Optical
Engineering (4th Edition), McGraw-Hill, 2008
Copyright © 2012 by Keith J. Kasunic
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2.2 Optical Fundamentals
2.2 Summary
• Once aberrations are reduced to an “acceptable”
p
level,, lens designers
g
create prescriptions for each of the lenses (elements) in the system.
• Even if aberrations
could be completely
removed, a phenomenon
known as diffraction
limits the smallest
attainable blur size.
Chief ray
Marginal
g
ray
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2.3 Optical Fundamentals
Diffraction
• The best performance attainable from a lens is known as “diffraction
limited”, i.e., not limited by aberrations
• A diffraction-limited (WFE = 0) lens has the smallest-possible blur size.
Bessel J1(x)/x
Airy disk size
(angular , or
physical B)
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24
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2.3 Optical Fundamentals
Image Quality
• The criterion for diffraction-limited
i
imaging
i was established
t bli h d b
by JJohn
h
Strutt (Lord Rayleigh) using a perfect
optic’s ability to resolve 2 point
sources such as stars or laser beams
• He found that points are resolved
when the peak of one sits on the first
zero off the
th other
th  S = 1
1.22
22 x f/#.
f/#
• WFE of as much as /4 waves
peak-to-valley
p
y (≈
( /14 RMS)) is also
difficult to distinguish from WFE = 0,
and is also called diffraction limited.
 f/#
Sparrow criterion
S
1.22 f/#
Rayleigh criterion
WFE ≤ 0.25 PV
≤ 0.07 RMS
• The WFE of an optical system is
“Quarter-wave” optics =
thus a key measure of resolving
“diffraction-limited” optics
power and image quality.
Copyright © 2012 by Keith J. Kasunic
2-25
2.3 Optical Fundamentals
Wavefront Error
• WFE can be specified as either peak-to-valley (PV) or root-mean-square
(RMS).
• For example, California geography with Death Valley (-282 feet) and Mt.
Whitney elevations (+14,494 ft) gives a PV deviation of 14,776 ft (4.4 km)
• The
Th RMS deviation
d i ti averaged
d over allll off California
C lif i iis much
h smaller.
ll
• The relationship between PV WFE and RMS depends on the aberration.
• By removing outliers such as 6-sigma deviations, the RMS WFE is a
much better statistical measure of image quality
quality.
WFE PV  5 WFE RMS
For random fabrication errors
Credit: Optimax Inc.
Copyright © 2012 by Keith J. Kasunic
2-26
2.3 Optical Fundamentals
Root-sum-square
(RSS) addition for
uncorrelated
errors
Wavefront Error
• WFE budgets summarize
the total WFE expected in a
system (including lenses).
• The
Th RMS WFE budget
b d t to
t
the right includes residual
lens design WFE
(aberrations) as well as
optomechanical terms.
• Defocus is often used for
WFE budget comparisons
(WPV = 3.5WRMS), such that
/14 ((0.071 waves)) RMS ≈
/4 (0.25 waves) PV.
Copyright © 2012 by Keith J. Kasunic
WFE s  WFE12  WFE 22  ...
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