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

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References

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    [CrossRef]
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2008

2007

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]

2006

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]

2000

1997

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]

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

B. D. Stone and G. W. Forbes, “Differential ray tracing in inhomogeneous media,” J. Opt. Soc. Am. A14(10), 2824–2836 (1997).
[CrossRef]

1996

1982

1980

1971

Baer, E.

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]

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]

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]

Beadie, G.

Blankenbecler, R.

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

Bociort, F.

Ebeling, T.

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]

Fleet, E.

Forbes, G. W.

Ghatak, A. K.

Gordon, J. M.

Hiltner, A.

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]

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]

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]

Jin, Y.

Kamdar, A. R.

Kazmierczak, T.

Krishna, K. S. R.

Kumar, D. V.

Lane, P. A.

Manhart, P. K.

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

Moore, D. T.

Mueller, C. D.

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]

Nazarenko, S.

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]

Ponting, M.

Rosenberg, A.

Sands, P. J.

Schuman, T. L.

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]

Sharma, A.

Shirk, J. S.

Stone, B. D.

Tai, H.

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]

Tarkhanov, V. I.

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]

Yang, Y.

Appl. Opt.

J. Appl. Polym. Sci.

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]

J. Opt. A, Pure Appl. Opt.

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]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

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

Opt. Express

Polym. Eng. Sci.

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]

Other

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).

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Figures (5)

Fig. 1
Fig. 1

Schematic of the manufacturing process for building a GRIN lens. Individual films consist of a nanolayered composition of at least 2 different polymers. Each film can have a different index of refraction. The films are stacked to conform to a desired index profile, molded, and then polished to produce GRIN lenses. AFM image, right, has thicker nanolayered pairs (~188 nm) than are used in GRIN films and is shown only to illustrate the layered structure.

Fig. 2
Fig. 2

The lens is defined by a center thickness tc, surface radius of curvature RL, and a gradient index radius of curvature RG, both radii defined relative to the vertex of the lens on the left hand side (z = 0). RG and RL are positive as shown. In the inset is the distance Δz traveled by a ray at height y from the vertex plane to the surface of the lens.

Fig. 3
Fig. 3

Computed longitudinal focal shifts as a function of wavelength. Left: a single homogeneous lens (solid line) and several near-optimized GRIN lenses that bracket the GRIN achromat. Right: the GRIN achromat plotted on vertical scale 10,000x smaller than the plot on the left.

Fig. 4
Fig. 4

GRIN profiles in a plano-convex lens. The index of refraction is plotted, left, along the center of the lens on the optical axis (z) and, right, across the rear surface of the lens (x) at z = 1 mm. These profiles are for an idealized material made from two materials of base index 1.48 and 1.7, and are calculated for λ = 587.6 nm.

Fig. 5
Fig. 5

Focal length dependence on three GRIN parameters. Optimized focal lengths are shown for GRIN lenses with varying RG, V, and δn (as represented by n2).

Tables (1)

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Table 1 Results from Four Test Cases Comparing a Full Ray Trace to the Approximation in Eq. (12)

Equations (12)

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φ( y )= k o Δz( y )+ k o Δz( y ) tc n[ r( y,z ) ]dz
r( y,z )= R G | R G | y 2 + ( R G z) 2
n( r )= n 0 +a(r R G )
n( y,z )= n 0 +a R G [ z R G + y 2 2 R G 2 +ϑ{ ( y z 2 / R G 3 ),( z 3 / R G 3 ) } ]
Δz( y )=y[ y 2 R L +ϑ{ ( y/ R L ) 3 } ].
φ( y ) k o ( n 0 t c a t c 2 2 ) k o 2 ( n 0 1 R L a t c R G ) y 2 + k o 4 ( a 2 R L 2 a R L R G ) y 4 .
φ f ( y )= φ o + k o ( f f 2 + y 2 )= φ o k o f 2 ( y f ) 2 +ϑ{ k o f ( y/f ) 4 }
f( λ )= ( n 0 ( λ )1 R L a( λ ) t c R G ) 1
f( λ ) nonGRIN = ( n 0 ( λ )1 R L ) 1
1 f blue 1 f red = [ n 0 ( λ blue ) n 0 ( λ red ) ] R L [ n 0 ( λ blue ) n 0 ( λ red ) ][ n 1 ( λ blue ) n 1 ( λ red ) ] R G
R L = R G Δ n 0 Δ n 0 Δ n 1
f balanced = R G / Δ n 1 n 1 ( λ red )1 Δ n 1 n 0 ( λ red )1 Δ n 0

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