optics for engineers chapter 5 · chapter 5 charles a. dimarzio northeastern university feb. 2014....
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Optics for Engineers
Chapter 5
Charles A. DiMarzio
Northeastern University
Feb. 2014
The Story So Far
• Small–Angle Approximation
sin θ = θ = tan θ and cos θ = 1
• Perfect Imaging
– Lens Equation (Image Location)
– Magnification (Image Size)
– Matrix Optics and Other Bookkeeping Tricks
– Point Images as Point, or . . .
– Image Position, X ′, Independent of Pupil Position, X1
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–1
Using Snell’s Law Exactly
• Example: Single Convex Air–to–Glass Interface• Paraxial Rays Follow Small–Angle Approximation• Edge Rays May Focus Quite Differently• Rays Do Not Intersect at a Single Point (or at all in 3D)• Large “Shot Pattern” at “Paraxial” Focus• “Best” Focus Translated and Depth of Focus Increased
−10 0 10−1
0
1
z, Axial Position
x, H
eigh
t
8 9 10
−0.2
0
0.2
z, Axial Position
x, H
eigh
t
Complete Ray Trace Expanded View Near Image
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–2
Looking Ahead: Diffraction
• Diffraction Theory (Ch. 8) Predicts a Minimum Spot Size
– Rooted in Fundamental Physics
– ≈ λDpupil
z
• Ray Tracing Result Below this Limit is “Good Enough”
– Characterized as “Diffraction–Limited”
• Larger Ray–Tracing Result Indicates Degraded Imaging
– Can Characterize Roughly by “XDL”
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–3
Ray Tracing: Overview
• Setup: Launch a Fan of Rays (eg. Fill FOV and Pupil)
• Loop On Rays
– Loop On Elements (Like Matrix But No Approximations)
∗ Translation (Straight–Line Propagation)
∗ Refraction or Reflection (Interfaces)
– Close (End the Ray Calculation)
• Report (eg. Spot Size vs. Field Position)
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–4
Ray Tracing: Translation (1)
• Parametric Eq. for Ray
x = x0 + `v
x
y
z
=
x0y0z0
+ `
u
v
w
• Surface Eq. (Eg. Sphere, Centered on Axis)
(x− xc) · (x− xc) = r2 x2 + y2 + (z − zc)2 = r2
• Combine to Find Intersection
v · v`2 +2(x0 − xc) · v`+ (x0 − xc) · (x0 − xc)− r2 = 0
u2`2 + v2`2 + w2`2 +2x0u`+2y0v`+2(z0 − zc)w`+
x20 + y20 + (z0 − zc)2 − r2 = 0
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–5
Ray Tracing: Translation (2)
• Solution (Quadratic in `)
aq`2 + bq`+ cq = 0,
aq = v · v = 1 bq = 2(x0 − xc) · v
cq = (x0 − xc) · (x0 − xc)− r2
• Zero to Two Real Solutions
` =−bq ±
√b2q − 4aqcq
2aq
• Pick the “Right” One and Find Intersection
xA = x0 + `v
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–6
Ray Tracing: Refraction
• Find the Normal in Order to Apply Snell’s Law
n =x− xc√
(x− xc) · (x− xc)
• At the New Origin from Translation Calculation
xA
• Compute the New Direction (Snell’s Law in Vector Notation)
v′ =n
n′v+
√1−(n
n′
)2 [1− (v · n)2
]−
n
n′v · n
n
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–7
Ray Tracing: Close
• After Iterating Translation and Refraction as Needed. . .
• Answer Some Question
– “Where is Intersection with Paraxial Focal Plane?” . . .
xB · z = zclose
(xA + `v1) · z = zclose
– or Any of a Collection of More Complicated Questions
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–8
Ray Tracing: Report (and More)
• Spot–Diagram (eg. vs. Field Position or Depth)
• Through–Focus Spot–Diagrams
• RMS or Maximum Spot Size (eg. XDL)
• Optical Path Length (eg. vs. Field Position)
• Many More
• Advanced Ideas (Many Commercial Programs)
– Optimization (eg. Vary Radii of Curvature and Distances)
– Use Vendor’s Stock Lenses
– Use Vendor’s Existing Tools
• Commercial Optical Designers
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–9
Ray Tracing
Take–Away Message
• Ray Tracing Gives Exact Answers (Except for Diffraction)
• Paraxial Rays Obey the Small–Angle Equations of Ch. 3.
• Diffraction–Limited System
– Spot Less than zλ/D
– Such a Design is “Good Enough.”
• Commercial Programs Exist
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–10
Ellipsoidal Mirror (1)
• Path:
– S to Surface to S′
• Fermat’s Principle:
– Minimal Time
• Imaging:
– All Paths Minimal 0 10 20 30−10
0
10
zx
SS’
b (s+s’)/2
|s−s’|/2
√(z − s)2 + x2 + y2 +
√(z − s′
)2+ x2 + y2 = s+ s′
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–11
Ellipsoidal Mirror (2)
• Vertex at Origin(z − a
a
)2+(x
b
)2+(y
b
)2= 1
a =s+ s′
2
b2 =
(s+ s′
2
)2−(s− s′
2
)2= ss′
1
f=
1
s+
1
s′=
s′ + s
s′s=
2a
b2
f =b2
2a
• Spherical Mirror (s = s′)
z2 + x2 + y2 = r2
r = s = 2f
• Parabolic Mirror
z =x2
4s′+
y2
4s′(s → ∞)
z =x2
4f+
y2
4f
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–12
Ellipsoidal Mirror (3)
• Ellipse(z − a
a
)2+(x
b
)2+(y
b
)2= 1
f =b2
2a• Best–Fit Sphere
r = 2f
• Best–Fit Parabola
z =x2
4f+
y2
4f
• Ellipse Perfect for
One Point Object
• Sphere or Parabola
May be Best Overall
0 10 20 30−10
0
10
z
xSS’F
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–13
Mirror Aberrations: Definitions
• Match Second Derivatives at Origin (or See Previous Slide)
• Perfect Ellipsoid Defined by z, and ∆ Represents OPL Error
zsphere = z +∆sphere/2 zpara = z +∆para/2
(z − a)2 = a2 −(a
b
)2 (x2 + y2
)(Ellipsoid)(
z +∆sphere
2− r
)2= r2 − x2 − y2 (Sphere, r = 2f)
z +∆para
2=
x
4f(Paraboloid)
• Solve for ∆ for Sphere or Parabola
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–14
Mirror Aberrations: OPL Errors
• Sphere or Parabola ok if ∆ � λ
−6 −4 −2 0 2 4−15
−10
−5
0
5
10
15
Error
x
f/1
f/1
NA=0.9
NA=0.9
0.8
0.8
0.5
0.5
SphereParab.
10−5
100
10−1
100
101
|Error||x
|Errors for s = 25 and s′ = 5 Log Scale
Relative to Ellipsoid Errors vary with |x|4
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–15
Summary of Mirror Aberrations
• An ellipsoidal mirror is ideal to image one point to another.
• For any other pair of points, aberrations will exist.
• A paraboloidal mirror is ideal to image a point at its focus
to infinity.
• Spherical and paraboloidal mirrors may be good approxima-
tions to ellipsoidal ones for certain situations and the aber-
rations can be computed as surface errors.
• Aberrations are fundamental to optical systems. Although
it is possible to correct them perfectly for one object point,
in an image with non–zero pupil and field of view, there will
always be some aberration.
• Aberrations generally increase with increasing field of view
and increasing numerical aperture.
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–16
Seidel Aberrations and OPL
• Small–Angle Approximation
sin θ ≈ θ
• Next–Best Approximation (Third Order)
sin θ ≈ θ +θ3
3!
• Wavefront Aberrations: ∆ = Error in OPL (eg. Tilt)
d∆tilt
dx1= a1 ≈ δθ
• Approach: ∆ vs. Field Position and Pupil Position
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–17
General Expression for Error
• Pupil Definitions, ρ, φ
x1 = ρdp/2cosφ
y1 = ρdp/2 sinφ
• Field Point: (x,0)
– Defines Coordinates
• Expand ∆ in ρ, φ, x
– Even Orders to 4th
∆ = a0 +b0x
2 + b1ρ2 + b2ρx cosφ +
c0x4 + c1ρ
4 + c2x2ρ2 cos2 φ +
c3x2ρ2 + c4x
3ρ cosφ +c5xρ
3 cos3 φ + . . .
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–18
The Easy Terms
• Constant Term (The Very Easy One)
– “Piston” (Just a Phase Change)
a0
• Tilt (Linear Terms Displace the Image)
• Quadratic Term
– Defocus
b1ρ2
∗ Can be Corrected
∗ Does not Affect Image Quality
• That’s the End of the Easy Ones
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–19
Spherical Aberration
• Previously Analyzed for Mirrors
• First Quartic Term
∆sa = c1ρ4 (Spherical Aberration)
• Analysis
– ρ2 is Focus
– c1ρ2ρ2 Means Focus Varies With ρ2
– Different Focus for Different Parts of Pupil: Blur
– Blur Occurs Even for x = 0
1200min 14 Feb 2014
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–20
Distortion
• Quartic Terms
∆d = c4x3ρ cosφ (Distortion)
• Analysis
– ρ cosφ Is Wavefront Tilt
– c4x3ρ cosφ Means Tilt Varies with x3
– Tilt Increases (c4 > 0) or Decreases (c4 < 0) as x3
– No Error at x = 0
Object Barrel Pincushion
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–21
Coma
• Quartic Terms
∆c = c5xρ3 cos3 φ (Coma)
Image Rel. to
Field Point (x)
Pupil
• Analysis
– ρ cosφ Is Wavefront
Tilt
– c5xρ2 cos2 φρ cosφ
Means Tilt Varies
∗ Linearly with x
(No Error at x = 0)
∗ Quadratically with
ρ cosφ (Symmetric
in Pupil)
– Comet–Like Image of
a Point
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–22
Field Curvature andAstigmatism
• Quartic Terms
∆fca = c2x2ρ2 cos2 φ+ c3x
2ρ2
(Field Curvature and Astigma-
tism)
• Analysis
– ρ2 is Focus
– c3x2ρ2 Means Focus
Varies with c3x2
(Field Curvature)
– c2x2 cos2 φρ2 Means
Focus Varies with
c2x2 cos2 φ
(Astigmatism in cos2 φ)
– No Effect at x = 0
– Astigmatism Increases
with x2
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–23
Astigmatism Examples (RayTracing)
−0.0200.02−1.32−1.3
−1.28
y
x
−0.0200.02−1.37−1.36−1.35−1.34−1.33
y
x
−0.0200.02−1.42−1.4
−1.38
y
x
z = 8.0 z = 8.3 z = 8.6 (sag image)
−0.0200.02−1.47−1.46−1.45−1.44−1.43
y
x
−0.0200.02−1.52−1.5
−1.48
y
x
−0.0200.02−1.56−1.54−1.52
y
x
z = 9.2 z = 9.5 (tan image) z = 9.8
−0.04−0.0200.020.04−1.62−1.6
−1.58−1.56
y
x
−0.05 0 0.05
−1.65
−1.6
y
x
8 10 120
0.05
0.1
z = 10.1 z = 10.3 Summary
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–24
“Deliberate Astigmatism”
• Setup
– Cylindrical Lens
– (With Spherical?)
– Ellipsoidal Lens
• Result
– Astigmatism On Axis
– Different Paraxial Foci
• Some Applications
– Eyes and Eyeglasses
– CD Player Focusing
∗ Quadrant Detector
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–25
Seidel Aberrations Summary
x0 x1 x2 x3
ρ1 Tilt Distortionρ2 Focus F. C. & Astig.ρ3 Comaρ4 Spherical
Expressions for aberrations. Aberrations are characterized
according to their dependence on x and ρ.
On Axis the Only Aberration is Spherical
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–26
Aberrations
Take–Away Message
• Seidel Aberrations Cause Distortion and Blurring
• Only Spherical for On–Axis Object
• Aberrations Increase With Field of View and NA
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–27
Spherical Aberration in a ThinLens: Coddington Factors
• Given s and s′ What is the Best Thin Lens?
1
f=
1
s+
1
s′1
f= (n− 1)
(1
r1−
1
r2
)
• Two Unknowns, r1 and r2
• Definition: Coddington Position Factor (f , p: Lens Use)
p =s′ − s
s′ + s=
1+m
1−m
• Coddington Shape Factor (f , q: Lens Manufacture)
q =r2 − r1r2 + r1
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–28
Spherical Aberration in a ThinLens: Computing Aberration
• Equations for Surface Radii
1
f= (n− 1)
(1
r1−
1
r2
)
r1 = 2fq
q +1(n− 1) r2 = −2f
q
q − 1(n− 1)
• Longitudinal Aberration a Function of Height in Pupil
Diopters Ls =1
s′(x1)−
1
s′(0)=
x21
8f3
1
n (n− 1)
(n+2
n− 1q2 +4(n+1) pq + (3n+2) (n− 1) p2 +
n3
n− 1
)
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–29
Spherical Aberration in a ThinLens: Aberration Distances
• Focal Change in Diopters
1
s′(x1)−
1
s′(0)=
s′(0)− s′(x1)
s′(x1)s′(0)≈
s′(0)− s′(x1)
s′(0)2
• Displacement of Focal Position
∆s′ (x1) ≈[s′ (0)
]2Ls (x1)
• Transverse Displacement
∆x′ (x1) = x1∆s′ (x1)s′(0)
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–30
Spherical Aberration in a ThinLens: Minimizing Aberration
• Set Derivative to ZerodLs
dq= 0
• Solve for Best q
qopt = −2(n2 − 1
)p
n+2
• Transverse Aberration: Varies with NA3
∆x (x1) =x31s
′ (0)
8f3
[−np2
n+2+(
n
n− 1
)]
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–31
Spherical Aberration Examples
0 5 1010
0
102
104
q, Shape Factor∆ x,
Tra
nsve
rse
Abe
rrat
ion,
µ
n = 1.5
n = 2.4
n = 4.0
0.50 µ m1.06 µ m
10.59 µ m
s=100.0 cm, s’=4.0 cm, x1=0.4 cm.
100
101
102
10−2
100
102
104
x1, mm
∆ x,
µ m
Minimum May be Good Enough ∆X ∝ x31(or not) D.L. ∝ 1/x1
≈ f/5 Lens. ≈Plano–Convex Best for Glass, Meniscus Best for Ge
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–32
Spherical Aberration in a ThinLens: Designing the Lens
+ signs show Plano–Convex, Biconvex, Convex–Plano
−1.5 −1 −0.5 0 0.5 1 1.5−6
−4
−2
0
2
4
6
p, Position Factor
q, S
hape
Fac
tor
n=1.5 2.4 4.0
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–33
Guideline: Share the Bending
Take–Away Message• Share the Bending for Best Aberration
• Watch Principal Planes
– IR Detector Lens Example
0 2 4
−1
0
1
z, Axial Position
x, H
eigh
t
0 2 4
−1
0
1
z, Axial Position
x, H
eigh
t
A. Correct Orientation B. Reversed
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–34
IR Detector Lens Problem
• Small Detector
• Inside Dewar
• Germanium Lens
• Meniscus
• Short Focal Length
• High NA (Small Spot)
• Principal Planes
– Outside Lens
– Badly Positioned
• Bad Solution
– Reverse the Lens
– Focal Point on Detector
– Bad Aberrations
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–35
Chromatic Aberration
• Focal Length Depends on Index of Refraction
• Index of Refraction Depends on Wavelength
– See Glass Map in Ch. 1
• Different Colors Have Different Focal Lengths
– Important for White Light Spectrum or a Portion of It
– Important for Multi–Wavelength Systems
(eg. Fluorescence, λexcitation 6= λemission)
– Important for Short Pulses (δf = 1/δt): Remember
δλ
λ=
δf
f=
1
fδt=
1
Cycles per Pulse
• Correction is Possible
• Reflective Optics Eliminate Chromatic Aberration
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–36
Lens Design Ideas
When One Doesn’t Work . . .
Use Two (or more)
and Share the Bending
(The More the Better)
or “Let George Do It”
(Use Commercial Lenses)
or Try an Aspheric
1300min 18 Feb 2014
Feb. 2014 c©C. DiMarzio (Based on Optics for Engineers, CRC Press) slides5r1–37