## Achromatic GRIN singlet lens design |

Optics Express, Vol. 21, Issue 4, pp. 4970-4978 (2013)

http://dx.doi.org/10.1364/OE.21.004970

Acrobat PDF (1408 KB)

### Abstract

Gradient refractive index (GRIN) materials are attractive candidates for improved optical design, especially in compact systems. For GRIN lenses cut from spherically symmetric GRIN material, we derive an analogue of the “lens maker’s” equation. Using this equation, we predict and demonstrate via ray tracing that an achromatic singlet lens can be designed, where the chromatic properties of the GRIN counterbalance those of the lens shape. Modeling the lens with realistic materials and realistic fabrication geometries, we predict we can make an achromatic singlet with a 19 mm focal length using a matrix of known polymers.

© 2013 OSA

## 1. Introduction

1. D. T. Moore, “Design of singlets with continuously varying indices of refraction,” J. Opt. Soc. Am. **61**(7), 886–894 (1971). [CrossRef]

3. V. I. Tarkhanov, “Lens with a spherical gradient of refractive index, ideally focusing for an object at a finite distance,” J. Opt. A, Pure Appl. Opt. **8**(6), 610–615 (2006). [CrossRef]

4. D. T. Moore, “Gradient-index optics: a review,” Appl. Opt. **19**(7), 1035–1038 (1980). [CrossRef] [PubMed]

5. G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express **16**(15), 11540–11547 (2008). [PubMed]

5. G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express **16**(15), 11540–11547 (2008). [PubMed]

7. Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci. **103**(3), 1834–1841 (2007). [CrossRef]

5. G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express **16**(15), 11540–11547 (2008). [PubMed]

8. K. S. R. Krishna and A. Sharma, “Chromatic aberrations of radial gradient-index lenses. I. Theory,” Appl. Opt. **35**(7), 1032–1036 (1996). [CrossRef] [PubMed]

9. F. Bociort, “Chromatic paraxial aberration coefficients for radial gradient-index lenses,” J. Opt. Soc. Am. A **13**(6), 1277–1284 (1996). [CrossRef]

10. P. J. Sands, “Inhomogeneous lenses. II. Chromatic paraxial aberrations,” J. Opt. Soc. Am. **61**(6), 777–783 (1971). [CrossRef] [PubMed]

11. P. J. Sands, “Inhomogeneous lenses. 5. Chromatic paraxial aberrations of lenses with axial or cylindrical index distributions,” J. Opt. Soc. Am. **61**(11), 1495–1500 (1971). [CrossRef]

12. P. K. Manhart and R. Blankenbecler, “Fundamentals of macro axial gradient index optical design and engineering,” Opt. Eng. **36**(6), 1607–1621 (1997). [CrossRef]

10. P. J. Sands, “Inhomogeneous lenses. II. Chromatic paraxial aberrations,” J. Opt. Soc. Am. **61**(6), 777–783 (1971). [CrossRef] [PubMed]

10. P. J. Sands, “Inhomogeneous lenses. II. Chromatic paraxial aberrations,” J. Opt. Soc. Am. **61**(6), 777–783 (1971). [CrossRef] [PubMed]

11. P. J. Sands, “Inhomogeneous lenses. 5. Chromatic paraxial aberrations of lenses with axial or cylindrical index distributions,” J. Opt. Soc. Am. **61**(11), 1495–1500 (1971). [CrossRef]

**61**(6), 777–783 (1971). [CrossRef] [PubMed]

13. ZEMAX software, Zemax Development Corp, www.zemax.com

## 2. Theory

*ϕ*(

*y*) along one axis

*y*perpendicular to the propagation direction

*z*:where

*k*is the vacuum wavevector of the light, Δ

_{o}*z*(

*y*) is the distance in air from the vertex plane to the lens surface at height

*y*,

*t*is the center thickness of the lens,

_{c}*r*(

*y,z*) is proportional to the distance from the GRIN center of curvature at point (

*y,z*) within the lens, and

*n*(

*r*) is the index of refraction at

*r*. The first term is the accumulated phase in air before striking the curved front surface, while the second represents the phase from traversing the GRIN.

*r*(

*y,z*) fromwhere

*R*is the radius of curvature of the GRIN contour at the lens vertex, with the usual sign convention

_{G}*R*>0 if the center lies to the right of the lens and

_{G}*R*<0 otherwise. Note that for

_{G}*R*<0,

_{G}*r*takes on negative values within the lens. As mentioned above, we model a simple linear relationship for the index of refraction:where

*n*is the index of refraction of the lens at its vertex and

_{0}*a*is the rate of index change with distance. (Because it will come up later in the text, we point out that for all values of

*R*the index of refraction

_{G}*n*at the back vertex of our lenses is given by

_{1}*n*=

_{1}*n*–

_{0}*a t*.) Assuming that

_{c}*R*>>(

_{G}*y,z*), which implies paraxial (y) and thin lens (z) assumptions, and inserting Eq. (2) into Eq. (3) gives an approximate GRIN profile along the integralThe form for Δ

*z*depends on the radius of curvature of the front surface. Denoting the radius of curvature of the lens

*R*and assuming that

_{L}*R*>>

_{L}*y*, again a paraxial assumption:Substituting the expressions in Eqs. (4) and (5) into Eq. (1) and computing the integral, we arrive at an approximate wave frontBy comparison, a wavefront which would focus perfectly at a focal distance

*f*is written as:From the second terms of Eqs. (6) and (7), we find an analytic approximation for the focal length

*f*(

*λ*) of the plano-convex GRIN lens:where we call out the dependence of variables

*a*, n

_{0}and

*f*on wavelength, λ.

*f*= (1/

*f*

_{1}+ 1/

*f*

_{2})

^{−1}. We can see how the geometric optical power, the first term, compares with the GRIN optical power, the second term. Critically, the power of the GRIN component depends not on a single index value, but rather the

*rate of change*in index via the slope parameter,

*a*. The GRIN power also grows as the GRIN curvature, 1/R

_{G}, becomes stronger. (In the limit of R

_{G}→ 0, Eq. (8) is invalid, as small R

_{G}violates the approximation R

_{G}>> (y,z).) Note that the GRIN power can be both positive and negative. If

*a*and

*R*have opposite signs then the GRIN adds focal power to the lens. On the other hand, if they have the same sign then they serve to weaken the lens power.

_{G}*n*is easy enough to understand – whatever material is at the vertex of the lens will exhibit some level of optical dispersion which is described by

_{0}*n*(

_{0}*λ*). The wavelength dependence of

*a*(

*λ*) is similar. At the back vertex of the lens (

*z*=

*t*and

_{c}*y*= 0) the index at some reference wavelength

*λ*is defined by [

_{ref}*n*(

_{0}*λ*) –

_{ref}*a*(

*λ*)

_{ref}*t*]. Just as for the front vertex, this index function is associated with a

_{c}*material*composition at the back vertex. This different material composition will exhibit its own dispersion characteristic, likely of different shape than that of the front vertex. Suppose the back material has both a higher index of refraction and steeper dispersion than the front vertex. Then

*a*(

*λ*) would be negative-valued, creating the higher index. Also,

_{ref}*a*(

*λ*) would have greater magnitude than

_{blue}*a*(

*λ*), leading to higher dispersion at the back vertex.

_{red}## 3. Achromat design

*f*for the same lens at two different wavelengths

*λ*and

_{blue}*λ*:where we use Eq. (8) twice and remember from the comment after Eq. (3) that (

_{red}*a t*) is equal to the difference in index between the front and back vertices (

_{c}*n*–

_{0}*n*). If we restrict ourselves to materials with normal dispersion and assume a plano-convex lens, then the first term in Eq. (10) will always be positive. To obtain a chromatically balanced lens, therefore, the second term must also be positive, and equal to the first term.

_{1}*n*–

_{0}^{blue}*n*) would be greater than that at the back vertex (

_{0}^{red}*n*–

_{1}^{blue}*n*) and the GRIN radius of curvature

_{1}^{red}*R*would have to be greater than zero to get a balanced lens. Taking the same materials but reversing their order, so that the high-index material is located at the back vertex, requires the sign of

_{G}*R*to change in order to maintain a chromatic balance.

_{G}*R*and material properties fixed and solve for the radius of curvature

_{G}*R*which sets the focal length difference in Eq. (10) equal to zero:In Eq. (11) we adopt the shorthand Δ

_{L}*n*= [

_{j}*n*(

_{j}*λ*) -

_{blue}*n*(

_{j}*λ*)] at surface j. We express the balanced focal length, where f

_{red}*= f*

_{red}*, by combining Eq. (8) at λ*

_{blue}*with Eq. (11):*

_{red}## 4. Achromat simulations

^{®}. While ZEMAX has the capability to trace rays through gradient index materials, and furthermore has a built in capability to model a spherical GRIN profile, it does not possess the native ability to model dispersion in a spherical GRIN lens. Therefore we were required to develop our own software model which could communicate dispersive GRIN information to ZEMAX’s ray trace engine. Each base material’s dispersion is modeled by Cauchy’s formula, as described in [15] on p. 17, Eq. (1).37-1.39, with coefficients calculated solely from index and abbe number. This integrated model was validated by comparing ZEMAX-computed wavefronts transmitted through GRIN test cases to those calculated from independently-developed GRIN propagation code, based on published algorithms [16

16. B. D. Stone and G. W. Forbes, “Differential ray tracing in inhomogeneous media,” J. Opt. Soc. Am. A **14**(10), 2824–2836 (1997). [CrossRef]

17. A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. **21**(6), 984–987 (1982). [CrossRef] [PubMed]

_{G}consistent with Eq. (3). The two materials' index, Abbe values (n, V) represent a fairly typical polymer (1.48, 60) and a high index dispersive polymer (1.70, 20) unless noted otherwise. An appropriate value of index slope, a, is chosen so that the index never exceeds the limits of the base polymers throughout the lens volume. Each lens is 1 mm thick; however to first order the lens thickness value is immaterial as long as Δn

_{0}-Δn

_{1}remains unchanged. Lens shape is plano-convex, as illustrated in Fig. 2. We optimize each lens by fixing the GRIN profile and allowing the software's optimization routine to select a surface power that shows least chromatic effect.

*V*to compute

_{j}*Δn*=

_{j}*V*/ (

_{j}*n*– 1), with

_{j}*n*as the d-line wavelength of material

_{j}*j*. There’s no unambiguous way to compute

*n*(

*λ*) from

_{red}*n, V*, however, so rather than using Eq. (12) to report the balanced focal length we report the focal length at the d-line as calculated from Eq. (8) and the curvature computed in Eq. (11). Included in Table 1 are focal lengths calculated from Eq. (8) and those determined by a ray trace through the ZEMAX model. Achromatic focal length match can be achieved both for standard material pairs displaying increasing dispersion with index and for more exotic materials pairs with decreasing dispersion with index, and in each case both for negative and positive GRIN curvature. Table 1 enumerates these four different combinations, where GRIN curvature is held constant at 20 mm but can be positive or negative. The exotic case of decreasing dispersion with index is included for academic interest; such materials are not currently in use. Agreement between ray traced and theoretically predicted focal lengths is within ± 2% of the paraxial theory.

_{G}is negative and R

_{L}positive. This higher power results from rays encountering a more rapidly varying radial GRIN profile.

_{G}is fixed at 20 mm. The left plot shows the focal shift for both a non-GRIN lens with R

_{L}= 27 mm (solid line), and two less than fully optimized GRIN singlets with R

_{L}= 26 and 28 mm (2 dashed lines). As R

_{L}is varied, the GRIN can both over-correct (R

_{L}= 28 mm) and under-correct (R

_{L}= 26 mm) the chromatic focal shift. We find the best overall focal shift at R

_{L}= 27.095 mm, depicted in the right plot in Fig. 3 (note the expanded vertical scale).

_{0}and n

_{1}, index at the front and back vertex of the lens, in Table 1 obscures an important consideration in a real spherically-varying GRIN lens. While one extreme index appears at a surface vertex, the other typically appears at a surface edge. Consider a lens spanning a maximum material index range of 1.48 to 1.70, with R

_{G}= −8 mm and a = −0.18. As Fig. 4 shows, the minimum index occurs at the vertex of the front surface (Z = 0, left plot), but the maximum index occurs at the

*edges*of the back surface (X = ± 2 mm, right plot). This boundary issue is incorporated into our ZEMAX model.

_{G}) and second blended material (represented by its index, n

_{2}, and Abbe number, V) are varied. The graph clearly shows a near-linear dependence of

*f*on R

_{G}, further validating the prediction in Eq. (12). Since the Abbe number and base index of one polymer are held fixed (n = 1.48, V = 60), this search focuses on the δn and δV of the system. Clear trends emerge: increased optical power results from decreasing R

_{G}, increasing δn, and increasing δV. Increasing n

_{2}, V from 1.57, 35 to a higher and more dispersive 1.7, 20 more than doubles optical power.

## 5. Conclusion

_{G}is desirable. Chromatic effects in the GRIN are stronger for small R

_{G}, resulting in a higher power achromatized system. Higher power lenses result when R

_{G}is opposite in sign to the surface radius, since rays encounter a more rapidly varying radial GRIN profile. Third, placing the lower dispersion material at the highest power lens surface allows the achromat to achieve the highest overall optical power.

## Acknowledgments

## References and links

1. | D. T. Moore, “Design of singlets with continuously varying indices of refraction,” J. Opt. Soc. Am. |

2. | J. M. Gordon, “Spherical gradient-index lenses as perfect imaging and maximum power transfer devices,” Appl. Opt. |

3. | V. I. Tarkhanov, “Lens with a spherical gradient of refractive index, ideally focusing for an object at a finite distance,” J. Opt. A, Pure Appl. Opt. |

4. | D. T. Moore, “Gradient-index optics: a review,” Appl. Opt. |

5. | G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express |

6. | C. D. Mueller, S. Nazarenko, T. Ebeling, T. L. Schuman, A. Hiltner, and E. Baer, “Novel structures by microlayer coextrusion - talc-filled PP, PC/SAN, and HDPE/LLDPE,” Polym. Eng. Sci. |

7. | Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci. |

8. | K. S. R. Krishna and A. Sharma, “Chromatic aberrations of radial gradient-index lenses. I. Theory,” Appl. Opt. |

9. | F. Bociort, “Chromatic paraxial aberration coefficients for radial gradient-index lenses,” J. Opt. Soc. Am. A |

10. | P. J. Sands, “Inhomogeneous lenses. II. Chromatic paraxial aberrations,” J. Opt. Soc. Am. |

11. | P. J. Sands, “Inhomogeneous lenses. 5. Chromatic paraxial aberrations of lenses with axial or cylindrical index distributions,” J. Opt. Soc. Am. |

12. | P. K. Manhart and R. Blankenbecler, “Fundamentals of macro axial gradient index optical design and engineering,” Opt. Eng. |

13. | ZEMAX software, Zemax Development Corp, www.zemax.com |

14. | G. Beadie and J. S. Shirk, “Effects of diffraction and partial reflection in multilayered gradient index polymer lenses,” in |

15. | R. Ditteon, |

16. | B. D. Stone and G. W. Forbes, “Differential ray tracing in inhomogeneous media,” J. Opt. Soc. Am. A |

17. | A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. |

**OCIS Codes**

(080.3630) Geometric optics : Lenses

(110.2760) Imaging systems : Gradient-index lenses

(220.3630) Optical design and fabrication : Lenses

(080.5692) Geometric optics : Ray trajectories in inhomogeneous media

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: November 20, 2012

Revised Manuscript: January 14, 2013

Manuscript Accepted: January 16, 2013

Published: February 21, 2013

**Citation**

Richard A. Flynn, E. F. Fleet, Guy Beadie, and James S. Shirk, "Achromatic GRIN singlet lens design," Opt. Express **21**, 4970-4978 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4970

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### References

- D. T. Moore, “Design of singlets with continuously varying indices of refraction,” J. Opt. Soc. Am.61(7), 886–894 (1971). [CrossRef]
- J. M. Gordon, “Spherical gradient-index lenses as perfect imaging and maximum power transfer devices,” Appl. Opt.39(22), 3825–3832 (2000). [CrossRef] [PubMed]
- V. I. Tarkhanov, “Lens with a spherical gradient of refractive index, ideally focusing for an object at a finite distance,” J. Opt. A, Pure Appl. Opt.8(6), 610–615 (2006). [CrossRef]
- D. T. Moore, “Gradient-index optics: a review,” Appl. Opt.19(7), 1035–1038 (1980). [CrossRef] [PubMed]
- G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express16(15), 11540–11547 (2008). [PubMed]
- C. D. Mueller, S. Nazarenko, T. Ebeling, T. L. Schuman, A. Hiltner, and E. Baer, “Novel structures by microlayer coextrusion - talc-filled PP, PC/SAN, and HDPE/LLDPE,” Polym. Eng. Sci.37(2), 355–362 (1997). [CrossRef]
- Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci.103(3), 1834–1841 (2007). [CrossRef]
- K. S. R. Krishna and A. Sharma, “Chromatic aberrations of radial gradient-index lenses. I. Theory,” Appl. Opt.35(7), 1032–1036 (1996). [CrossRef] [PubMed]
- F. Bociort, “Chromatic paraxial aberration coefficients for radial gradient-index lenses,” J. Opt. Soc. Am. A13(6), 1277–1284 (1996). [CrossRef]
- P. J. Sands, “Inhomogeneous lenses. II. Chromatic paraxial aberrations,” J. Opt. Soc. Am.61(6), 777–783 (1971). [CrossRef] [PubMed]
- P. J. Sands, “Inhomogeneous lenses. 5. Chromatic paraxial aberrations of lenses with axial or cylindrical index distributions,” J. Opt. Soc. Am.61(11), 1495–1500 (1971). [CrossRef]
- P. K. Manhart and R. Blankenbecler, “Fundamentals of macro axial gradient index optical design and engineering,” Opt. Eng.36(6), 1607–1621 (1997). [CrossRef]
- ZEMAX software, Zemax Development Corp, www.zemax.com
- G. Beadie and J. S. Shirk, “Effects of diffraction and partial reflection in multilayered gradient index polymer lenses,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2010), paper FThU3.
- R. Ditteon, Modern Geometrical Optics (John Wiley & Sons, 1998).
- B. D. Stone and G. W. Forbes, “Differential ray tracing in inhomogeneous media,” J. Opt. Soc. Am. A14(10), 2824–2836 (1997). [CrossRef]
- A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt.21(6), 984–987 (1982). [CrossRef] [PubMed]

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