Abstract

Generating a prescribed irradiance distribution given a source distribution is an inverse problem that sits at the heart of illumination design. The growing prevalence of freeform optics has inspired several design methods for obtaining a prescribed irradiance distribution possessing no symmetry. Up to now, these methods have relied exclusively on freeform optical surfaces for generating freeform irradiances. This paper presents a design method that, for the first time, applies gradient-index (GRIN) optics to solving this inverse problem. Using a piecewise-continuous freeform gradient-index (F-GRIN) profile, a single optic with two planar surfaces can be designed to produce a far-field prescribed irradiance distribution from a point source. The design process is herein presented along with two design examples which demonstrate some of the unique properties of F-GRIN illumination optics.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

2019 (7)

S. Sorgato, J. Chaves, H. Thienpont, and F. Duerr, “Design of illumination optics with extended sources based on wavefront tailoring,” Optica 6(8), 966 (2019).
[Crossref]

K. Desnijder, W. Ryckaert, P. Hanselaer, and Y. Meuret, “Luminance spreading freeform lens arrays with accurate intensity control,” Opt. Express 27(23), 32994 (2019).
[Crossref]

C. Bösel and H. Gross, “Compact freeform illumination system design for pattern generation with extended light sources,” Appl. Opt. 58(10), 2713 (2019).
[Crossref]

B. G. Assefa, T. Saastamoinen, M. Pekkarinen, V. Nissinen, J. Biskop, M. Kuittinen, J. Turunen, and J. Saarinen, “Realizing freeform lenses using an optics 3D-printer for industrial based tailored irradiance distribution,” OSA Continuum 2(3), 690 (2019).
[Crossref]

L. B. Romijn, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, “Freeform lens design for a point source and far-field target,” J. Opt. Soc. Am. A 36(11), 1926 (2019).
[Crossref]

L. L. Doskolovich, D. A. Bykov, A. A. Mingazov, and E. A. Bezus, “Optimal mass transportation and linear assignment problems in the design of freeform refractive optical elements generating far-field irradiance distributions,” Opt. Express 27(9), 13083 (2019).
[Crossref]

D. A. Bykov, L. L. Doskolovich, A. A. Mingazov, and E. A. Bezus, “Optimal mass transportation problem in the design of freeform optical elements generating far-field irradiance distributions for plane incident beam,” Appl. Opt. 58(33), 9131 (2019).
[Crossref]

2018 (2)

2017 (2)

2016 (5)

2015 (2)

K. Brix, Y. Hafizogullari, and A. Platen, “Designing illumination lenses and mirrors by the numerical solution of Monge-Ampère equations,” J. Opt. Soc. Am. A 32(11), 2227 (2015).
[Crossref]

M. Li, D. Li, C. Zhang, K. E Z. Hong, and C. Li, “Improved zonal wavefront reconstruction algorithm for Hartmann type test with arbitrary grid patterns,” Proc. SPIE 9623, 962319 (2015).
[Crossref]

2014 (3)

2013 (1)

2012 (1)

2011 (2)

D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36(6), 918 (2011).
[Crossref]

A. Timinger, J. Unterhinninghofen, S. Junginger, and A. Hofmann, “Tolerancing free-form optics for illumination,” Proc. SPIE 8170, 817006 (2011).
[Crossref]

2006 (1)

V. Oliker, “Freeform optical systems with prescribed irradiance properties in near-field,” Proc. SPIE 6342, 634211 (2006).
[Crossref]

2004 (1)

P. Benítez, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

2003 (1)

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” Proc. SPIE 5186, 133–142 (2003).
[Crossref]

2002 (1)

1982 (1)

1980 (1)

1975 (1)

1972 (1)

1955 (1)

H. W. Kuhn, “The Hungarian method for the assignment problem,” Nav. Res. Logist. Q. 2(1-2), 83–97 (1955).
[Crossref]

Anselm, C.

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” Proc. SPIE 5186, 133–142 (2003).
[Crossref]

Assefa, B. G.

Bäuerle, A.

Benítez, P.

Bentley, J.

Berens, M.

Bezus, E. A.

Biskop, J.

Borisova, K. V.

Bösel, C.

Bräuer, A.

Brix, K.

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 injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

Bruneton, A.

Bykov, 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 injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

Chaves, J.

Crouse, D. F.

D. F. Crouse, “On implementing 2D rectangular assignment algorithms,” IEEE Trans. Aerosp. Electron. Syst. 52(4), 1679–1696 (2016).
[Crossref]

Desnijder, K.

Doskolovich, L. L.

Duerr, F.

S. Sorgato, J. Chaves, H. Thienpont, and F. Duerr, “Design of illumination optics with extended sources based on wavefront tailoring,” Optica 6(8), 966 (2019).
[Crossref]

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Mi nano, and F. Duerr, “Design of Freeform Illumination Optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
[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 injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

Dupuy, C. G.

D. M. Schut, C. G. Dupuy, and J. P. Harmon, “Inks for 3D printing gradient refractive index (GRIN) optical components,” US 9,447,299 (2016).

Feng, Z.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Mi nano, and F. Duerr, “Design of Freeform Illumination Optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

Z. Feng, B. D. Froese, and R. Liang, “Freeform illumination optics construction following an optimal transport map,” Appl. Opt. 55(16), 4301 (2016).
[Crossref]

Froese, B. D.

Ghatak, A. K.

Gross, H.

Hafizogullari, Y.

Hanselaer, P.

Harmon, J. P.

D. M. Schut, C. G. Dupuy, and J. P. Harmon, “Inks for 3D printing gradient refractive index (GRIN) optical components,” US 9,447,299 (2016).

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 injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

Hofmann, A.

A. Timinger, J. Unterhinninghofen, S. Junginger, and A. Hofmann, “Tolerancing free-form optics for illumination,” Proc. SPIE 8170, 817006 (2011).
[Crossref]

IJzerman, W.

J. ten Thije Boonkkamp, C. Prins, W. IJzerman, and T. Tukker, “The Monge-Ampère Equation for Freeform Optics,” in Imaging and Applied Optics 2015, (OSA, 2015), p. FTh3B.4.

IJzerman, W. L.

Junginger, S.

A. Timinger, J. Unterhinninghofen, S. Junginger, and A. Hofmann, “Tolerancing free-form optics for illumination,” Proc. SPIE 8170, 817006 (2011).
[Crossref]

Kazanskiy, N. L.

Knoflach, C.

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” Proc. SPIE 5186, 133–142 (2003).
[Crossref]

Kuhn, H. W.

H. W. Kuhn, “The Hungarian method for the assignment problem,” Nav. Res. Logist. Q. 2(1-2), 83–97 (1955).
[Crossref]

Kuittinen, M.

Kumar, D. V.

Kunkel, W. M.

Leger, J. R.

Li, C.

M. Li, D. Li, C. Zhang, K. E Z. Hong, and C. Li, “Improved zonal wavefront reconstruction algorithm for Hartmann type test with arbitrary grid patterns,” Proc. SPIE 9623, 962319 (2015).
[Crossref]

Li, D.

M. Li, D. Li, C. Zhang, K. E Z. Hong, and C. Li, “Improved zonal wavefront reconstruction algorithm for Hartmann type test with arbitrary grid patterns,” Proc. SPIE 9623, 962319 (2015).
[Crossref]

Li, H.

Li, M.

M. Li, D. Li, C. Zhang, K. E Z. Hong, and C. Li, “Improved zonal wavefront reconstruction algorithm for Hartmann type test with arbitrary grid patterns,” Proc. SPIE 9623, 962319 (2015).
[Crossref]

Liang, R.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Mi nano, and F. Duerr, “Design of Freeform Illumination Optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

Z. Feng, B. D. Froese, and R. Liang, “Freeform illumination optics construction following an optimal transport map,” Appl. Opt. 55(16), 4301 (2016).
[Crossref]

Liu, P.

Liu, X.

Loosen, P.

Marchand, E. W.

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

Meuret, Y.

Mi nano, J. C.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Mi nano, and F. Duerr, “Design of Freeform Illumination Optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

Michaelis, D.

Miñano, J. C.

Mingazov, A. A.

Moiseev, M. A.

Moore, D. T.

Müller, G.

Muschaweck, J.

Muschaweck, J. A.

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” Proc. SPIE 5186, 133–142 (2003).
[Crossref]

Nissinen, V.

Oliker, V.

V. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method (SQM),” Opt. Express 25(4), A58 (2017).
[Crossref]

V. Oliker, “Freeform optical systems with prescribed irradiance properties in near-field,” Proc. SPIE 6342, 634211 (2006).
[Crossref]

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 injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

Pekkarinen, M.

Platen, A.

Pohl, W.

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” Proc. SPIE 5186, 133–142 (2003).
[Crossref]

Prins, C.

J. ten Thije Boonkkamp, C. Prins, W. IJzerman, and T. Tukker, “The Monge-Ampère Equation for Freeform Optics,” in Imaging and Applied Optics 2015, (OSA, 2015), p. FTh3B.4.

Ries, H.

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” Proc. SPIE 5186, 133–142 (2003).
[Crossref]

H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002).
[Crossref]

Romijn, L. B.

Ryckaert, W.

Saarinen, J.

Saastamoinen, T.

Schmidt, G.

Schreiber, P.

Schruben, J. S.

Schut, D. M.

D. M. Schut, C. G. Dupuy, and J. P. Harmon, “Inks for 3D printing gradient refractive index (GRIN) optical components,” US 9,447,299 (2016).

Sharma, A.

Sorgato, S.

Southwell, W.

Stollenwerk, J.

Sulman, M. M.

Takaki, N.

ten Thije Boonkkamp, J.

J. ten Thije Boonkkamp, C. Prins, W. IJzerman, and T. Tukker, “The Monge-Ampère Equation for Freeform Optics,” in Imaging and Applied Optics 2015, (OSA, 2015), p. FTh3B.4.

ten Thije Boonkkamp, J. H. M.

Thienpont, H.

Timinger, A.

A. Timinger, J. Unterhinninghofen, S. Junginger, and A. Hofmann, “Tolerancing free-form optics for illumination,” Proc. SPIE 8170, 817006 (2011).
[Crossref]

Timinger, A. L.

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” Proc. SPIE 5186, 133–142 (2003).
[Crossref]

Tukker, T.

J. ten Thije Boonkkamp, C. Prins, W. IJzerman, and T. Tukker, “The Monge-Ampère Equation for Freeform Optics,” in Imaging and Applied Optics 2015, (OSA, 2015), p. FTh3B.4.

Turunen, J.

Unterhinninghofen, J.

A. Timinger, J. Unterhinninghofen, S. Junginger, and A. Hofmann, “Tolerancing free-form optics for illumination,” Proc. SPIE 8170, 817006 (2011).
[Crossref]

Völl, A.

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 injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

Wester, R.

Wu, R.

Xu, L.

Yang, T.

Z. Hong, K. E

M. Li, D. Li, C. Zhang, K. E Z. Hong, and C. Li, “Improved zonal wavefront reconstruction algorithm for Hartmann type test with arbitrary grid patterns,” Proc. SPIE 9623, 962319 (2015).
[Crossref]

Zhang, C.

M. Li, D. Li, C. Zhang, K. E Z. Hong, and C. Li, “Improved zonal wavefront reconstruction algorithm for Hartmann type test with arbitrary grid patterns,” Proc. SPIE 9623, 962319 (2015).
[Crossref]

Zhang, Y.

Zheng, Z.

Appl. Opt. (5)

IEEE Trans. Aerosp. Electron. Syst. (1)

D. F. Crouse, “On implementing 2D rectangular assignment algorithms,” IEEE Trans. Aerosp. Electron. Syst. 52(4), 1679–1696 (2016).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Laser Photonics Rev. (1)

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Mi nano, and F. Duerr, “Design of Freeform Illumination Optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

Nav. Res. Logist. Q. (1)

H. W. Kuhn, “The Hungarian method for the assignment problem,” Nav. Res. Logist. Q. 2(1-2), 83–97 (1955).
[Crossref]

Opt. Eng. (1)

P. Benítez, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

Opt. Express (10)

T. Yang, N. Takaki, J. Bentley, G. Schmidt, and D. T. Moore, “Efficient representation of freeform gradient-index profiles for non-rotationally symmetric optical design,” Opt. Express 28(10), 14788 (2020).
[Crossref]

W. M. Kunkel and J. R. Leger, “Gradient-index design for mode conversion of diffracting beams,” Opt. Express 24(12), 13480 (2016).
[Crossref]

R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22(S2), A552 (2014).
[Crossref]

V. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method (SQM),” Opt. Express 25(4), A58 (2017).
[Crossref]

A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477 (2012).
[Crossref]

R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with L2 Monge-Kantorovich theory for the Monge-Ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161 (2014).
[Crossref]

C. Bösel and H. Gross, “Ray mapping approach for the efficient design of continuous freeform surfaces,” Opt. Express 24(13), 14271 (2016).
[Crossref]

D. A. Bykov, L. L. Doskolovich, A. A. Mingazov, E. A. Bezus, and N. L. Kazanskiy, “Linear assignment problem in the design of freeform refractive optical elements generating prescribed irradiance distributions,” Opt. Express 26(21), 27812 (2018).
[Crossref]

L. L. Doskolovich, D. A. Bykov, A. A. Mingazov, and E. A. Bezus, “Optimal mass transportation and linear assignment problems in the design of freeform refractive optical elements generating far-field irradiance distributions,” Opt. Express 27(9), 13083 (2019).
[Crossref]

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Figures (11)

Fig. 1.
Fig. 1. Freeform illumination using a freeform gradient-index (F-GRIN) optic. The design objective is to solve the inverse problem of determining the refractive index profile $n\left (x, y, z\right )$ that yields the prescribed irradiance. The color map shows differences in refractive index.
Fig. 2.
Fig. 2. Linear GRIN refractive index profile and ray path. (a) The refractive index gradient direction $\hat {\rho }$ lies in the $x$-$y$ plane and is oriented at an angle $\theta _{G}$ with respect to the $y$-axis. (b) The ray path $\rho \left (z\right )$ through a linear GRIN has the form of a hyperbolic cosine and lies in the $\rho$-$z$ plane.
Fig. 3.
Fig. 3. Beam angular characteristics are approximately conserved when transmitting through a linear GRIN medium, such as for (a) collimated, (b) diverging, or (c) converging bundles of rays. The color map shows differences in refractive index.
Fig. 4.
Fig. 4. Summary of the F-GRIN design process for producing a prescribed irradiance.
Fig. 5.
Fig. 5. Target mapping by solving the linear assignment problem. The chosen cost function is to minimize the total three-dimensional Euclidean distance between node pairs. Demonstrated in two-dimensions, black nodes represent array points, and colored nodes represent target points.
Fig. 6.
Fig. 6. The effect of different linear GRIN parameters on the ray trajectory. Radial contours depict ray positions for different gradient magnitudes $\alpha$. Azimuthal contours depict ray positions for different gradient directions $\theta _{G}$. The center point is the ray position for a homogeneous medium. The ray trajectories are evaluated at $z=1$ m for a $15.8^{\circ }$ angle of incidence at the GRIN.
Fig. 7.
Fig. 7. Example linear GRIN array midpoint design. Each array element has a unique gradient magnitude $\alpha$ and direction $\theta _{G}$ in order to illuminate its corresponding mapped point in the target. The refractive index profile is depicted in the $x$-$y$ plane and possesses no axial dependence. This array was used in reconstructing the final piecewise-continuous F-GRIN profile for the flower design in Fig. 9.
Fig. 8.
Fig. 8. F-GRIN reconstruction and interpolation. (a) A linear GRIN array is created as a midpoint in the design process. (b) Reconstruction is performed with a modified version of the Southwell algorithm. (c) The reconstructed result then undergoes bicubic interpolation to obtain a continuous refractive index profile. The color maps show differences in refractive index and are not set to the same scale.
Fig. 9.
Fig. 9. Two piecewise-continuous F-GRIN designs that produce a prescribed irradiance distribution, evaluated at $z=1$ m. (a) Two different targets were considered, both of which have holes and sharp discontinuities. A target throw ratio of $0.5$ was considered for both cases. (b) The designed F-GRIN illumination optics have both refractive index and gradient discontinuities with a total refractive index change $\Delta n=0.1$. The refractive index profile is depicted in the $x$-$y$ plane and possesses no axial dependence. The F-GRIN optics have planar surfaces. (c) The produced relative irradiance distribution for each design was evaluated using a Monte Carlo ray trace with $10^{6}$ rays.
Fig. 10.
Fig. 10. Refractive index and gradient discontinuities in the flower F-GRIN design in Fig. 9. Gradient discontinuities in the F-GRIN profile impart phase discontinuities which create holes and sharp discontinuities in the irradiance. Discontinuities are depicted by black lines in the expanded view. The stair-step nature of the discontinuities is a lasting effect from the linear GRIN array midpoint. The refractive index profile is depicted in the $x$-$y$ plane and possesses no axial dependence.
Fig. 11.
Fig. 11. Relative irradiance distribution at different evaluation distances $z$ produced by the “U of R” F-GRIN design. The different distributions scale in size with $z$ according to the target throw ratio of $0.5$. With design mapping performed at distance $z_{t}=1$ m, the ray-based irradiance is maintained into the far-field.

Equations (26)

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P = s 0 s 1 n d s
L = n ( x , y , z ) ( x 2 + y 2 + z 2 ) 1 / 2
( n r ) = n
n ( ρ ) = n 0 + α ρ
n ρ ( 1 + ρ ˙ 2 ) + n ρ ¨ = 0
ρ ( z ) = n 0 α [ 1 1 + β 2 cosh ( α 1 + β 2 n 0 z + c ) 1 ]
β = sin θ i n 0 2 sin 2 θ i .
d ρ ( z ) d z = sinh ( α 1 + β 2 n 0 z + c ) .
C = ( x a x t ) 2 + ( y a y t ) 2 + ( z a z t ) 2
L = n ( x , y , z ) ( 1 + x ˙ 2 + y ˙ 2 ) 1 / 2
d d z ( L x ˙ ) = L x d d z ( L y ˙ ) = L y .
( n z x ˙ n x ) ( 1 + x ˙ 2 + y ˙ 2 ) + n x ¨ = 0 ( n z y ˙ n y ) ( 1 + x ˙ 2 + y ˙ 2 ) + n y ¨ = 0.
n ρ ( 1 + ρ ˙ 2 ) + n ρ ¨ = 0
{ ρ ( z = 0 ) = 0 d ρ d z | z = 0 = β .
ρ ¨ = 1 + ρ ˙ 2 κ + ρ
v d v d ρ = 1 + v 2 κ + ρ .
v d v 1 + v 2 = d ρ κ + ρ
v 2 = ( κ + ρ ) 2 c 1
ρ ˙ 2 = ( κ + ρ ) 2 c 1.
ρ ˙ 2 = 1 γ 2 ( ρ + κ + γ ) ( ρ + κ γ ) ρ ˙ = ± 1 γ ( ρ + κ + γ ) ( ρ + κ γ )
d ρ ( ρ + κ + γ ) ( ρ + κ γ ) = ± 1 γ d z .
d ρ ( ρ + κ γ ) ( ρ + κ + γ ) = d ρ 2 γ ( ρ + κ γ ) ( ρ + κ γ 2 γ ) 2 + 1 .
d ρ ( ρ + κ γ ) ( ρ + κ + γ ) = 2 d u u 2 + 1 = 2 sinh 1 ( u ) = 2 sinh 1 ( ρ + κ γ 2 γ ) .
ρ ( z ) = 2 γ sinh 2 ( z 2 γ + c 2 ) κ + γ
ρ ( z ) = γ cosh ( z γ + c ) κ .
ρ ( z ) = n 0 α [ 1 1 + β 2 cosh ( α 1 + β 2 n 0 z + c ) 1 ] .