Optical Design with Freeform Surfaces

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Optical Design with Freeform Surfaces Joseph M.
Howard, Ph.D.
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• Outline
• Behavior of Optical Systems
• Systems with rotational symmetry
• Systems with broken symmetry
• Correcting bad behavior (i.e. optical design
  methods)
• First Principles
• Brute Force
• Copy Cat
• Examples of concepts with freeform surfaces
• The Freeform Optics Manifesto

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• 1. Behavior of Optical Systems
• A. Systems with rotational symmetry

Wave aberration function
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• 1. Behavior of Optical Systems
• A. Systems with rotational symmetry

Wave aberration function

• Even order terms in wavefront aberration
  function
• Larger field (h) or aperture (r) will increase
  blur.
• Larger values of coefficients (w1, w2, etc.)
  also increase blur.

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1. Behavior of Optical Systems (cont.) B.
Systems with broken symmetry
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1. Behavior of Optical Systems (cont.) B.
Systems with broken symmetry

• Even AND Odd terms in wavefront blur function
• Field (y) and aperture (p) coordinates are now
  decoupled into their individual y and z
  components.

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• 1. Behavior of Optical Systems (cont.)
• Regardless of symmetry, a Taylor expansion is the
  fundamental mathematical model used to describe
  the behavior of optical imaging systems
• This suggests that the best optics used to
  minimize this blur function will also be of
  this form (i.e. aspheric)

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• Outline
• Behavior of Optical Systems
• Systems with rotational symmetry
• Systems with broken symmetry
• Correcting bad behavior (i.e. optical design
  methods)
• First Principles
• Brute Force
• Copy Cat
• Examples of concepts with freeform surfaces
• The Freeform Optics Manifesto

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2. Correcting bad behavior (i.e. optical design
methods) A. First principles
Rotational Symmetry
Object space
Optical system
Image space
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2. Correcting bad behavior (i.e. optical design
methods) A. First principles
Rotational Symmetry
Plane symmetry
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2. Correcting bad behavior (i.e. optical design
methods) A. First principles
Rotational Symmetry
Plane symmetry
All coefficients (A,B, w1, w2, etc.) are
expressions of lens system construction
parameters (e.g. thickness, spacing, curvature,
conic, etc.)
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2. Correcting bad behavior (cont.) Example 1
Ritchey - Chretien Telescope Construction
parameters
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2. Correcting bad behavior (cont.) Example 1
Ritchey - Chretien Telescope Construction
parameters
0
B focal length
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2. Correcting bad behavior (cont.) Example 1
Ritchey - Chretien Telescope Construction
parameters
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2. Correcting bad behavior (cont.) Example 1
Ritchey - Chretien Telescope Construction
parameters
Remaining Degrees of Freedom d1, d2
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2. Correcting bad behavior (cont.) Example 2
Unobstructed three-mirror telescope (plane
symmetry)
18 construction parameters (16 for object at ?)
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2. Correcting bad behavior (cont.) Example 2
Unobstructed three-mirror telescope (cont.) 18
DOF ? 9 DOF (finite conjugates) ? 7 DOF
(telescope)
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2. Correcting bad behavior (cont.) Example 2
Unobstructed three-mirror telescope (cont.) 18
DOF ? 9 DOF (finite conjugates) ? 7 DOF
(telescope)
tant3 -((GI(d2 m)tantim d1(2d3tant1
mtantim))pow(d2GI d3GI (d1
GI)m,-1))/2. cost1 cos(atan(tant1)) cost2
cos(atan(tant2)) cost3 cos(atan(tant3)) S22I
(pow(cost1,-1)pow(GI,-1))/2. S22II -((d3GI
d1m d2m GIm)pow(cost2,-1)pow(d2,-1)pow
(d1 GI,-1)pow(m,-1))/2. S22III -((d2GI
d3GI d1m GIm)pow(cost3,-1)pow(d2,-1)pow(
d3,-1)pow(GI,-1))/2. S11I cost1cost1S22I
S11II cost2cost2S22II S11III
cost3cost3S22III S122I (pow(d1,-1)pow(GI,-2)
(d1(-(d2mtant1) d2GItantim
4GImtantim) (d3tant1 2mtantim)pow(d1,2)
(d2 2m)tantimpow(GI,2))pow(d1d3 -
d2m,-1))/4. S122II (pow(d1,-1)pow(d2,-2)pow(
d1 GI,-2)pow(m,-2) (m(2d3tant1
mtantim)pow(d1,3) - 2d2m(d2
2m)tantimpow(GI,2) pow(d1,2)(d3m(-2d2tan
t1 GI(2tant1 tantim)) 2GItant1pow(d3,2)
(2GItantim - d2(2tant1
tantim))pow(m,2)) d1(m(d3 m)tantimpow(GI,
2) 2tant1pow(d2,2)pow(m,2)
d2GI(-2d3mtant1 2d3GItantim
- 2tant1pow(m,2) - 5tantimpow(m,2)))))/8. S12
2III (pow(d2,-2)pow(d3,-2)pow(GI,-2)(-(d3m(
2d3tant1 mtantim)pow(d1,3)) - d2m(d2
m)(d2 - d3 3m)tantimpow(GI,2)
d1GI(-(d3GIm(d3 m)tantim)
pow(d2,2)(- 2d3mtant1 d3GItantim -
4tantimpow(m,2))d2(d3m(2mtant1
2GItantim mtantim) (2mtant1 -
GItantim)pow(d3,2) - 6tantimpow(m,3)))
pow(d1,2)(-(d3GI(d3m(2tant1 tantim)
2tant1pow(d3,2) 2tantimpow(m,2)))
d2(2d3m(mtant1 GItantim)
2GItant1pow(d3,2) - 3tantimpow(m,3))))pow(
-(d1d3) d2m,-1))/8. S111III
3cost3cost3S122III S111II
3cost2cost2S122II S111I 3cost1cost1S122I
While the equations to the left appear tedious,
they have only to be characterized ONCE and are
applicable for ALL three mirror systems with
plane symmetry.
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2. Correcting bad behavior (cont.) Design Note
Both of the previous examples predict
well-corrected optical imaging systems. Both
also require reflective aspheres, typically
conics, and they are generally non-rotationally
symmetric for unobstructed systems.
That is, the local surface curvature is different
in orthogonal directions. Conic constants can
differ as well.
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2. Correcting bad behavior (cont.) Design Note
Both of the previous examples predict
well-corrected optical imaging systems. Both
also require reflective aspheres, typically
conics, and they are generally non-rotationally
symmetric for unobstructed systems.
That is, the local surface curvature is different
in orthogonal directions. Conic constants can
differ as well.
The fundamental surface type for analytically
corrected systems is the general asymmetric
asphere
Which, not surprisingly, is the form of the
aberration equation previously listed.
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2. Correcting bad behavior (cont.) B. Brute
force
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• 2. Correcting bad behavior (cont.)
• B. Brute force
• Instead of reducing the DOF analytically, one can
  numerically search for systems using downhill
  or global optimization methods

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• 2. Correcting bad behavior (cont.)
• B. Brute force
• Instead of reducing the DOF analytically, one can
  numerically search for systems using downhill
  or global optimization methods
• Additionally, one can start with a limited number
  of DOF, e.g. number of lenses or mirrors in the
  system, and then add DOFs (additional mirrors,
  aspheric surfaces, break symmetry) until the
  requirements are met

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• 2. Correcting bad behavior (cont.)
• B. Brute force
• Instead of reducing the DOF analytically, one can
  numerically search for systems using downhill
  or global optimization methods
• Additionally, one can start with a limited number
  of DOF, e.g. number of lenses or mirrors in the
  system, and then add DOFs (additional mirrors,
  aspheric surfaces, break symmetry) until the
  requirements are met
• Disadvantages systems can have too many DOFs to
  converge quickly, and unknown solutions may be
  missed due to the local nature of downhill search
  methods

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• 2. Correcting bad behavior (cont.)
• B. Brute force
• Instead of reducing the DOF analytically, one can
  numerically search for systems using downhill
  or global optimization methods
• Additionally, one can start with a limited number
  of DOF, e.g. number of lenses or mirrors in the
  system, and then add DOFs (additional mirrors,
  aspheric surfaces, break symmetry) until the
  requirements are met
• Disadvantages systems can have too many DOFs to
  converge quickly, and unknown solutions may be
  missed due to the local nature of downhill search
  methods
• Most popular

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2. Correcting bad behavior (cont.) C. Copy Cat
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• 2. Correcting bad behavior (cont.)
• C. Copy Cat
• Start with a known system that is similar to the
  needs of the task at hand
• Patent literature
• Published papers
• In-house design

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• 2. Correcting bad behavior (cont.)
• C. Copy Cat
• Start with a known system that is similar to the
  needs of the task at hand
• Patent literature
• Published papers
• In-house design
• Optimize locally to force the system to meet
  the different requirements

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• 2. Correcting bad behavior (cont.)
• C. Copy Cat
• Start with a known system that is similar to the
  needs of the task at hand
• Patent literature
• Published papers
• In-house design
• Optimize locally to force the system to meet
  the different requirements
• Advantages Generally quick to converge, if the
  new system is close to the old one. This
  heritage approach tends to please conservative
  project managers.

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• Freeform optics designers toolkit
• (available in most lens design software)
• Conics, Cylinders, Cones, Toroids, Biconics,
  General aspheres
• General asymmetric asphere (i.e. Taylor expansion
  in x,y)
• Zernikes circle polynomial surfaces
• Radial spline and Asymmetric bi-cubic spline
  surfaces
• Faceted surfaces, Fresnel surfaces, Binary optics
• Grid surfaces (defined from a 2-d matrix)
• User defined surfaces catch all to do anything
  that can be programmed.

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3. Examples of systems with freeform surfaces
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3. Examples of systems with freeform
surfaces Hubble Primary mirror figure
error (as well as off-axis astigmatism) has
required freeform correctors for follow-on
instruments.
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3. Examples (cont.) Hubble WF/PC 3 (currently
not scheduled for service)
Pickoff mirror
M1 M2 Anamorphic Aspheres for correction
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3. Examples (cont.) Constellation-X
cone
Nested Cone mirrors for x-ray imaging
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3. Examples (cont.) Wasserman-Wolf aplanatic
telescope (spherical primary)
Free from spherical and coma through all
orders. M2 and M3 are splines defined by
solution of coupled linear differential equations
L. Mertz spherical primary telescopes are other
examples of this class.
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3. Examples (cont.) Image slicing
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3. Examples (cont.) Beam shapers
Useful for laser applications, or even
coronographs
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3. Examples (cont.) Consumer Optics Thin form
factor projection TV optics
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4. The Freeform Optics Manifesto Motivation
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• 4. The Freeform Optics Manifesto
• Motivation
• Experienced designers are generally conservative
  in choosing component prescriptions, especially
  when faced with time and budget constraints

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• 4. The Freeform Optics Manifesto
• Motivation
• Experienced designers are generally conservative
  in choosing component prescriptions, especially
  when faced with time and budget constraints
• As such, fabrication methods of freeform optics
  (and their limits) need to be understood by the
  designer, who can then incorporate them into the
  design process itself (e.g. a maximum departure
  from best fit sphere, maximum local slope)

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• 4. The Freeform Optics Manifesto
• Motivation
• Experienced designers are generally conservative
  in choosing component prescriptions, especially
  when faced with time and budget constraints
• As such, fabrication methods of freeform optics
  (and their limits) need to be understood by the
  designer, who can then incorporate them into the
  design process itself (e.g. a maximum departure
  from best fit sphere, maximum local slope)
• Naturally, testing methods of freeform optics
  (and their limits) also need to be understood by
  the design community, to establish confidence in
  the component in meeting its requirements.

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The Freeform Optics Manifesto
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• The Freeform Optics Manifesto
• DESIGN Develop design tools to facilitate a
  design for manufacturing approach to lens
  design, where solutions can be geared for a
  specific process.

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• The Freeform Optics Manifesto
• DESIGN Develop design tools to facilitate a
  design for manufacturing approach to lens
  design, where solutions can be geared for a
  specific process.
• FABRICATION Create the need for freeform
  surfaces within the design community by
  communicating their availability as a degree of
  freedom, and when possible, participating in the
  design process itself.

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• The Freeform Optics Manifesto
• DESIGN Develop design tools to facilitate a
  design for manufacturing approach to lens
  design, where solutions can be geared for a
  specific process.
• FABRICATION Create the need for freeform
  surfaces within the design community by
  communicating their availability as a degree of
  freedom, and when possible, participating in the
  design process itself.
• TESTING Ensure testing methods will keep pace
  with fabrication capability of freeform optics,
  establishing confidence within the design
  community for using freeform optics.

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