Optical Design with Freeform Surfaces Joseph M. Howard, Ph.D.

1. Optical Design with Freeform Surfaces Joseph M. Howard, Ph.D.

2. 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

3. 1. Behavior of Optical Systems • A. Systems with rotational symmetry: “Wave aberration function”

4. 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.

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

6. 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.

7. 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)

8. 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

9. 2. Correcting bad behavior (i.e. optical design methods) • A. “First principles” Rotational Symmetry Object space Optical system Image space

10. 2. Correcting bad behavior (i.e. optical design methods) • A. “First principles” Rotational Symmetry Plane symmetry

11. 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.)

12. 2. Correcting bad behavior (cont.) Example 1: Ritchey - Chretien Telescope Construction parameters:

13. 2. Correcting bad behavior (cont.) Example 1: Ritchey - Chretien Telescope Construction parameters: 0 B = focal length

14. 2. Correcting bad behavior (cont.) Example 1: Ritchey - Chretien Telescope Construction parameters:

15. 2. Correcting bad behavior (cont.) Example 1: Ritchey - Chretien Telescope Construction parameters: Remaining Degrees of Freedom: d1, d2

16. 2. Correcting bad behavior (cont.) Example 2: Unobstructed three-mirror telescope (plane symmetry) 18 construction parameters (16 for object at )

17. 2. Correcting bad behavior (cont.) Example 2: Unobstructed three-mirror telescope (cont.) 18 DOF  9 DOF (finite conjugates)  7 DOF (telescope)

18. 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*(2*d3*tant1 + m*tantim))*pow(d2*GI + d3*GI + (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 = -((d3*GI + d1*m + d2*m + GI*m)*pow(cost2,-1)*pow(d2,-1)*pow(d1 + GI,-1)*pow(m,-1))/2.; S22III = -((d2*GI + d3*GI + d1*m + GI*m)*pow(cost3,-1)*pow(d2,-1)*pow(d3,-1)*pow(GI,-1))/2.; S11I = cost1*cost1*S22I; S11II = cost2*cost2*S22II; S11III = cost3*cost3*S22III; S122I = (pow(d1,-1)*pow(GI,-2)*(d1*(-(d2*m*tant1) + d2*GI*tantim + 4*GI*m*tantim) + (d3*tant1 +2*m*tantim)*pow(d1,2) + (d2 + 2*m)*tantim*pow(GI,2))*pow(d1*d3 - d2*m,-1))/4.; S122II = (pow(d1,-1)*pow(d2,-2)*pow(d1 + GI,-2)*pow(m,-2)* (m*(2*d3*tant1 + m*tantim)*pow(d1,3) - 2*d2*m*(d2 + 2*m)*tantim*pow(GI,2) + pow(d1,2)*(d3*m*(-2*d2*tant1 + GI*(2*tant1 + tantim)) + 2*GI*tant1*pow(d3,2) + (2*GI*tantim - d2*(2*tant1 + tantim))*pow(m,2)) + d1*(m*(d3 + m)*tantim*pow(GI,2) + 2*tant1*pow(d2,2)*pow(m,2) + d2*GI*(-2*d3*m*tant1 + 2*d3*GI*tantim - 2*tant1*pow(m,2) - 5*tantim*pow(m,2)))))/8.; S122III = (pow(d2,-2)*pow(d3,-2)*pow(GI,-2)*(-(d3*m*(2*d3*tant1 + m*tantim)*pow(d1,3)) - d2*m*(d2 + m)*(d2 - d3 + 3*m)*tantim*pow(GI,2) + d1*GI*(-(d3*GI*m*(d3 + m)*tantim) + pow(d2,2)*(- 2*d3*m*tant1 + d3*GI*tantim - 4*tantim*pow(m,2))+d2*(d3*m*(2*m*tant1 + 2*GI*tantim + m*tantim) + (2*m*tant1 - GI*tantim)*pow(d3,2) - 6*tantim*pow(m,3))) + pow(d1,2)*(-(d3*GI*(d3*m*(2*tant1 + tantim) + 2*tant1*pow(d3,2) + 2*tantim*pow(m,2))) + d2*(2*d3*m*(m*tant1 + GI*tantim) + 2*GI*tant1*pow(d3,2) - 3*tantim*pow(m,3))))*pow(-(d1*d3) + d2*m,-1))/8.; S111III = 3*cost3*cost3*S122III; S111II = 3*cost2*cost2*S122II; S111I = 3*cost1*cost1*S122I; 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.

19. 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.

20. 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.

21. 2. Correcting bad behavior (cont.) B. “Brute force”

22. 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

23. 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

24. 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

25. 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

26. 2. Correcting bad behavior (cont.) C. “Copy Cat”

27. 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

28. 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

29. 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.

30. 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) • Zernike’s 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.

31. 3. Examples of systems with freeform surfaces

32. 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.

33. 3. Examples (cont.) Hubble WF/PC 3(currently not scheduled for service) Pickoff mirror M1 & M2 Anamorphic Aspheres for correction

34. 3. Examples (cont.) Constellation-X cone Nested Cone mirrors for x-ray imaging

35. 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.

36. 3. Examples (cont.): Image slicing

37. 3. Examples (cont.) Beam shapers Useful for laser applications, or even coronographs

38. 3. Examples (cont.) Consumer Optics: Thin form factor projection TV optics

39. 4. The Freeform Optics Manifesto Motivation

40. 4. The Freeform Optics Manifesto • Motivation • Experienced designers are generally conservative in choosing component prescriptions, especially when faced with time and budget constraints

41. 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)

42. 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.

43. The Freeform Optics Manifesto

44. 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.

45. 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.

46. 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.

47. THANK YOU Go Navy!! Beat Army!!!