Abstract

Diffractive surface relief elements made of lossy materials exhibit phase-dependent absorption, which not only reduces the efficiency but also distorts the signal if the surface profile is realized on the basis of a phase-only design. We introduce an extension of the iterative Fourier transform algorithm, which accounts for such phase-dependent absorption, and present examples of its application to the design of diffractive beam splitters. The operator required for taking absorption into account is chosen to maximize the efficiency of the found design.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane structures,” Optik 35, 237–246 (1971).
  2. O. Bryngdald and F. Wyrowski, “Digital holography–computer-generated holograms,” Prog. Opt. 27, 1–86 (1990).
  3. F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [Crossref]
  4. J. Turunen and F. Wyrowski, eds. Diffractive Optics for Industrial and Commercial Applications (Wiley-VCH, 1997).
  5. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [Crossref]
  6. F. Wyrowski, “Diffractive optical elements: iterative calculations of quantized blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [Crossref]
  7. B. Nöhammer, C. David, H. P. Herzig, and J. Gobrecht, “Optimized staircase profiles for diffractive optical devices made from absorbing materials,” Opt. Lett. 28, 1087–1089 (2003).
    [Crossref]
  8. S. T. Wu, “Absorption measurements of liquid crystals in the ultraviolet, visible, and infrared,” J. Appl. Phys. 84, 4462–4465 (1998).
    [Crossref]
  9. R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt. 46, 8118–8133 (2007).
    [Crossref]
  10. J. Kischkat, S. Peters, B. Gruska, M. Semtsiv, M. Chashnikova, M. Klinkmüller, O. Fedosenko, S. Machulik, A. Aleksandrova, G. Monastyrskyi, Y. Flores, and W. T. Masselink, “Mid-infrared optical properties of thin films of aluminum oxide, titanium dioxide, silicon dioxide, aluminum nitride, and silicon nitride,” Appl. Opt. 51, 6789–6798 (2012).
    [Crossref]
  11. The Center for X-Ray Optics, http://www.cxro.lbl.gov/ .
  12. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
    [Crossref]
  13. A. D. Verhoeven, H. Aagedal, F. Wyrowski, and J. Turunen, “Upper bound of signal-relevant efficiency of constrained diffractive elements,” J. Opt. Soc. Am. A 33, 2425–2430 (2016).
    [Crossref]
  14. F. Wyrowski, “Upper bound of the efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [Crossref]

2016 (1)

2012 (1)

2007 (1)

2003 (1)

1998 (2)

S. T. Wu, “Absorption measurements of liquid crystals in the ultraviolet, visible, and infrared,” J. Appl. Phys. 84, 4462–4465 (1998).
[Crossref]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[Crossref]

1991 (2)

F. Wyrowski, “Upper bound of the efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
[Crossref]

F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[Crossref]

1990 (2)

O. Bryngdald and F. Wyrowski, “Digital holography–computer-generated holograms,” Prog. Opt. 27, 1–86 (1990).

F. Wyrowski, “Diffractive optical elements: iterative calculations of quantized blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
[Crossref]

1989 (1)

1971 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane structures,” Optik 35, 237–246 (1971).

Aagedal, H.

Aleksandrova, A.

Borghi, R.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[Crossref]

Bryngdahl, O.

F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[Crossref]

Bryngdald, O.

O. Bryngdald and F. Wyrowski, “Digital holography–computer-generated holograms,” Prog. Opt. 27, 1–86 (1990).

Chashnikova, M.

Cincotti, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[Crossref]

David, C.

Di Fabrizio, E.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[Crossref]

Fedosenko, O.

Flores, Y.

Gentili, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane structures,” Optik 35, 237–246 (1971).

Gobrecht, J.

Gori, F.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[Crossref]

Gruska, B.

Herzig, H. P.

Jonasz, M.

Kischkat, J.

Kitamura, R.

Klinkmüller, M.

Machulik, S.

Masselink, W. T.

Monastyrskyi, G.

Nöhammer, B.

Peters, S.

Pilon, L.

Santarsiero, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane structures,” Optik 35, 237–246 (1971).

Semtsiv, M.

Turunen, J.

Verhoeven, A. D.

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[Crossref]

Wu, S. T.

S. T. Wu, “Absorption measurements of liquid crystals in the ultraviolet, visible, and infrared,” J. Appl. Phys. 84, 4462–4465 (1998).
[Crossref]

Wyrowski, F.

Appl. Opt. (3)

J. Appl. Phys. (1)

S. T. Wu, “Absorption measurements of liquid crystals in the ultraviolet, visible, and infrared,” J. Appl. Phys. 84, 4462–4465 (1998).
[Crossref]

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

Opt. Commun. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[Crossref]

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane structures,” Optik 35, 237–246 (1971).

Prog. Opt. (1)

O. Bryngdald and F. Wyrowski, “Digital holography–computer-generated holograms,” Prog. Opt. 27, 1–86 (1990).

Rep. Prog. Phys. (1)

F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[Crossref]

Other (2)

J. Turunen and F. Wyrowski, eds. Diffractive Optics for Industrial and Commercial Applications (Wiley-VCH, 1997).

The Center for X-Ray Optics, http://www.cxro.lbl.gov/ .

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Figures (12)

Fig. 1.
Fig. 1. Complex plane visualization of the phase-amplitude constraint (5) imposed on the transmission function for κ / Δ n = 0.1 . (a) All phase values are allowed. (b) The phase is restricted to four quantized values ϕ q = { 0 , π / 2 , π , 3 π / 2 } .
Fig. 2.
Fig. 2. Effect of the phase-amplitude constraint in the uniformity of beam splitter gratings. (a) A simulated 4 × 4 array generated by a phase-only element. (b) The array produced with the phase-only design when absorption at the level κ / Δ n = 0.1 is present. Some higher diffraction orders around the array, representing the field U freedom in Eq. (7), are also shown.
Fig. 3.
Fig. 3. Visualization of the system: a transparency t is illuminated by U in and generates an output field U out , which is the sum of a signal field U signal and an error field U error inside the signal window W , and equals U freedom outside W .
Fig. 4.
Fig. 4. Projection of angles onto the (continuous) phase-amplitude constraint for κ / Δ n = 0.2 . (a) Direct projection, the projection is normal to the constraint curve up to the point where the tangent line crosses the curve itself. (b) Radial projection, the projection operation affects the amplitude but not the angle.
Fig. 5.
Fig. 5. Flowchart of the algorithm. In both stages the material constraint is applied to maximize SRE as specified in Eq. (12). The constraint at the output is first limited to phase-only optimization [Eq. (16)] and upon stagnation replaced by phase-amplitude [Eq. (18)] to improve uniformity.
Fig. 6.
Fig. 6. Phase profiles of optimum triplicators under different levels of absorption: the solid line shows the optimum profile in the dielectric case ( κ = 0 ) . The dashed lines illustrate the profiles at different absorption levels.
Fig. 7.
Fig. 7. Comparison of the analytical solution (Gori extended) to the results of IFTA with direct and radial projection for a 3 × 1 on-axis signal. The dotted line indicates the theoretical upper bound as provided by the signal relevant efficiency theorem [13].
Fig. 8.
Fig. 8. Estimated size of the desired signal at which the zeroth diffraction order will become dominating for on-axis design as described by Eq. (25). Beyond this limit the algorithm might need to actively suppress the zeroth order, resulting in a lower efficiency.
Fig. 9.
Fig. 9. Performance of the algorithm for an on-axis 5 × 5 beam splitter: efficiency (top) and signal-to-noise ratio (bottom) as a function of the absorption level.
Fig. 10.
Fig. 10. (a) A phase map of a typical 4 × 4 beam splitter operating on-axis, designed by direct projection. (b) An expectation value histogram of the phase values. The value of N at ϕ = 0 , which is 2900 in the shown scale, is left out for clarity.
Fig. 11.
Fig. 11. Same as Fig. 9, but for an off-axis beam splitter with the 5 × 5 array centered at ( m , n ) = ( 15,15 ) .
Fig. 12.
Fig. 12. Inward spiral shows the constraint of Eq. (A1) with κ / Δ n = 0.2 . The dashed line shows the tangent that goes through the point 1 + 0 i , the line perpendicular to this represent normals to this, and the line originating from 1 + 0 i is inclined at an angle arctan ( κ / Δ n ) .

Equations (82)

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n ^ = 1 + Δ n + i κ .
n ^ ( x , y ) = { n ^ if    0 z h ( x , y ) 1 if    z > h ( x , y ) .
| t ( x , y ) | = exp [ ( 2 π / λ ) κ h ( x , y ) ]
ϕ ( x , y ) = ( 2 π / λ ) Δ n h ( x , y )
A c = { exp ( ( i κ / Δ n ) ϕ ) | ϕ 0 } .
U out = L { U in t } ,
U out = α U signal + U error + U freedom ,
η SRE = α U signal 2 U in 2 = | U out | U signal | 2 U in 2 U signal 2 .
η = α U signal + U error 2 U in 2 = α U signal 2 + U error 2 U in 2 .
SNR = α U signal 2 U error 2 = | α | 2 U signal 2 U out α U signal W 2 .
t ub = argmin t A c lim r t r L 1 { U signal } U in 2 ,
ϕ proj = { θ arctan ( κ / Δ n ) if    0 θ arctan ( κ / Δ n ) θ M 0 otherwise ,
1 + ( κ / Δ n ) 2 cos [ θ M + arctan ( κ / Δ n ) ] exp ( θ M κ / Δ n ) = 1 .
t ( x , y ) = ( m , n ) = T m n exp [ i 2 π ( m x + n y ) ] ,
T m n = 0 1 t ( x , y ) exp [ i 2 π ( m x + n y ) ] d x d y
T m n = { D m n exp ( i arg T m n ) when ( m , n ) W 0 otherwise ,
t ( x , y ) = exp ( i κ / Δ n ) exp [ ( i κ / Δ n ) ϕ proj ( x , y ) ] .
T m n = { A | D m n | exp ( i arg T m n ) if ( m , n ) W T m n otherwise ,
A = | T | | D | D 2 .
ϕ ( x ) = arctan [ a sin ( 2 π x ) ] ,
ϕ ( x ) = { 0 if    x [ R , R ] ϕ ( x ) + ϕ c if    x [ R , R ] ,
( 1 α ) [ cosh ( α ϕ ) a cos ( 2 π x ) sinh ( α ϕ ) ] sin ϕ = ( 1 + α ) [ sinh ( α ϕ ) a cos ( 2 π x ) cosh ( α ϕ ) ] cos ϕ ,
f = 0 1 | t ( x ) | 2 d x = 1 exp ( 4 π κ / Δ n ) 4 π κ / Δ n .
η 0 = | 0 1 t ( x ) d x | 2 = [ 1 exp ( 2 π κ / Δ n ) ] 2 4 π 2 ( 1 + κ 2 / Δ n 2 ) .
N > f η 0 .
η Abs = η noAbs × η Carrier .
A c ( ϕ ) = exp [ ( i κ / Δ n ) ϕ ] ,
θ = ϕ + arctan ( κ / Δ n ) .
x ( ϕ ) = R { A c ( ϕ ) } = exp [ ( κ / Δ n ) ϕ ] cos ϕ y ( ϕ ) = I { A c ( ϕ ) } = exp [ ( κ / Δ n ) ϕ ] sin ϕ ,
θ 0 = arctan [ x ( 0 ) y ( 0 ) ] = arctan ( κ / Δ n ) ,
t d x d ϕ | ϕ M + X 0 = 1 t d y d ϕ | ϕ M + Y 0 = 0 ,
exp ( κ Δ n ϕ M ) ( κ Δ n cos ϕ M sin ϕ M ) t + exp ( κ Δ n ϕ M ) cos ϕ M = 1 , exp ( κ Δ n ϕ M ) ( κ Δ n sin ϕ M + cos ϕ M ) t + exp ( κ Δ n ϕ M ) sin ϕ M = 0 ,
1 + κ Δ n exp ( κ Δ n ϕ M ) ( cos ϕ M + sin ϕ M ) = 0
1 + ( κ / Δ n ) 2 cos [ ϕ M + arctan ( κ / Δ n ) ] exp ( κ Δ n ϕ M ) = 1 .
θ M = ϕ M + arctan ( κ / Δ n ) .
t ( x ) = exp [ ( i α ) ϕ ( x ) ] ,
t ( x ) = exp [ ( i α ) ϕ e ] [ cosh ( α ϕ o ) sinh ( α ϕ o ) ] × [ cos ( ϕ o ) + i sin ( ϕ o ) ] .
T n = 1 / 2 1 / 2 t ( x ) exp ( i 2 π n x ) d x .
T 0 = x R d x + x R exp [ ( i α ) ϕ e ] [ cosh ( α ϕ o ) sinh ( α ϕ o ) ] × ( cos ϕ o + i sin ϕ o ) d x ,
T 1 = x R exp ( i 2 π x ) d x + x R exp [ ( i α ) ϕ e ] [ cosh ( α ϕ o ) sinh ( α ϕ o ) ] × ( cos ϕ o + i sin ϕ o ) [ cos ( 2 π x ) + i sin ( 2 π x ) ] d x .
T 0 = x R d x + x R exp [ ( i α ) ϕ e ] × [ cosh ( α ϕ o ) cos ϕ o i sinh ( α ϕ o ) sin ϕ o ] d x
T 1 = x R exp ( i 2 π x ) d x + x R exp [ ( i α ) ϕ e ] sin ( 2 π x ) × [ i cosh ( α ϕ o ) sin ϕ o sinh ( α ϕ o ) cos ϕ o ] d x + x R exp [ ( i α ) ϕ e ] cos ( 2 π x ) × [ cosh ( α ϕ o ) cos ϕ o i sinh ( α ϕ o ) sin ϕ 0 ] d x .
T 1 = x R i sin ( 2 π x ) d x + x R exp [ ( i α ) ϕ e ] sin ( 2 π x ) × [ i cosh ( α ϕ o ) sin ϕ o sinh ( α ϕ o ) cos ϕ o ] d x .
T 0 = 2 R + 2 R 1 / 2 exp [ ( i α ) ϕ e ] × [ cosh ( α ϕ o ) cos ϕ o i sinh ( α ϕ o ) sin ϕ o ] d x
T 1 = R R i cos ( 2 π x ) d x + 2 R 1 / 2 exp [ ( i α ) ϕ e ] cos ( 2 π x ) × [ i cosh ( α ϕ o ) sin ϕ o sinh ( α ϕ o ) cos ϕ o ] d x .
| T 0 | 2 R + 2 R 1 / 2 exp ( α ϕ e ) × [ cosh ( α ϕ o ) | cos ϕ o | + | sinh ( α ϕ o ) sin ϕ o | ] d x
| T 1 | sin ( 2 π R ) / π + 2 R 1 / 2 exp ( α ϕ e ) | cos ( 2 π x ) | × [ cosh ( α ϕ o ) | sin ϕ o | + | sinh ( α ϕ o ) cos ϕ o | ] d x ,
I 1 = I P + I Q ,
| I 2 | = I | P | + I | Q | ,
I P = 0 1 / 2 | P | exp ( i ϕ p ) d x ,
I | P | = 0 1 / 2 | P | d x ,
| I 1 | 2 = | I P | 2 + | I Q | 2 + 2 | I P | | I Q | ,
| I 2 | 2 = I | P | 2 + I | Q | 2 + 2 I | P | I | Q | ,
| I P | 2 = I | P | 2 and | I Q | 2 = I | Q | 2 .
| I P | 2 = δ 2 k = 1 N | P k | 2 + δ 2 k = 1 N h k | P k | | P h | exp [ i ( ϕ p , k ϕ p , h ) ] = δ 2 k = 1 N | P k | 2 + 2 δ 2 k = 1 N h > k | P k | | P h | cos ( ϕ p , k ϕ p , h )
I | P | 2 = δ 2 k = 1 N | P k | 2 + 2 δ 2 k = 1 N h > k | P k | | P h | .
cos ( ϕ p , k ϕ p , h ) = 1    ( h , k ) ,
cos ( ϕ q , k ϕ q , h ) = 1    ( h , k )
P = exp [ ( i α ) ϕ e ] cosh ( α ϕ o ) cos ϕ o ,
Q = i exp [ ( i α ) ϕ e ] sinh ( α ϕ o ) sin ϕ o ,
ϕ p = ϕ e + arg ( cos ϕ o ) = ϕ e + ϕ cos ,
ϕ q = ϕ e + arg [ sinh ( α ϕ o ) sin ϕ o ] + 3 π / 2 = ϕ e + ϕ sinhsin + 3 π / 2 .
| T 0 | = 2 R + R 1 / 2 exp ( α ϕ e ) × [ cosh ( α ϕ o ) | cos ϕ o | + | sinh α ϕ o sin ϕ o | ] d x
ϕ e + ϕ cos = ϕ c 1 ,
ϕ e + ϕ sinhsin + 3 π / 2 = ϕ c 2 ,
P = exp [ ( i α ) ϕ e ] cosh ( α ϕ o ) sin ϕ o cos ( 2 π x ) ,
Q = exp [ ( i α ) ϕ e ] sinh ( α ϕ o ) cos ϕ o cos ( 2 π x ) ,
ϕ p = ϕ e + arg ( sin ϕ o ) = ϕ e + ϕ sin ,
ϕ q = ϕ e + arg [ sinh ( α ϕ o ) cos ϕ o ] + π / 2 = ϕ e + ϕ sinhcos + π / 2 ,
| T 1 | = sin ( 2 π R ) / π + 2 R 1 / 2 exp ( α ϕ e ) | cos ( 2 π x ) | × [ cosh ( α ϕ o ) | sin ϕ o | + | sinh ( α ϕ o ) cos ϕ o | ] d x
ϕ e + ϕ sin = ϕ c 3 ,
ϕ e + ϕ sinhcos + π / 2 = ϕ c 4 ,
T 0 = 2 R + 2 R 1 / 2 exp ( α ϕ e ) exp ( i ϕ cos ) × [ cosh ( α ϕ o ) cos ϕ o + sinh ( α ϕ o ) sin ϕ o ] d x ,
T 1 = sin ( 2 π R ) / π + 2 R 1 / 2 exp ( α ϕ e ) exp ( i ϕ cos ) cos ( 2 π x ) × [ cosh ( α ϕ o ) sin ϕ o + sinh ( α ϕ o ) cos ϕ o ] d x .
δ F = lim ϵ 0 [ F ( ϕ o + ϵ ) F ( ϕ o ) ] = 0 .
F ( ϕ o ) = 2 R + a sin ( 2 π R ) / π + 2 R 1 / 2 exp ( α ϕ e ) exp ( i ϕ cos ) × [ cosh ( α ϕ 0 ) cos ϕ o + sinh ( α ϕ o ) sin ϕ o ] d x 2 a R 1 / 2 exp ( α ϕ e ) exp ( i ϕ cos ) cos ( 2 π x ) × [ cosh ( α ϕ o ) sin ϕ o + sinh ( α ϕ o ) cos ϕ o ] d x
F ( ϕ o + ϵ ) 2 R + a sin ( 2 π R ) / π + 2 R 1 / 2 exp ( α ϕ e ) × { [ cosh ( α ϕ o ) + ε α sinh ( α ϕ o ) ] ( cos ϕ o ε sin ϕ o ) + [ sinh ( α ϕ o ) + ε α cosh ( α ϕ o ) ( sin ϕ o + ε cos ϕ o ) } d x 2 a R 1 / 2 exp ( α ϕ e ) cos ( 2 π x ) × { [ cosh ( α ϕ o ) + ε α sinh ( α ϕ o ) ] ( sin ϕ o + ε cos ϕ o ) + [ sinh ( α ϕ o ) + ε α cosh ( α ϕ o ) ] ( cos ϕ o ε sin ϕ o ) } d x .
δ F = 2 ε R 1 / 2 exp ( α ϕ e ) × { cosh ( α ϕ o ) sin ϕ o + α sinh ( α ϕ o ) cos ϕ o + sinh ( α ϕ o ) cos ϕ o + α cosh ( α ϕ o ) sin ϕ o a cos ( 2 π x ) [ cosh ( α ϕ o ) cos ϕ o + α sinh ( α ϕ o ) sin ϕ o sinh ( α ϕ o ) sin ϕ o + α cosh ( α ϕ o ) cos ϕ o ] } d x = 0 .
( 1 α ) [ cosh ( α ϕ o ) a cos ( 2 π x ) sinh ( α ϕ o ) ] sin ϕ o = ( 1 + α ) [ sinh ( α ϕ o ) a cos ( 2 π x ) cosh ( α ϕ o ) ] cos ϕ o .
ϕ ( x ) = { 0 if    x [ R , R ] ϕ ( x ) + ϕ c if    x [ R , R ] .
sin ϕ o = a cos ( 2 π x ) cos ϕ o ,
ϕ o = arctan [ a cos ( 2 π x ) ] .

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