Abstract

We use first-order optical principles to examine the ability of gradient index (GRIN) lenses to correct chromatic aberrations. We consider radial GRIN lenses with flat surfaces, with a flat diffractive surface, and with curved surfaces. We model the GRIN material system as a locally varying, subwavelength blend of three materials. In this model, we demonstrate that the color-correcting properties of each lens type can be expressed solely in terms of the dispersion properties of the base materials. We find, at this level of approximation, that the material condition for a two-material GRIN achromat with curved surfaces is identical to that for a homogeneous doublet achromat comprised of the same two materials. For the more general case of three-material, ternary GRIN elements, we use the theory to develop a figure-of-merit-based optimization approach. This allows us to identify promising material combinations without first fabricating a GRIN element. The optimization approach can be applied to alternate GRIN geometries and arbitrary glass catalogs. We use our model to search a large, commercial glass catalog to identify the best achromatic glass combinations for the three different GRIN lenses described above. Significant numerical effort was required to identify which glass combinations performed best. Ternary glass combinations are necessary to achieve good achromatic performance for flat geometries. Diffraction combined with a graded-index enables improved color correction for the same optical power or nearly a factor of two increase in power for the same level of color correction. Glass pairs that perform well as an achromatic doublet also perform well chromatically when blended in a GRIN singlet.

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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2017 (2)

2016 (3)

2015 (1)

2013 (3)

R. A. Flynn, E. F. Fleet, G. Beadie, and J. S. Shirk, “Achromatic GRIN singlet lens design,” Opt. Express 21, 4970–4978 (2013).
[Crossref] [PubMed]

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

1998 (1)

J. A. Bennett, “Peter Dolland answers Jesse Ramsden,” Sphaera 8, 5 (1998).

1991 (1)

1988 (1)

1985 (1)

1971 (1)

Baer, E.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Beadie, G.

Bennett, J. A.

J. A. Bennett, “Peter Dolland answers Jesse Ramsden,” Sphaera 8, 5 (1998).

Brister, A.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Brocker, D. E.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

Buff, A.

Buralli, D. A.

Campbell, S. D.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

Chamma, K.

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Part One (Dover Publications, Inc., 1985).

Corsetti, J. A.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

Dupuy, C.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

Fang, K.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

Fleet, E. F.

Flynn, R. A.

George, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd Ed. (Roberts & Company Publishers, 1988).

Harmon, P.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

Houde-Walter, S. N.

Ji, S.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Kalinowski, M.

Kang, M.

Kirk, A.

Mackey, M.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Mait, J.

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Ternary versus binary material systems for gradient index optics,” Proc. SPIE 10181, 1018108 (2017).
[Crossref]

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” Proc. SPIE 9822, 98220Q (2016).
[Crossref]

Mait, J. N.

Markel, V. A.

Mayer, T. S.

McCarthy, P.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

Melvin, M.

Milojkovic, P.

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Ternary versus binary material systems for gradient index optics,” Proc. SPIE 10181, 1018108 (2017).
[Crossref]

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” Proc. SPIE 9822, 98220Q (2016).
[Crossref]

J. N. Mait, G. Beadie, R. A. Flynn, and P. Milojkovic, “Dispersion design in gradient index elements using ternary blends,” Opt. Express 24, 29295–29301 (2016).
[Crossref] [PubMed]

J. N. Mait, G. Beadie, P. Milojkovic, and R. A. Flynn, “Chromatic analysis and design of a first-order radial GRIN lens,” Opt. Express 23, 22069–22086 (2015).
[Crossref] [PubMed]

Moore, D. T.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

S. N. Houde-Walter and D. T. Moore, “Gradient-index profile control by field-assisted ion exchange in glass,” Appl. Opt. 24, 4326–4333 (1985).
[Crossref] [PubMed]

Morris, G. M.

Musgraves, J. D.

Park, S.-K.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

Ponting, M.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Richardson, K.

Rivero-Baleine, C.

Sands, P. J.

Schmidt, G. R.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

Shirk, J. S.

Sisken, L.

Smith, C.

Stone, T.

Visconti, A. J.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

Wachtel, P.

Werner, D. H.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, Inc., 1988).

Yin, K.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Opt. Express (3)

Opt. Mater. Express (1)

Proc. SPIE (2)

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” Proc. SPIE 9822, 98220Q (2016).
[Crossref]

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Ternary versus binary material systems for gradient index optics,” Proc. SPIE 10181, 1018108 (2017).
[Crossref]

Sphaera (1)

J. A. Bennett, “Peter Dolland answers Jesse Ramsden,” Sphaera 8, 5 (1998).

Other (6)

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

J. W. Goodman, Introduction to Fourier Optics, 3rd Ed. (Roberts & Company Publishers, 1988).

A. E. Conrady, Applied Optics and Optical Design, Part One (Dover Publications, Inc., 1985).

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, Inc., 1988).

“Optical glass - overview,” https://www.us.schott.com/advanced_optics/english/download/index.html .

“Mgrinx64_radial_v102.dll,” https://github.com/GrinLens/Zemax-Radial-and-Ball-GRIN/releases .

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Figures (10)

Fig. 1
Fig. 1 Achromatic lenses. (a) Wood. (b) Hybrid diffractive-Wood. (c) GRIN (refractive-Wood).

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Fig. 2
Fig. 2 Dispersion map representation of binary and ternary material blends. Binary blends exist on the periphery of the region indicated. Ternary blends exist within its interior. The red vertical line indicates a graded ternary blend whose index slope is constant at the design wavelength.

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Fig. 3
Fig. 3 Dispersion map representation of ternary material blends for quantized volume ratios.

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Fig. 4
Fig. 4 Histogram of triplet combinations for achromatic Wood lenses from the coarse search, plotted on a semi-log10 scale. Vertical dashed lines indicate boundaries for the top 10 and top 100 FOM values, out of the 357,760 glass combinations.

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Fig. 5
Fig. 5 Index versus index slope representation of selected Wood lenses. The base glasses form vertices of triangles, with the lens solution forming a line that transects the triangle. The best-performing material triplet is indicated by its solid black perimeter. Labels indicate the triplet’s rank in Table 1.

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Fig. 6
Fig. 6 Wood lens designs based on two different material systems: (top) Material system from simple triangle search, (bottom) Top-ranked solution from Table 1. Lens design details discussed in the text.

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Fig. 7
Fig. 7 Histogram of triplet combinations for achromatic diffractive Wood lenses from the coarse search, plotted on a semi-log10 scale. Vertical dashed lines indicate boundaries for the top 10 and top 100 FOM values, out of the 357,760 glass combinations.

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Fig. 8
Fig. 8 Index versus index slope representation of selected diffractive Wood lens FOM solutions. The base glasses form vertices of triangles, with the lens solution forming a line which transects the triangle. The best-performing solution is outlined by the triangle with solid lines. Each triangle is labeled with its solution’s corresponding rank in Table 3.

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Fig. 9
Fig. 9 Histogram of triplet combinations for achromatic GRIN singlet lenses from the coarse search, plotted on a semi-log10 scale. Vertical dashed lines indicate boundaries for the top 10 and top 100 FOM values, out of the 357,760 glass combinations.

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Fig. 10
Fig. 10 Index versus index slope representation of the best-performing blend for an achromatic GRIN singlet. The vertices of the triangle are defined by the base glasses. The optimal blend for the lens is defined by a line that transects the triangle. The blend is labeled with the triplet’s rank in Table 5.

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Tables (6)

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Table 1 Material Triplet Rankings for an Achromatic Wood Lens.

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Table 2 Parameters for Achromatic Wood Lenses.

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Table 3 Material Triplet Rankings for an Achromatic Diffractive Wood Lens.

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Table 4 Parameters for Achromatic Diffractive Wood Lenses.

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Table 5 Material Triplet Rankings for an Achromatic GRIN Singlet Lens.

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Table 6 Parameters for Achromatic GRIN Singlet Lenses.

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Equations (79)

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ϕ R ( λ ) = ( 1 R f − 1 R b ) [ n ( λ ) − 1 ] ,
d ϕ R ( λ ) d λ = ϕ ˙ R ( λ ) , = ϕ R ( λ ) V R ( λ ) ,
V ( λ ) = ϕ ( λ ) ϕ ˙ ( λ ) .
V R ( λ ) = n ( λ ) − 1 n ˙ ( λ ) .
ϕ D ( λ ) = 8 M λ D 2 .
V D ( λ ) = λ .
ϕ G ( λ ) = ϕ R ( λ ) C [ Φ ( λ ) ] + [ ϕ W ( λ ) + ϕ R 2 ( λ ) ] Snc [ Φ ( λ ) ] .
Φ ( λ ) = ( 8 t 2 D 2 ) [ | n edg ( λ ) − n ctr ( λ ) | n ctr ( λ ) ] .
n ( r , λ ) = n ctr ( λ ) + [ n edg ( λ ) − n ctr ( λ ) ] ( r D ) 2 .
ϕ G ( λ ) = ϕ R ( λ ) + ϕ W ( λ ) ,
ϕ ˙ G ( λ ) = ϕ R ( λ ) V R ( λ ) + ϕ W ( λ ) V W ( λ ) ,
ϕ R ( λ ) = ( 1 R f − 1 R b ) [ n ctr ( λ ) − 1 ] ,
ϕ W ( λ ) = ( 8 t D 2 ) [ n ctr ( λ ) − n edg ( λ ) ] .
V W ( λ ) = n ctr ( λ ) − n edg ( λ ) n ˙ ctr ( λ ) − n ˙ edg ( λ ) .
ϕ G ( λ ) = ϕ ctr ( λ ) + ϕ edg ( λ ) ,
ϕ ctr ( λ ) = [ 1 R f − ( 1 R b − 8 t D 2 ) ] [ n ctr ( λ ) − 1 ] ,
ϕ edg ( λ ) = [ ( 1 R b − 8 t D 2 ) − 1 R b ] [ n edg ( λ ) − 1 ] .
n 2 ( r , λ ) = n a 2 ( λ ) + γ b ( r ) [ n b 2 ( λ ) − n a 2 ( λ ) ] + γ c ( r ) [ n c 2 ( λ ) − n a 2 ( λ ) ] ,
γ a ( r ) + γ b ( r ) + γ c ( r ) = 1 .
n ˙ ( r , λ ) = [ 1 n ( r , λ ) ] { n a ( λ ) n ˙ a ( λ ) + γ b ( r ) [ n b ( λ ) n ˙ b ( λ ) − n a ( λ ) n ˙ a ( λ ) ] + γ c ( r ) [ n c ( λ ) n ˙ c ( λ ) − n a ( λ ) n ˙ a ( λ ) ] } .
n ctr ( λ ) = { n a 2 ( λ ) + γ b ( 0 ) [ n b 2 ( λ ) − n a 2 ( λ ) ] + γ c ( 0 ) [ n c 2 ( λ ) − n a 2 ( λ ) ] } 1 / 2 ,
n edg ( λ ) = { n a 2 ( λ ) + γ b ( D 2 ) [ n b 2 ( λ ) − n a 2 ( λ ) ] + γ c ( D 2 ) [ n c 2 ( λ ) − n a 2 ( λ ) ] } 1 / 2 ,
n ˙ ctr ( λ ) = [ 1 n ctr ( λ ) ] { n a ( λ ) n ˙ a ( λ ) + γ b ( 0 ) [ n b ( λ ) n ˙ b ( λ ) − n a ( λ ) n ˙ a ( λ ) ] + γ c ( 0 ) [ n c ( λ ) n ˙ c ( λ ) − n a ( λ ) n ˙ a ( λ ) ] } .
n ˙ edg ( λ ) = [ 1 n edg ( λ ) ] { n a ( λ ) n ˙ a ( λ ) + γ b ( D 2 ) [ n b ( λ ) n ˙ b ( λ ) − n a ( λ ) n ˙ a ( λ ) ] + γ c ( D 2 ) [ n c ( λ ) n ˙ c ( λ ) − n a ( λ ) n ˙ a ( λ ) ] } .
ϕ ( λ ) = ϕ 1 ( λ ) + ϕ 2 ( λ ) ,
ϕ ˙ ( λ ) = ϕ ˙ 1 ( λ ) + ϕ ˙ 2 ( λ ) , = ϕ 1 ( λ ) V 1 ( λ ) + ϕ 2 ( λ ) V 2 ( λ ) .
ϕ ˙ 1 ( λ ) = − ϕ ˙ 2 ( λ ) ,
ϕ ( λ ) = C 1 h 1 ( λ ) + C 2 h 2 ( λ ) .
C 2 h ˙ 2 ( λ 0 ) = − C 1 h ˙ 1 ( λ 0 ) .
C 1 = h ˙ 2 ( λ 0 ) ϕ ( λ 0 ) h 1 ( λ 0 ) h ˙ 2 ( λ 0 ) − h 2 ( λ 0 ) h ˙ 1 ( λ 0 ) ,
C 2 = − h ˙ 1 ( λ 0 ) ϕ ( λ 0 ) h 1 ( λ 0 ) h ˙ 2 ( λ 0 ) − h 2 ( λ 0 ) h ˙ 1 ( λ 0 ) .
ϕ ( λ ) ϕ ( λ 0 ) = h 1 ( λ ) h ˙ 2 ( λ 0 ) − h 2 ( λ ) h ˙ 1 ( λ 0 ) h 1 ( λ 0 ) h ˙ 2 ( λ 0 ) − h 2 ( λ 0 ) h ˙ 1 ( λ 0 ) .
h 1 ( λ ) = h 2 ( λ ) + Δ n .
ϕ ( λ ) = C 1 Δ n .
ϕ ( λ ) = 8 t D 2 n ctr ( λ ) − 8 t D 2 n edg ( λ ) .
C 1 = 8 t D 2 ,
C 2 = 8 t D 2 ,
h 1 ( λ ) = n ctr ( λ ) ,
h 2 ( λ ) = n edg ( λ ) .
n ctr ( λ ) = n edg ( λ ) + Δ n .
ϕ ( λ ) = 8 t Δ n / D 2 .
ϕ ( λ ) = 8 M λ D 2 + 8 t D 2 [ n ctr ( λ ) − n edg ( λ ) ] .
C 1 = 8 M λ 0 D 2 ,
C 2 = 8 t D 2 ,
h 1 ( λ ) = λ λ 0 ,
h 2 ( λ ) = n ctr ( λ ) − n edg ( λ ) .
ϕ ( λ ) = ( 1 R f − 1 R b ) [ n ctr ( λ ) − 1 ] + 8 t D 2 [ n ctr ( λ ) − n edg ( λ ) ] .
C 1 = 1 R f − ( 1 R b − 8 t D 2 ) ,
C 2 = ( 1 R b − 8 t D 2 ) − 1 R b ,
h 1 ( λ ) = n ctr ( λ ) − 1 ,
h 2 ( λ ) = n edg ( λ ) − 1 .
m 12 = [ h 1 ( λ 0 ) − h 2 ( λ 0 ) h ˙ 1 ( λ 0 ) h ˙ 2 ( λ 0 ) ] 2 σ [ h 1 ( λ ) − h 2 ( λ ) h ˙ 1 ( λ 0 ) h ˙ 2 ( λ 0 ) ] + δ σ , ≈ ϕ ( λ 0 ) σ [ ϕ ( λ ) ϕ ( λ 0 ) ] ,
σ [ g ( λ ) ] = { ( 1 N λ ) ∑ i = 1 N λ [ g ( λ i ) − g ¯ ] 2 } 1 / 2 ,
g ¯ = ( 1 N λ ) ∑ i = 1 N λ g ( λ i ) ,
n ^ = ( 1 N ) ∑ i = 1 N [ | n a ( λ i ) − n b ( λ i ) | + | n ˙ a ( λ i ) − n ˙ b ( λ i ) | + | n ¨ a ( λ i ) − n ¨ b ( λ i ) | ] .
m continuum = m 12 ( ∊ γ 1 + ∊ γ ) ,
∊ γ = { [ γ b ( 0 ) − γ b ( D / 2 ) ] 2 + [ γ c ( 0 ) − γ c ( D / 2 ) ] 2 0.05 } 5 .
m WL = [ n ctr ( λ 0 ) − n edg ( λ 0 ) ] 2 σ [ n ctr ( λ ) − n edg ( λ ) ] .
m DWL = [ 1 − n ctr ( λ 0 ) − n edg ( λ 0 ) λ 0 [ n ˙ ctr ( λ 0 ) − n ˙ edg ( λ 0 ) ] ] 2 σ [ λ λ 0 − n ctr ( λ ) − n edg ( λ ) λ 0 [ n ˙ ctr ( λ 0 ) − n ˙ edg ( λ 0 ) ] ] .
m GSL = [ ( n ctr ( λ 0 ) − 1 ) − n ˙ ctr ( λ 0 ) n ˙ edg ( λ 0 ) ( n edg ( λ 0 ) − 1 ) ] 2 σ [ ( n ctr ( λ ) − 1 ) − n ˙ ctr ( λ 0 ) n ˙ edg ( λ 0 ) ( n edg ( λ ) − 1 ) ] .
ϕ G ( λ ) = ϕ R ( λ ) C [ Φ ( λ ) ] + [ ϕ W ( λ ) + ϕ R 2 ( λ ) ] Snc [ Φ ( λ ) ] ,
Snc [ Φ ( λ ) ] = S [ Φ ( λ ) ] Φ ( λ ) .
s G = sgn [ n ctr ( λ ) − n edg ( λ ) ] ,
C [ Φ ( λ ) ] = cos Φ ( λ ) ,
S [ Φ ( λ ) ] = sin Φ ( λ ) ,
C [ Φ ( λ ) ] = cosh Φ ( λ ) ,
S [ Φ ( λ ) ] = sinh Φ ( λ ) .
ϕ R ( λ ) = ( 1 R f − 1 R b ) [ n ctr ( λ ) − 1 ] .
ϕ R 2 ( λ ) = [ n ctr ( λ ) − 1 ] 2 t R f R b n ctr ( λ ) ,
ϕ W ( λ ) = ( 8 t D 2 ) [ n ctr ( λ ) − n edg ( λ ) ] .
Φ ( λ ) = ( 8 t 2 D 2 ) [ | n edg ( λ ) − n ctr ( λ ) | n ctr ( λ ) ] ,
ϕ ˙ G ( λ ) = { C [ Φ ( λ ) ] ϕ ˙ R ( λ ) + ϕ R ( λ ) C ˙ [ Φ ( λ ) ] } + { Snc [ Φ ( λ ) ] [ ϕ ˙ W ( λ ) + ϕ ˙ R 2 ( λ ) ] + [ ϕ W ( λ ) + ϕ R 2 ( λ ) ] S ˙ nc [ Φ ( λ ) ] } = ϕ R ( λ ) [ C [ Φ ( λ ) ] V R ( λ ) + C ˙ [ Φ ( λ ) ] ] + ϕ R 2 ( λ ) [ Snc [ Φ ( λ ) ] V R 2 ( λ ) + S ˙ nc [ Φ ( λ ) ] ] + ϕ W ( λ ) [ Snc [ Φ ( λ ) ] V W ( λ ) + S ˙ nc [ Φ ( λ ) ] ] , = ϕ R ( λ ) V Rth ( λ ) + ϕ R 2 ( λ ) V R 2 th ( λ ) + ϕ W ( λ ) V Wth ( λ ) ,
V Rth ( λ ) = V R ( λ ) C [ Φ ( λ ) ] − s G V R ( λ ) Φ 2 ( λ ) Snc [ Φ ( λ ) ] [ Φ ˙ ( λ ) / Φ ( λ ) ] ,
V R 2 th ( λ ) = V R 2 ( λ ) Snc [ Φ ( λ ) ] + V R 2 ( λ ) { C [ Φ ( λ ) − Snc [ Φ ( λ ) ] } [ Φ ˙ ( λ ) / Φ ( λ ) ] ,
V Wth ( λ ) = V W ( λ ) Snc [ Φ ( λ ) ] + V W ( λ ) { C [ Φ ( λ ) ] − Snc [ Φ ( λ ) ] } [ Φ ˙ ( λ ) / Φ ( λ ) ] ,
V R ( λ ) = n ctr ( λ ) − 1 n ˙ ctr ( λ ) ,
V R 2 ( λ ) = { n ctr ( λ ) [ n ctr ( λ ) − 1 ] n ctr ( λ ) + 1 } [ 1 n ˙ ctr ( λ ) ] ,
V W ( λ ) = n ctr ( λ ) − n edg ( λ ) n ˙ ctr ( λ ) − n ˙ edg ( λ ) .
Φ ˙ ( λ ) Φ ( λ ) = ( 1 2 ) [ n edg ( λ ) n ctr ( λ ) − n edg ( λ ) ] [ n ˙ ctr ( λ ) n ctr ( λ ) − n ˙ edg ( λ ) n edg ( λ ) ] .

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