Extended depth-of-field lenses and methods for their design, optimization and manufacturing
View Patent ↗For rotationally symmetric aspheric lenses, one can establish lens design and layout based upon the phase delay function exp[−iφ(ρ)]. An embodiment of the invention is a method for calculating a corresponding variation in focal length denoted by f(ρ). According to an aspect, one can also assert a shape for the focal length f(ρ) and thereafter calculate a phase delay function in order to synthesize a novel lens. New EDoF lens designs are obtained by selection of an inner and outer focal length connected by a simple curve that can be approximate by a polynomial. From the selected f(ρ), one can synthesize a finished EDoF lens design and fabricate the lens. Another aspect of this invention is directed to a method to tailor prior-art EDoF lenses so that their performance over some range is improved.
1. A method for making an aspheric lens, comprising:
a) selecting a radially varying focal length value f(ρ) for the lens, where ρis the lens radius; and
determining a corresponding radially varying phase delay function φ(ρ); and
relating f(ρ) and φ(ρ) via the expression
φ
(
ρ
)
=
∫
0
ρ
-
k
ρ
[
f
2
(
ρ
1
)
+
ρ
1
2
]
1
/
2
ⅆ
ρ
1
,
where ρ 1 is the variable of integration running from zero to ρ and k is a constant.
2. The method of claim 1 , further comprising fabricating the lens.
3. The method of claim 1 , further comprising modifying a performance metric of the lens as a function of f(ρ).
4. The method of claim 3 , further comprising modifying a Strehl ratio of the lens.
5. The method of claim 3 , further comprising modifying a Point Spread Function of the lens.
6. The method of claim 1 , further comprising modifying a depth-of-field value of the lens as a function of f(ρ).
7. The method of claim 1 , further comprising determining an extended depth-of-field of the lens by selecting an inner object distance s 1 and an outer object distance s 2 .
8. The method of claim 1 , further comprising determining a polynomial representation of f(ρ) and obtaining a curve of f versus ρ over a desired range of ρ.
9. The method of claim 8 , further comprising modifying a portion of the curve to provide improved lens performance over a desired object distance sub-range and determining a corresponding modified polynomial representation of f(ρ).
10. The method of claim 9 , further comprising modifying a portion of the curve to provide improved lens performance over a desired object distance sub-range and determining a corresponding modified polynomial representation of f(ρ) mod .
11. The method of claim 10 , further comprising determining a modified phase delay function φ(ρ) mod of the lens from the modified focal length f(ρ) mod .
12. The method of claim 11 , wherein
φ
(
ρ
)
mod
=
∫
0
ρ
-
k
ρ
[
f
2
(
ρ
1
)
mod
+
ρ
1
2
]
1
/
2
ⅆ
ρ
1
,
where ρ 1 is the variable of integration running from zero to ρ.
13. The method of claim 1 , comprising making a rotationally symmetric, aspheric lens having an extended depth of field.
14. A method for optimizing the design of a rotationally symmetric, aspheric, extended depth of field (EDoF) lens specified by a known phase delay function φ L (ρ), where ρ is the lens radius, comprising:
determining a value of the radially-varying focal length f L (ρ) of the lens from the known phase delay function φ L (ρ), wherein said lens has a given range of focal lengths over the range of ρ;
determining a new value of a radially-varying focal length f new (ρ) of the lens that is different than f L (ρ), wherein f new (ρ) incorporates the given range of focal lengths over a range of ρ new , where ρ new is less than ρ; and
determining a new value of a phase delay function φ new (ρ new ).
15. The method of claim 14 , wherein the value of f new (ρ) is constant over a range between about 0.8ρ≦ρ new <ρ.
16. The method of claim 14 , further comprising expressing f new (ρ) as a curve that can be approximated by an n th -order polynomial, where 1≦n ≦6 including non-integer values.
17. The method of claim 16 , wherein the curve is a parabola.
18. The method of claim 16 , further comprising hyper-optimizing the optimized design, comprising:
generating a plurality of curves based on at least two different values of n;
determining a value of a lens performance metric over a desired range of object distances for each of the curves;
selecting the values of the lens performance metrics for two of the curves;
selecting a better range of performance metric values for a corresponding range of object distances over a radial portion ρ 1 of a first one of the two curves and selecting a better range of performance metric values for a corresponding range of different object distances over a radial portion ρ 2 of a second one of the two curves;
generating a new curve f new-hyper (ρ) that includes the radial portions of the first and the second curves and, which, smoothly connects the first curve at ρ 1 to the second curve at ρ 2 ;
determining a polynomial expression for the new, hyper-optimized function f new-hyper (ρ); and
directly determining a hyper-optimized phase delay function φ hyper (ρ) as a function of the lens radius ρ.
19. The method of claim 18 , wherein
φ
hyper
(
ρ
)
=
∫
0
ρ
-
k
ρ
[
f
new
-
hyper
(
ρ
1
)
+
ρ
1
2
]
1
/
2
ⅆ
ρ
1
,
where ρ 1 is the variable of integration running from zero to ρ.
20. A method for designing an aspheric lens that includes a diffraction limited lens and a phase mask, having an extended depth of field (EDoF), comprising:
determining a phase delay function φ P (ρ) of the diffraction limited lens, where ρis the lens radius;
determining a total phase delay function φ L (ρ) of the EDoF lens;
determining a phase delay function φ E (ρ) of the phase mask to provide a desired EDoF, where φ L (ρ)=φ P (ρ)+φ E (ρ);
determining a radially-varying focal length function f L (ρ) of the EDoF lens from the function φ L (ρ);
determining a value of a lens performance metric for the EDoF lens;
modifying f L (ρ) so as to tailor the design to improve the performance metric; and
determining a tailored phase delay function φ T (ρ).
21. The method of claim 20 , further comprising recalculating the value of the lens performance metric for the tailored EDoF lens.
22. The method of claim 21 , further comprising modifying the tailored f (ρ) value to further tailor the lens performance.
23. The method of claim 22 , further comprising recalculating the value of the lens performance metric for the furthered tailored EDoF lens.
24. The method of claim 20 , wherein determining f L (ρ) comprises determining a polynomial representation of f L (ρ) and obtaining a curve of f L versus ρ for a desired range of ρ.
25. The method of claim 24 , further comprising modifying a portion of the curve to provide improved lens performance over a desired object distance sub-range and determining a corresponding modified polynomial representation of f L (ρ) .
26. The method of claim 25 , further comprising determining a modified phase delay function φ L (ρ) of the lens from the modified focal length f L (ρ).
27. The method of claim 24 , wherein the step of modifying f L (ρ) so as to tailor the design to improve the performance metric comprises decreasing the slope of the curve of f L (ρ) corresponding to values of ρ over which improved performance is desired.