Abstract

From the expression for optical power of a radial first-order graded-index (GRIN) lens with curved surfaces, we derive an expression for chromatic aberration. Our expressions for optical power and chromatic aberration are valid under the paraxial approximation. By applying a series of further simplifying assumptions, namely a thin lens and thin GRIN, we derive a set of equations with which one can design an achromatic GRIN lens. We also derive expressions for the dispersive property of a GRIN element. Our analysis enables us to derive the relationship between material pairs that indicate their suitability as a material pair for a GRIN achromat. We use this relationship to search a standard glass catalog for attractive GRIN material pairs for a particular achromat design. We compare the optical performance of our GRIN design to that of a conventional homogeneous doublet and demonstrate that our approach is capable of identifying material pairs that perform well for achromatic GRIN lenses which would not generally be considered for conventional achromatic design. We also demonstrate our approach is capable of designing GRIN achromats with superior performance.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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2013 (2)

J. A. Corsetti, P. McCarthy, and D. T. Moore, “Color correction in the infrared using gradient-index materials,” Opt. Eng. 52, 112109 (2013).
[Crossref]

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]

2010 (1)

M. Ponting, A. Hiltner, and E. Baer, “Polymer nanostructures by forced assembly: process, structure, properties,” Macromol. Symp. 294, 19–32 (2010).
[Crossref]

2004 (1)

H. Liu, T. Maekawa, N. M. Patrikalakis, E. M. Sachs, and W. Cho, “Methods for feature-based design of heterogeneous solids,” Computer-Aided Design 36, 1141–1159 (2004).
[Crossref]

1997 (1)

1996 (3)

1988 (1)

K. M. Jones, S. Lundgren, and A. Chakravorty, “A calculus of variations demonstration: The gradient index lens,” Am. J. Phys. 56, 1099 (1988).
[Crossref]

1985 (1)

1982 (1)

1980 (2)

1971 (3)

1970 (1)

Baer, E.

M. Ponting, A. Hiltner, and E. Baer, “Polymer nanostructures by forced assembly: process, structure, properties,” Macromol. Symp. 294, 19–32 (2010).
[Crossref]

Beadie, G.

Bobrov, S.

G. Greisukh, S. Bobrov, and S. Stepanov, Optics of Diffractive and Gradient-Index Elements and Systems (SPIE Optical Engineering Press, 1997).

Bociort, F.

Bossard, J. A.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover Publications, 1968).

Chakravorty, A.

K. M. Jones, S. Lundgren, and A. Chakravorty, “A calculus of variations demonstration: The gradient index lens,” Am. J. Phys. 56, 1099 (1988).
[Crossref]

Cho, W.

H. Liu, T. Maekawa, N. M. Patrikalakis, E. M. Sachs, and W. Cho, “Methods for feature-based design of heterogeneous solids,” Computer-Aided Design 36, 1141–1159 (2004).
[Crossref]

Corsetti, J. A.

J. A. Corsetti, P. McCarthy, and D. T. Moore, “Color correction in the infrared using gradient-index materials,” Opt. Eng. 52, 112109 (2013).
[Crossref]

Fleet, E. F.

Flynn, R. A.

Forbes, G. W.

Ghatak, A. K.

Greisukh, G.

G. Greisukh, S. Bobrov, and S. Stepanov, Optics of Diffractive and Gradient-Index Elements and Systems (SPIE Optical Engineering Press, 1997).

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987).

Hiltner, A.

M. Ponting, A. Hiltner, and E. Baer, “Polymer nanostructures by forced assembly: process, structure, properties,” Macromol. Symp. 294, 19–32 (2010).
[Crossref]

Houde-Walter, S. N.

Johnson, J. D.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Jones, K. M.

K. M. Jones, S. Lundgren, and A. Chakravorty, “A calculus of variations demonstration: The gradient index lens,” Am. J. Phys. 56, 1099 (1988).
[Crossref]

Krishna, K. S. R.

Kumar, D. V.

Liu, H.

H. Liu, T. Maekawa, N. M. Patrikalakis, E. M. Sachs, and W. Cho, “Methods for feature-based design of heterogeneous solids,” Computer-Aided Design 36, 1141–1159 (2004).
[Crossref]

Lundgren, S.

K. M. Jones, S. Lundgren, and A. Chakravorty, “A calculus of variations demonstration: The gradient index lens,” Am. J. Phys. 56, 1099 (1988).
[Crossref]

Maekawa, T.

H. Liu, T. Maekawa, N. M. Patrikalakis, E. M. Sachs, and W. Cho, “Methods for feature-based design of heterogeneous solids,” Computer-Aided Design 36, 1141–1159 (2004).
[Crossref]

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic Press, 1978).

Mayer, T. S.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

McCarthy, P.

J. A. Corsetti, P. McCarthy, and D. T. Moore, “Color correction in the infrared using gradient-index materials,” Opt. Eng. 52, 112109 (2013).
[Crossref]

Moore, D. T.

Muise, R.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Musgraves, J. D.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Nishizawa, K.

Patrikalakis, N. M.

H. Liu, T. Maekawa, N. M. Patrikalakis, E. M. Sachs, and W. Cho, “Methods for feature-based design of heterogeneous solids,” Computer-Aided Design 36, 1141–1159 (2004).
[Crossref]

Podraza, N.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Pogrebnyakov, A.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Ponting, M.

M. Ponting, A. Hiltner, and E. Baer, “Polymer nanostructures by forced assembly: process, structure, properties,” Macromol. Symp. 294, 19–32 (2010).
[Crossref]

Richardson, K.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Rivero-Baleine, C.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Rogers, S.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Sachs, E. M.

H. Liu, T. Maekawa, N. M. Patrikalakis, E. M. Sachs, and W. Cho, “Methods for feature-based design of heterogeneous solids,” Computer-Aided Design 36, 1141–1159 (2004).
[Crossref]

Sands, P. J.

Sharma, A.

Shin, H. J.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Shirk, J. S.

Smolyaninov, I. I.

I. I. Smolyaninov, “Graded index metamaterial lens,” International Patent Application, WO2013/032758 (7March2013).

Stepanov, S.

G. Greisukh, S. Bobrov, and S. Stepanov, Optics of Diffractive and Gradient-Index Elements and Systems (SPIE Optical Engineering Press, 1997).

Stone, B. D.

Turpin, J.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Werner, D. H.

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

Wood, R. W.

R. W. Wood, Physical Optics (Macmillan, 1905).

Am. J. Phys. (1)

K. M. Jones, S. Lundgren, and A. Chakravorty, “A calculus of variations demonstration: The gradient index lens,” Am. J. Phys. 56, 1099 (1988).
[Crossref]

Appl. Opt. (6)

Computer-Aided Design (1)

H. Liu, T. Maekawa, N. M. Patrikalakis, E. M. Sachs, and W. Cho, “Methods for feature-based design of heterogeneous solids,” Computer-Aided Design 36, 1141–1159 (2004).
[Crossref]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Macromol. Symp. (1)

M. Ponting, A. Hiltner, and E. Baer, “Polymer nanostructures by forced assembly: process, structure, properties,” Macromol. Symp. 294, 19–32 (2010).
[Crossref]

Opt. Eng. (1)

J. A. Corsetti, P. McCarthy, and D. T. Moore, “Color correction in the infrared using gradient-index materials,” Opt. Eng. 52, 112109 (2013).
[Crossref]

Opt. Express (1)

Other (8)

R. W. Wood, Physical Optics (Macmillan, 1905).

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987).

http://www.schott.com/advanced_optics/english/download/schott-optical-glass-pocket-catalog-january-2014-row.pdf . (Accessed March 5, 2015).

E. W. Marchand, Gradient Index Optics (Academic Press, 1978).

I. I. Smolyaninov, “Graded index metamaterial lens,” International Patent Application, WO2013/032758 (7March2013).

D. H. Werner, T. S. Mayer, C. Rivero-Baleine, N. Podraza, K. Richardson, J. Turpin, A. Pogrebnyakov, J. D. Musgraves, J. A. Bossard, H. J. Shin, R. Muise, S. Rogers, and J. D. Johnson, “Adaptive phase change metamaterials for infrared aperture control,” in Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems, J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds., Proc. SPIE8165, 81651H (2011).
[Crossref]

H. A. Buchdahl, Optical Aberration Coefficients (Dover Publications, 1968).

G. Greisukh, S. Bobrov, and S. Stepanov, Optics of Diffractive and Gradient-Index Elements and Systems (SPIE Optical Engineering Press, 1997).

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Figures (4)

Fig. 1
Fig. 1 GRIN lens.
Fig. 2
Fig. 2 GRIN lens represented as doublet comprised of a homogeneous refractive lens and a Wood lens.
Fig. 3
Fig. 3 Optical performance of achromats. Rayfan for homogeneous doublet achromat design using (a) glass pair 1 and (b) glass pair 2. Rayfan for GRIN achromat design using (c) glass pair 1 and (d) glass pair 2. The color of each graph is representative of its wavelength: red (λC), green (λd), and blue (λF).
Fig. 4
Fig. 4 Volume fraction as a function of lens radius for the GRIN achromats. (a) Glass Pair 1. (b) Glass Pair 2.

Tables (7)

Tables Icon

Table 1 Design Coefficients for Dispersion.

Tables Icon

Table 2 Design Coefficients for Optical Power.

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Table 3 Glass pairs selected using homogeneous doublet (pair 1) and GRIN (pair 2) achromatic criteria.

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Table 4 Surface descriptions for homogeneous achromat designs.

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Table 5 Surface descriptions for GRIN achromat designs.

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Table 6 Additional parameters for GRIN achromat designs.

Tables Icon

Table 7 Calculated spot size in μm for designed achromats.

Equations (56)

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n ( r , z , λ ) = = 0 L m = 0 M Γ m ( λ ) z r 2 m ,
Γ m ( λ ) = k = 0 K ν k m ( λ λ 0 ) k ,
d n ( r , z , λ ) d r = 2 = 0 L m = 0 M m Γ m ( λ ) z r 2 m 1 ,
d n ( r , z , λ ) d z = = 0 L m = 0 M Γ m ( λ ) z 1 r 2 m ,
d n ( r , z , λ ) d λ = = 0 L m = 0 M d Γ m ( λ ) d λ z r 2 m .
n ( r , λ ) = n 0 ( λ ) + Γ ( λ ) r 2 .
n ( 0 , λ ) = n ctr ( λ ) ,
n ( D / 2 , λ ) = n edg ( λ ) , = n ctr ( λ ) + Γ ( λ ) ( D / 2 ) 2 ,
Γ ( λ ) = ( 2 D ) 2 [ n edg ( λ ) n ctr ( λ ) ] .
s G = sgn [ n ctr ( λ 0 ) n edg ( λ 0 ) ] ,
ϕ ( λ ) = 1 f ( λ ) .
ϕ ( λ ) = ϕ R ( λ ) C [ Φ ( λ ) ] + [ ϕ G ( λ ) + ϕ R 2 ( λ ) ] [ S [ Φ ( λ ) ] Φ ( λ ) ] ,
C [ ϕ ( λ ) ] = cosh ϕ ( λ ) ,
S [ ϕ ( λ ) ] = sinh ϕ ( λ ) .
C [ ϕ ( λ ) ] = cos ϕ ( λ ) ,
S [ ϕ ( λ ) ] = sin 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 G ( λ ) = s G Φ 2 ( λ ) n ctr ( λ ) t ,
Φ ( λ ) = ( 8 t 2 D 2 ) [ | n edg ( λ ) n ctr ( λ ) | n ctr ( λ ) ] .
d ϕ ( λ ) d λ = ϕ R ( λ ) V R ( λ ) + ϕ R 2 ( λ ) V R 2 ( λ ) + ϕ G ( λ ) V G ( λ ) .
ϕ ( λ ) = ϕ G ( λ ) [ S [ Φ ( λ ) ] Φ ( λ ) ] ,
d ϕ ( λ ) d λ = ϕ G ( λ ) V G ( λ ) .
d n edg ( λ 0 ) d λ = [ d n ctr ( λ 0 ) d λ ] { 2 [ S [ Φ ( λ 0 ) ] Φ ( λ 0 ) ] [ n edg ( λ 0 ) n ctr ( λ 0 ) ] [ S [ Φ ( λ 0 ) ] Φ ( λ 0 ) C [ Φ ( λ 0 ) ] ] S [ Φ ( λ 0 ) ] Φ ( λ 0 ) + C [ Φ ( λ 0 ) ] } , = [ d n ctr ( λ 0 ) d λ ] [ n edg ( λ 0 ) n ctr ( λ 0 ) ] { 1 + 2 [ n ctr ( λ 0 ) n edg ( λ 0 ) 1 ] [ S [ Φ ( λ 0 ) ] Φ ( λ 0 ) S [ Φ ( λ 0 ) ] Φ ( λ 0 ) + C [ Φ ( λ ) ] ] } .
Ψ 1 ( λ ) = 1 n ctr ( λ ) d n ctr ( λ ) d λ 1 n edg ( λ ) d n edg ( λ ) d λ .
d ϕ ( λ 0 ) d λ = ( 8 t D 2 ) [ d n ctr ( λ 0 ) d λ d n edg ( λ 0 ) d λ ] .
ϕ ( λ ) = ϕ R ( λ ) C [ Φ ( λ ) + ϕ G ( λ ) ] [ S [ Φ ( λ ) ] Φ ( λ ) ] ,
d ϕ ( λ ) d λ = ϕ R ( λ ) V R ( λ ) + ϕ G ( λ ) V G ( λ ) ,
ϕ ( λ ) = ϕ R ( λ ) [ 1 s G Φ 2 ( λ ) 2 ] + ϕ G ( λ ) [ 1 s G Φ 2 ( λ ) 6 ] ,
d ϕ ( λ ) d λ = ϕ R ( λ ) V R ( λ ) + ϕ R ( λ ) V G ( λ ) ,
ϕ ( λ ) = ϕ R ( λ ) + ϕ R ( λ ) ,
d ϕ ( λ ) d λ = ϕ R ( λ ) V R ( λ ) + ϕ G ( λ ) V G ( λ ) .
ϕ ( λ ) = A ( λ ) ϕ R ( λ ) + B ( λ ) ϕ G ( λ ) ,
d ϕ ( λ ) d λ = ϕ R ( λ ) V R ( λ ) + ϕ G ( λ ) V G ( λ ) .
ϕ R ( λ 0 ) = ϕ 0 V R ( λ 0 ) A ( λ 0 ) V R ( λ 0 ) B ( λ 0 ) V G ( λ 0 ) ,
ϕ G ( λ 0 ) = ϕ 0 V G ( λ 0 ) A ( λ 0 ) V R ( λ 0 ) B ( λ 0 ) V G ( λ 0 )
ϕ G ( λ ) ϕ R ( λ ) = V G ( λ ) V R ( λ ) ,
| V G ( λ 0 ) V R ( λ 0 ) | = | [ d n ctr ( λ 0 ) / d λ n ctr ( λ 0 ) 1 ] [ n ctr ( λ 0 ) n edg ( λ 0 ) d n edg ( λ 0 ) / d λ d n ctr ( λ 0 ) / d λ ] | .
| V 1 ( λ 0 ) V 2 ( λ 0 ) | = [ n 1 ( λ 0 ) 1 n 2 ( λ 0 ) 1 ] | d n 2 ( λ 0 ) / d λ d n 1 ( λ 0 ) / d λ | .
( D R f D R b ) = ( D ϕ 0 ) d n ctr ( λ 0 ) / d λ d n edg ( λ 0 ) / d λ [ d n ctr ( λ 0 ) / d λ ] [ n edg ( λ 0 ) 1 ] [ d n edg ( λ 0 ) / d λ ] [ n ctr ( λ 0 ) 1 ]
( 8 t D ) = ( D ϕ 0 ) d n ctr ( λ 0 ) / d λ [ d n ctr ( λ 0 ) / d λ ] [ n edg ( λ 0 ) 1 ] [ d n edg ( λ 0 ) / d λ ] [ n ctr ( λ 0 ) 1 ]
d n edg ( λ 0 ) d λ = [ 1 + ( D 2 8 t ) ( 1 R f 1 R b ) ] [ d n ctr ( λ 0 ) d λ ] .
γ ( r ) = n 2 ( r , λ 0 ) n min 2 ( λ 0 ) n max 2 ( λ 0 ) n min 2 ( λ 0 ) .
ϕ ( λ ) = ϕ R ( λ ) cos Φ ( λ ) + [ ϕ G ( λ ) + ϕ R 2 ( λ ) ] [ sin Φ ( λ ) Φ ( λ ) ] ,
d ϕ ( λ ) d λ = { d ϕ R ( λ ) d λ cos Φ ( λ ) ϕ R ( λ ) [ sin Φ ( λ ) Φ ( λ ) ] [ Φ ( λ ) d Φ ( λ ) d λ ] } + { d [ Φ 2 ( λ ) n ctr ( λ ) / t ] d λ + d ϕ R 2 ( λ ) d λ } [ sin Φ ( λ ) Φ ( λ ) ] + [ Φ 2 ( λ ) n ctr ( λ ) t + ϕ R 2 ( λ ) ] d [ sin Φ ( λ ) / Φ ( λ ) ] d λ .
ϕ ( λ ) = ϕ R ( λ ) cosh Φ ( λ ) + [ ϕ G ( λ ) + ϕ R 2 ( λ ) ] [ sinh Φ ( λ ) Φ ( λ ) ] ,
d ϕ ( λ ) d λ = { d ϕ R ( λ ) d λ cosh Φ ( λ ) + ϕ R ( λ ) [ sinh Φ ( λ ) Φ ( λ ) ] [ Φ ( λ ) d Φ ( λ ) d λ ] } + { d [ Φ 2 ( λ ) n ctr ( λ ) / t ] d λ + d ϕ R 2 ( λ ) d λ } [ sinh Φ ( λ ) Φ ( λ ) ] + [ Φ 2 ( λ ) n ctr ( λ ) t + ϕ R 2 ( λ ) ] d [ sinh Φ ( λ ) / Φ ( λ ) ] d λ .
d ϕ R ( λ ) d λ = [ 1 n ctr ( λ ) 1 ] [ d n ctr ( λ ) d λ ] ϕ R ( λ ) ,
d ϕ R 2 ( λ ) d λ = [ 1 n ctr ( λ ) ] [ n ctr ( λ ) + 1 n ctr ( λ ) 1 ] [ d n ctr ( λ ) d λ ] ϕ R 2 ( λ ) ,
Φ ( λ ) d Φ ( λ ) d λ = [ Φ 2 ( λ ) 2 ] [ n edg ( λ ) n ctr ( λ ) n edg ( λ ) ] Ψ 1 ( λ ) .
d [ sin Φ ( λ ) / Φ ( λ ) ] d λ = [ cos Φ ( λ ) sin Φ ( λ ) Φ ( λ ) ] [ 1 Φ ( λ ) d Φ ( λ ) d λ ] ,
d [ sinh Φ ( λ ) / Φ ( λ ) ] d λ = [ cosh Φ ( λ ) sinh Φ ( λ ) Φ ( λ ) ] [ 1 Φ ( λ ) d Φ ( λ ) d λ ] .
1 V R ( λ ) = [ C [ Φ ( λ ) ] n ctr ( λ ) 1 ] [ d n ctr ( λ ) d λ ] + s G [ S [ Φ ( λ ) ] Φ ( λ ) ] [ Φ 2 ( λ ) 2 ] [ n edg ( λ ) n edg ( λ ) n ctr ( λ ) ] Ψ 1 ( λ ) ,
1 V R 2 ( λ ) = { n ctr ( λ ) + 1 n ctr ( λ ) [ n ctr ( λ ) 1 ] } [ S [ Φ ( λ ) ] Φ ( λ ) ] [ d n ctr ( λ ) d λ ] + ( 1 2 ) [ S [ Φ ( λ ) ] Φ ( λ ) C [ Φ ( λ ) ] ] [ n edg ( λ ) n edg ( λ ) n ctr ( λ ) ] Ψ 1 ( λ ) ,
1 V G ( λ ) = { 1 2 [ n edg ( λ ) n ctr ( λ ) ] } × ( [ d n ctr ( λ ) d λ ] { 2 [ S [ Φ ( λ ) ] Φ ( λ ) ] [ n edg ( λ ) n c t r ( λ ) ] [ S [ Φ ( λ ) ] Φ ( λ ) C [ Φ ( λ ) ] ] } [ d n edg ( λ ) d λ ] [ S [ Φ ( λ ) ] Φ ( λ ) + C [ Φ ( λ ) ] ] ) ,
Ψ 1 ( λ ) = 1 n ctr ( λ ) d n ctr ( λ ) d λ 1 n edg ( λ ) d n edg ( λ ) d λ .

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