Abstract

I present several classes of analytical and semi-analytical solutions for the design of high-speed rotary optical delay lines that use a combination of stationary and rotating curvilinear reflectors. Detailed analysis of four distinct classes of optical delay lines is presented. Particularly, I consider delay lines based on a single rotating reflector, a single rotating reflector and a single stationary reflector, two rotating reflectors, and two rotating reflectors and a single stationary reflector. I demonstrate that in each of these cases it is possible to design an infinite variety of the optical delay lines featuring linear dependence of the optical delay on the rotation angle. This is achieved via shape optimization of the rotating and stationary reflector surfaces. Moreover, in the case of two rotating reflectors a convenient spatial separation of the incoming and outgoing beams is possible. For the sake of example, all the blades presented in this paper are chosen to fit into a circle of 10cm diameter and these delay lines feature in excess of 600ps of optical delay. Finally, two prototypes of rotary delay lines were fabricated using CNC machining, and their optical properties are characterized.

© 2014 Optical Society of America

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References

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  1. R.E. Beselt, “Large Amplitude High Frequency Optical Delay,” Honeywell ASCa, US 7,899,281 (2011).
  2. D. C. Edelstein, R. B. Romney, and M. Scheuermann, “Rapid programmable 300 ps optical delay scanner and signal averaging system for ultrafast measurements,” Rev. Sci. Instrum. 62(3), 579 (1991).
    [Crossref]
  3. J. Ballif, R. Gianotti, Ph. Chavanne, R. Wälti, and R. P. Salathé, “Rapid and scalable scans at 21 m/s in optical low-coherence reflectometry,” Opt. Lett. 22(11), 757–759 (1997).
    [Crossref] [PubMed]
  4. J. Szydlo, N. Delachenal, R. Gianotti, R. Walti, H. Bleuler, and P. R. Salathe, “Air-turbine driven optical low coherence reflectometry at 28.6-kHz scan repetition rate,” Opt. Commun. 154(1–3), 1–4 (1998).
    [Crossref]
  5. G G. Lamouche, M. Dufour, B. Gauthier, V. Bartulovic, M. Hewko, and J.-P. Monchalin, “Optical delay line using rotating rhombic prisms,” Proc. SPIE 6429, 64292G (2007).
  6. A. L. Oldenburg, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Fast-Fourier-domain delay line for in vivo optical coherence tomography with a polygonal scanner,” Appl. Opt. 42(22), 4606–4611 (2003).
    [Crossref] [PubMed]
  7. P.-L. Hsiung, X. Li, C. Chudoba, I. Hartl, T. H. Ko, and J. G. Fujimoto, “High-speed path-length scanning with a multiple-pass cavity delay line,” Appl. Opt. 42(4), 640–648 (2003).
    [Crossref] [PubMed]
  8. L. Liu and N. G. Chen, “Double-pass rotary mirror array for fast scanning optical delay line,” Appl. Opt. 45(21), 5426–5431 (2006).
    [Crossref] [PubMed]
  9. X. Liu, M. J. Cobb, and X. Li, “Rapid scanning all-reflective optical delay line for real-time optical coherence tomography,” Opt. Lett. 29(1), 80–82 (2004).
    [Crossref] [PubMed]
  10. K. Locharoenrat and I. Jen Hsu, “Optical delay line for rapid scanning low-coherence reflectometer,” Intern. J. Inf. Electron. Eng. 2, 904–906 (2012).
  11. C.-L. Wnag, C.-L.Pan, “Scanning Optical Delay Device Having a Helicoid Reflecting Mirror,” National Science Council, Taiwan, US 5,907,423 (1999).
  12. T.D. Dorney, “Scanning Optical Delay Line Using a Reflective Element Arranged to Rotate,” US 7,046,412 (2006).
  13. G. J. Kim, Y. S. Jin, S. G. Jeon, and J. I. Kim, “Rotary Optical Delay Line,” Korea Electrotechnology Research Institute, US 7,453,619 (2008).
  14. J. Xu and X.-C. Zhang, “Circular involute stage,” Opt. Lett. 29(17), 2082–2084 (2004).
    [Crossref] [PubMed]
  15. G.-J. Kim, S.-G. Jeon, J.-I. Kim, and Y.-S. Jin, “High speed scanning of terahertz pulse by a rotary optical delay line,” Rev. Sci. Instrum. 79(10), 106102 (2008).
    [Crossref] [PubMed]

2012 (1)

K. Locharoenrat and I. Jen Hsu, “Optical delay line for rapid scanning low-coherence reflectometer,” Intern. J. Inf. Electron. Eng. 2, 904–906 (2012).

2008 (1)

G.-J. Kim, S.-G. Jeon, J.-I. Kim, and Y.-S. Jin, “High speed scanning of terahertz pulse by a rotary optical delay line,” Rev. Sci. Instrum. 79(10), 106102 (2008).
[Crossref] [PubMed]

2007 (1)

G G. Lamouche, M. Dufour, B. Gauthier, V. Bartulovic, M. Hewko, and J.-P. Monchalin, “Optical delay line using rotating rhombic prisms,” Proc. SPIE 6429, 64292G (2007).

2006 (1)

L. Liu and N. G. Chen, “Double-pass rotary mirror array for fast scanning optical delay line,” Appl. Opt. 45(21), 5426–5431 (2006).
[Crossref] [PubMed]

2004 (2)

X. Liu, M. J. Cobb, and X. Li, “Rapid scanning all-reflective optical delay line for real-time optical coherence tomography,” Opt. Lett. 29(1), 80–82 (2004).
[Crossref] [PubMed]

J. Xu and X.-C. Zhang, “Circular involute stage,” Opt. Lett. 29(17), 2082–2084 (2004).
[Crossref] [PubMed]

2003 (2)

A. L. Oldenburg, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Fast-Fourier-domain delay line for in vivo optical coherence tomography with a polygonal scanner,” Appl. Opt. 42(22), 4606–4611 (2003).
[Crossref] [PubMed]

P.-L. Hsiung, X. Li, C. Chudoba, I. Hartl, T. H. Ko, and J. G. Fujimoto, “High-speed path-length scanning with a multiple-pass cavity delay line,” Appl. Opt. 42(4), 640–648 (2003).
[Crossref] [PubMed]

1998 (1)

J. Szydlo, N. Delachenal, R. Gianotti, R. Walti, H. Bleuler, and P. R. Salathe, “Air-turbine driven optical low coherence reflectometry at 28.6-kHz scan repetition rate,” Opt. Commun. 154(1–3), 1–4 (1998).
[Crossref]

1997 (1)

J. Ballif, R. Gianotti, Ph. Chavanne, R. Wälti, and R. P. Salathé, “Rapid and scalable scans at 21 m/s in optical low-coherence reflectometry,” Opt. Lett. 22(11), 757–759 (1997).
[Crossref] [PubMed]

1991 (1)

D. C. Edelstein, R. B. Romney, and M. Scheuermann, “Rapid programmable 300 ps optical delay scanner and signal averaging system for ultrafast measurements,” Rev. Sci. Instrum. 62(3), 579 (1991).
[Crossref]

Ballif, J.

J. Ballif, R. Gianotti, Ph. Chavanne, R. Wälti, and R. P. Salathé, “Rapid and scalable scans at 21 m/s in optical low-coherence reflectometry,” Opt. Lett. 22(11), 757–759 (1997).
[Crossref] [PubMed]

Bartulovic, V.

G G. Lamouche, M. Dufour, B. Gauthier, V. Bartulovic, M. Hewko, and J.-P. Monchalin, “Optical delay line using rotating rhombic prisms,” Proc. SPIE 6429, 64292G (2007).

Bleuler, H.

J. Szydlo, N. Delachenal, R. Gianotti, R. Walti, H. Bleuler, and P. R. Salathe, “Air-turbine driven optical low coherence reflectometry at 28.6-kHz scan repetition rate,” Opt. Commun. 154(1–3), 1–4 (1998).
[Crossref]

Boppart, S. A.

A. L. Oldenburg, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Fast-Fourier-domain delay line for in vivo optical coherence tomography with a polygonal scanner,” Appl. Opt. 42(22), 4606–4611 (2003).
[Crossref] [PubMed]

Chavanne, Ph.

J. Ballif, R. Gianotti, Ph. Chavanne, R. Wälti, and R. P. Salathé, “Rapid and scalable scans at 21 m/s in optical low-coherence reflectometry,” Opt. Lett. 22(11), 757–759 (1997).
[Crossref] [PubMed]

Chen, N. G.

L. Liu and N. G. Chen, “Double-pass rotary mirror array for fast scanning optical delay line,” Appl. Opt. 45(21), 5426–5431 (2006).
[Crossref] [PubMed]

Chudoba, C.

P.-L. Hsiung, X. Li, C. Chudoba, I. Hartl, T. H. Ko, and J. G. Fujimoto, “High-speed path-length scanning with a multiple-pass cavity delay line,” Appl. Opt. 42(4), 640–648 (2003).
[Crossref] [PubMed]

Cobb, M. J.

X. Liu, M. J. Cobb, and X. Li, “Rapid scanning all-reflective optical delay line for real-time optical coherence tomography,” Opt. Lett. 29(1), 80–82 (2004).
[Crossref] [PubMed]

Delachenal, N.

J. Szydlo, N. Delachenal, R. Gianotti, R. Walti, H. Bleuler, and P. R. Salathe, “Air-turbine driven optical low coherence reflectometry at 28.6-kHz scan repetition rate,” Opt. Commun. 154(1–3), 1–4 (1998).
[Crossref]

Dufour, M.

G G. Lamouche, M. Dufour, B. Gauthier, V. Bartulovic, M. Hewko, and J.-P. Monchalin, “Optical delay line using rotating rhombic prisms,” Proc. SPIE 6429, 64292G (2007).

Edelstein, D. C.

D. C. Edelstein, R. B. Romney, and M. Scheuermann, “Rapid programmable 300 ps optical delay scanner and signal averaging system for ultrafast measurements,” Rev. Sci. Instrum. 62(3), 579 (1991).
[Crossref]

Fujimoto, J. G.

P.-L. Hsiung, X. Li, C. Chudoba, I. Hartl, T. H. Ko, and J. G. Fujimoto, “High-speed path-length scanning with a multiple-pass cavity delay line,” Appl. Opt. 42(4), 640–648 (2003).
[Crossref] [PubMed]

Gauthier, B.

G G. Lamouche, M. Dufour, B. Gauthier, V. Bartulovic, M. Hewko, and J.-P. Monchalin, “Optical delay line using rotating rhombic prisms,” Proc. SPIE 6429, 64292G (2007).

Gianotti, R.

J. Szydlo, N. Delachenal, R. Gianotti, R. Walti, H. Bleuler, and P. R. Salathe, “Air-turbine driven optical low coherence reflectometry at 28.6-kHz scan repetition rate,” Opt. Commun. 154(1–3), 1–4 (1998).
[Crossref]

J. Ballif, R. Gianotti, Ph. Chavanne, R. Wälti, and R. P. Salathé, “Rapid and scalable scans at 21 m/s in optical low-coherence reflectometry,” Opt. Lett. 22(11), 757–759 (1997).
[Crossref] [PubMed]

Hartl, I.

P.-L. Hsiung, X. Li, C. Chudoba, I. Hartl, T. H. Ko, and J. G. Fujimoto, “High-speed path-length scanning with a multiple-pass cavity delay line,” Appl. Opt. 42(4), 640–648 (2003).
[Crossref] [PubMed]

Hewko, M.

G G. Lamouche, M. Dufour, B. Gauthier, V. Bartulovic, M. Hewko, and J.-P. Monchalin, “Optical delay line using rotating rhombic prisms,” Proc. SPIE 6429, 64292G (2007).

Hsiung, P.-L.

P.-L. Hsiung, X. Li, C. Chudoba, I. Hartl, T. H. Ko, and J. G. Fujimoto, “High-speed path-length scanning with a multiple-pass cavity delay line,” Appl. Opt. 42(4), 640–648 (2003).
[Crossref] [PubMed]

Jen Hsu, I.

K. Locharoenrat and I. Jen Hsu, “Optical delay line for rapid scanning low-coherence reflectometer,” Intern. J. Inf. Electron. Eng. 2, 904–906 (2012).

Jeon, S.-G.

G.-J. Kim, S.-G. Jeon, J.-I. Kim, and Y.-S. Jin, “High speed scanning of terahertz pulse by a rotary optical delay line,” Rev. Sci. Instrum. 79(10), 106102 (2008).
[Crossref] [PubMed]

Jin, Y.-S.

G.-J. Kim, S.-G. Jeon, J.-I. Kim, and Y.-S. Jin, “High speed scanning of terahertz pulse by a rotary optical delay line,” Rev. Sci. Instrum. 79(10), 106102 (2008).
[Crossref] [PubMed]

Kim, G.-J.

G.-J. Kim, S.-G. Jeon, J.-I. Kim, and Y.-S. Jin, “High speed scanning of terahertz pulse by a rotary optical delay line,” Rev. Sci. Instrum. 79(10), 106102 (2008).
[Crossref] [PubMed]

Kim, J.-I.

G.-J. Kim, S.-G. Jeon, J.-I. Kim, and Y.-S. Jin, “High speed scanning of terahertz pulse by a rotary optical delay line,” Rev. Sci. Instrum. 79(10), 106102 (2008).
[Crossref] [PubMed]

Ko, T. H.

P.-L. Hsiung, X. Li, C. Chudoba, I. Hartl, T. H. Ko, and J. G. Fujimoto, “High-speed path-length scanning with a multiple-pass cavity delay line,” Appl. Opt. 42(4), 640–648 (2003).
[Crossref] [PubMed]

Lamouche, G G.

G G. Lamouche, M. Dufour, B. Gauthier, V. Bartulovic, M. Hewko, and J.-P. Monchalin, “Optical delay line using rotating rhombic prisms,” Proc. SPIE 6429, 64292G (2007).

Li, X.

X. Liu, M. J. Cobb, and X. Li, “Rapid scanning all-reflective optical delay line for real-time optical coherence tomography,” Opt. Lett. 29(1), 80–82 (2004).
[Crossref] [PubMed]

P.-L. Hsiung, X. Li, C. Chudoba, I. Hartl, T. H. Ko, and J. G. Fujimoto, “High-speed path-length scanning with a multiple-pass cavity delay line,” Appl. Opt. 42(4), 640–648 (2003).
[Crossref] [PubMed]

Liu, L.

L. Liu and N. G. Chen, “Double-pass rotary mirror array for fast scanning optical delay line,” Appl. Opt. 45(21), 5426–5431 (2006).
[Crossref] [PubMed]

Liu, X.

X. Liu, M. J. Cobb, and X. Li, “Rapid scanning all-reflective optical delay line for real-time optical coherence tomography,” Opt. Lett. 29(1), 80–82 (2004).
[Crossref] [PubMed]

Locharoenrat, K.

K. Locharoenrat and I. Jen Hsu, “Optical delay line for rapid scanning low-coherence reflectometer,” Intern. J. Inf. Electron. Eng. 2, 904–906 (2012).

Marks, D. L.

A. L. Oldenburg, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Fast-Fourier-domain delay line for in vivo optical coherence tomography with a polygonal scanner,” Appl. Opt. 42(22), 4606–4611 (2003).
[Crossref] [PubMed]

Monchalin, J.-P.

G G. Lamouche, M. Dufour, B. Gauthier, V. Bartulovic, M. Hewko, and J.-P. Monchalin, “Optical delay line using rotating rhombic prisms,” Proc. SPIE 6429, 64292G (2007).

Oldenburg, A. L.

A. L. Oldenburg, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Fast-Fourier-domain delay line for in vivo optical coherence tomography with a polygonal scanner,” Appl. Opt. 42(22), 4606–4611 (2003).
[Crossref] [PubMed]

Reynolds, J. J.

A. L. Oldenburg, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Fast-Fourier-domain delay line for in vivo optical coherence tomography with a polygonal scanner,” Appl. Opt. 42(22), 4606–4611 (2003).
[Crossref] [PubMed]

Romney, R. B.

D. C. Edelstein, R. B. Romney, and M. Scheuermann, “Rapid programmable 300 ps optical delay scanner and signal averaging system for ultrafast measurements,” Rev. Sci. Instrum. 62(3), 579 (1991).
[Crossref]

Salathe, P. R.

J. Szydlo, N. Delachenal, R. Gianotti, R. Walti, H. Bleuler, and P. R. Salathe, “Air-turbine driven optical low coherence reflectometry at 28.6-kHz scan repetition rate,” Opt. Commun. 154(1–3), 1–4 (1998).
[Crossref]

Salathé, R. P.

J. Ballif, R. Gianotti, Ph. Chavanne, R. Wälti, and R. P. Salathé, “Rapid and scalable scans at 21 m/s in optical low-coherence reflectometry,” Opt. Lett. 22(11), 757–759 (1997).
[Crossref] [PubMed]

Scheuermann, M.

D. C. Edelstein, R. B. Romney, and M. Scheuermann, “Rapid programmable 300 ps optical delay scanner and signal averaging system for ultrafast measurements,” Rev. Sci. Instrum. 62(3), 579 (1991).
[Crossref]

Szydlo, J.

J. Szydlo, N. Delachenal, R. Gianotti, R. Walti, H. Bleuler, and P. R. Salathe, “Air-turbine driven optical low coherence reflectometry at 28.6-kHz scan repetition rate,” Opt. Commun. 154(1–3), 1–4 (1998).
[Crossref]

Walti, R.

J. Szydlo, N. Delachenal, R. Gianotti, R. Walti, H. Bleuler, and P. R. Salathe, “Air-turbine driven optical low coherence reflectometry at 28.6-kHz scan repetition rate,” Opt. Commun. 154(1–3), 1–4 (1998).
[Crossref]

Wälti, R.

J. Ballif, R. Gianotti, Ph. Chavanne, R. Wälti, and R. P. Salathé, “Rapid and scalable scans at 21 m/s in optical low-coherence reflectometry,” Opt. Lett. 22(11), 757–759 (1997).
[Crossref] [PubMed]

Xu, J.

J. Xu and X.-C. Zhang, “Circular involute stage,” Opt. Lett. 29(17), 2082–2084 (2004).
[Crossref] [PubMed]

Zhang, X.-C.

J. Xu and X.-C. Zhang, “Circular involute stage,” Opt. Lett. 29(17), 2082–2084 (2004).
[Crossref] [PubMed]

Appl. Opt. (3)

A. L. Oldenburg, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Fast-Fourier-domain delay line for in vivo optical coherence tomography with a polygonal scanner,” Appl. Opt. 42(22), 4606–4611 (2003).
[Crossref] [PubMed]

P.-L. Hsiung, X. Li, C. Chudoba, I. Hartl, T. H. Ko, and J. G. Fujimoto, “High-speed path-length scanning with a multiple-pass cavity delay line,” Appl. Opt. 42(4), 640–648 (2003).
[Crossref] [PubMed]

L. Liu and N. G. Chen, “Double-pass rotary mirror array for fast scanning optical delay line,” Appl. Opt. 45(21), 5426–5431 (2006).
[Crossref] [PubMed]

Intern. J. Inf. Electron. Eng. (1)

K. Locharoenrat and I. Jen Hsu, “Optical delay line for rapid scanning low-coherence reflectometer,” Intern. J. Inf. Electron. Eng. 2, 904–906 (2012).

Opt. Commun. (1)

J. Szydlo, N. Delachenal, R. Gianotti, R. Walti, H. Bleuler, and P. R. Salathe, “Air-turbine driven optical low coherence reflectometry at 28.6-kHz scan repetition rate,” Opt. Commun. 154(1–3), 1–4 (1998).
[Crossref]

Opt. Lett. (3)

X. Liu, M. J. Cobb, and X. Li, “Rapid scanning all-reflective optical delay line for real-time optical coherence tomography,” Opt. Lett. 29(1), 80–82 (2004).
[Crossref] [PubMed]

J. Xu and X.-C. Zhang, “Circular involute stage,” Opt. Lett. 29(17), 2082–2084 (2004).
[Crossref] [PubMed]

J. Ballif, R. Gianotti, Ph. Chavanne, R. Wälti, and R. P. Salathé, “Rapid and scalable scans at 21 m/s in optical low-coherence reflectometry,” Opt. Lett. 22(11), 757–759 (1997).
[Crossref] [PubMed]

Proc. SPIE (1)

G G. Lamouche, M. Dufour, B. Gauthier, V. Bartulovic, M. Hewko, and J.-P. Monchalin, “Optical delay line using rotating rhombic prisms,” Proc. SPIE 6429, 64292G (2007).

Rev. Sci. Instrum. (2)

D. C. Edelstein, R. B. Romney, and M. Scheuermann, “Rapid programmable 300 ps optical delay scanner and signal averaging system for ultrafast measurements,” Rev. Sci. Instrum. 62(3), 579 (1991).
[Crossref]

G.-J. Kim, S.-G. Jeon, J.-I. Kim, and Y.-S. Jin, “High speed scanning of terahertz pulse by a rotary optical delay line,” Rev. Sci. Instrum. 79(10), 106102 (2008).
[Crossref] [PubMed]

Other (4)

C.-L. Wnag, C.-L.Pan, “Scanning Optical Delay Device Having a Helicoid Reflecting Mirror,” National Science Council, Taiwan, US 5,907,423 (1999).

T.D. Dorney, “Scanning Optical Delay Line Using a Reflective Element Arranged to Rotate,” US 7,046,412 (2006).

G. J. Kim, Y. S. Jin, S. G. Jeon, and J. I. Kim, “Rotary Optical Delay Line,” Korea Electrotechnology Research Institute, US 7,453,619 (2008).

R.E. Beselt, “Large Amplitude High Frequency Optical Delay,” Honeywell ASCa, US 7,899,281 (2011).

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Figures (8)

Fig. 1
Fig. 1 Schematic of a rotary optical delay line featuring a single rotating blade. By design, the blade edge is always perpendicular to the incoming light beam at all rotation angles. Light beam is arriving parallel to the OX axis and is reflected back by the blade along the same path.
Fig. 2
Fig. 2 (a) Schematic of a rotary optical delay line featuring a single rotating blade. Blade positions at various rotation angles are also presented as dotted lines. (b) Corresponding optical delay. (c) Schematic of a rotary delay line with 3 blades and (b) its corresponding optical delay.
Fig. 3
Fig. 3 (a) Schematic of a rotary optical delay line featuring a single rotating blade that forms a fixed inclination angle with the direction of incoming light and a linear stationary reflector. (b) Blade positions at various rotation angles (dotted curves). (c) Corresponding optical delay.
Fig. 4
Fig. 4 (a) Schematic of a linear rotary optical delay line featuring a straight rotating blade and a stationary curvilinear reflector. (b) Rotary stage design that can accommodate p = 7 straight blades. Blade positions at various rotation angles are presented as dotted lines. (c) Corresponding optical delay.
Fig. 5
Fig. 5 (a) Schematic of a linear rotary optical delay line featuring a composite rotating blade with two sub-blades. (b) An example of a rotary delay line with R d l = 5 c m , R i = 1 c m , R o = 3 c m . Blade positions at various rotation angles are presented as dotted lines. (c) Corresponding optical delay.
Fig. 6
Fig. 6 (a) Schematic of a rotary optical delay line featuring a composite rotating blade with two sub-blades and a stationary reflector. (b) An example of a rotary delay line. Blade positions at various rotation angles are presented as dotted lines. (c) Corresponding optical delay.
Fig. 7
Fig. 7 (a) Experimental realisation of the rotary delay line featuring one rotating and one stationary reflector. (b) Optical delay measured directly from the high-resolution images of the light path. (c) Optical delay.
Fig. 8
Fig. 8 (a) Experimental realisation of the rotary delay line featuring one rotating and one stationary reflector. (b) Optical delay measured directly from the high-resolution images of the light path. (c) Optical delay.

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

x( φ )=r( φ )cos( φ ) y( φ )=r( φ )sin( φ ).
x( φ )=r( φ )cos( φθ ) y( φ )=r( φ )sin( φθ ).
{ x( φ( θ ) )= x b ( θ ) y( φ( θ ) )= R i { tan( φ( θ )θ )= y x = R i x b ( θ ) r( φ( θ ) )= x 2 + y 2 = x b ( θ ) 2 + R i 2 ,
φ( t )=t+arctan( R i x b ( t ) )+πn r( φ( t ) )= x b ( t ) 2 + R i 2 , t[ 0, θ max ], n
tan( α b )= x b ( θ ) x b ( θ ) / θ R i .
x b ( θ )= x 0 + R i θ,
ΔT= 2 c ( x b ( θ ) x b ( 0 ) )= 2 R i c θ ; θ[ 0, θ max ].
x b ( 0 ) = 0 x 0 = 0 x b ( θ max ) = R d l 2 R i 2 R i = R d l θ max 2 + 1 .
Δ T max = 2 R d l c θ max θ max 2 + 1 .
Δ T max = 2 R dl c 2π p 2 + ( 2π ) 2 .
tan( α b )= x b ( θ ) x b ( θ ) / θ R i .
x b ( θ ) θ cot( α b ) x b ( θ ) R i =0,
x b ( θ )= x 0 exp( cot( α b )θ ) R i tan( α b ).
y= R i +( x x b ( θ ) )tan( 2 α b π ).
x sr ( θ )= x b ( θ )L( θ )cos( 2 α b ) y sr ( θ )= R i L( θ )sin( 2 α b ).
y sr ( θ ) / θ x sr ( θ ) / θ =tan( 2 α b π/2 ) L( θ ) θ = x b ( θ ) θ cos( 2 α b ).
L( θ )= L 0 + x b ( θ )cos( 2 α b ),
ΔT= 2 c ( x b ( θ )L( θ ) x b ( 0 )+L( 0 ) ) = 4 c sin 2 ( α b )( x b ( θ ) x b ( 0 ) ).
tan( α b ( θ ) )= x b ( θ ) x b ( θ ) / θ R i .
y= R i +( x x b ( θ ) )tan( 2 α b ( θ )π ).
x sr ( θ )= x b ( θ )L( θ )cos( 2 α b ( θ ) ) y sr ( θ )= R i L( θ )sin( 2 α b ( θ ) ).
y sr ( θ ) / θ x sr ( θ ) / θ =tan( 2 α b ( θ )π/2 ).
L( θ ) θ = x b ( θ ) θ cos( 2 α b ( θ ) ).
L( θ )= L 0 + 0 θ max dθ x b ( θ ) θ [ 1 ( x b ( θ ) x b ( θ ) / θ R i ) 2 ] [ 1+ ( x b ( θ ) x b ( θ ) / θ R i ) 2 ] 1 .
x b ( θ )L( θ )=βθ L 0 ,
x b ( θ ) θ = β 2 ( 1+ 1 tan 2 ( α b ( θ ) ) ).
x b ( θ ) θ = β 2 ( 1+ ( x b ( θ ) / θ R i ) 2 x b 2 ( θ ) ).
( x b ( θ ) θ ) 2 2( R i + x b 2 ( θ ) β ) x b ( θ ) θ +( x b 2 ( θ )+ R i 2 )=0,
x b ( θ ) θ = R i + x b 2 ( θ ) β ± ( x b 2 ( θ ) β ) 2 +( 2 R i β 1 ) x b 2 ( θ ) .
θ( x b )= C 0 + x b min x b dx [ R i + x 2 β ± ( x 2 β ) 2 +( 2 R i β 1 ) x 2 ] 1 ,
θ( x b )= C 0 + x b min x b dx [ R i + x 2 R i ] 1 = θ 0 +arctan( x b R i ).
x b ( θ )= R i tan( θ θ 0 ) L( θ )= L 0 + R i tan( θ θ 0 )2 R i θ.
ΔT= 2 c ( x b ( θ )L( θ ) x b ( 0 )+L( 0 ) )= 4 R i c θ.
ΔT= 4 R i c arccos( R i R dl ),
Δ T max = max R i ( ΔT ) 2.244 R dl c R i 0.652 R dl .
Δ T p = 4 R dl c ( 2π p )cos( 2π p ).
Δ T max p = max p=7 ( Δ T p ) 2.239 R dl c .
{ x 0 ( θ ) x 0 ( θ ) / θ R 0 =tan( α o ( θ ) ) x i ( θ ) x i ( θ ) / θ R i = α i =π/2 + α o 1 tan( α o ( θ ) ) { x 0 ( θ ) θ = R 0 + x 0 ( θ ) tan( α o ( θ ) ) x i ( θ ) θ = R i x i ( θ )tan( α o ( θ ) ) .
x i ( θ ) x o ( θ )=( R i R 0 )tan( 2 α o π 2 )= R i R 0 tan( 2 α o ) .
2 x i ( θ ) x o ( θ ) R i R 0 = tan 1 ( α o )tan( α o ) tan( α o ( θ ) )= x i ( θ ) x o ( θ ) R i R 0 ± ( x i ( θ ) x o ( θ ) R i R 0 ) 2 +1 .
ΔT= 1 c ( x i ( θ )+ x o ( θ )L( θ )( x i ( 0 )+ x o ( 0 )L( 0 ) ) )= R i + R o c θ,
L( θ )= ( x i ( θ ) x o ( θ ) ) 2 + ( R i R o ) 2 .
tan( α r ( θ ) )=cot( α i ( θ )+ α o ( θ ) )= 1cot( α i ( θ ) )cot( α o ( θ ) ) cot( α i ( θ ) )+cot( α o ( θ ) ) cot( α o ( θ ) )= x o ( θ ) / θ R o x o ( θ ) ; cot( α i ( θ ) )= x i ( θ ) / θ R i x i ( θ ) .
x r ( θ )= x o ( θ )+ L o ( θ )cos( π2 α o ( θ ) )= x i ( θ )+ L i ( θ )cos( 2 α i ( θ )π ) y r ( θ )= R o L o ( θ )sin( π2 α o ( θ ) )= R i + L i ( θ )sin( 2 α i ( θ )π ),
L o ( θ )= ( x i ( θ ) x o ( θ ) )tan( 2 α i ( θ ) )( R i R o ) ( tan( 2 α o ( θ ) )tan( 2 α i ( θ ) ) )cos( 2 α o ( θ ) ) L i ( θ )= ( x i ( θ ) x o ( θ ) )tan( 2 α o ( θ ) )( R i R o ) ( tan( 2 α o ( θ ) )tan( 2 α i ( θ ) ) )cos( 2 α i ( θ ) ) . x r ( θ )= x o ( θ )tan( 2 α o ( θ ) ) x i ( θ )tan( 2 α i ( θ ) )+( R i R o ) tan( 2 α o ( θ ) )tan( 2 α i ( θ ) ) y r ( θ )= R i cot( 2 α i ( θ ) ) R o cot( 2 α o ( θ ) )( x i ( θ ) x o ( θ ) ) cot( 2 α i ( θ ) )cot( 2 α o ( θ ) )
y r ( θ ) / θ x r ( θ ) / θ =tan( α r ( θ ) ),
tan( 2α )= 2tan( α ) / ( 1 tan 2 ( α ) ) cos( 2α )= ( 1 tan 2 ( α ) ) / ( 1+ tan 2 ( α ) ) .
y r ( θ ) / θ x r ( θ ) / θ | θ= θ 0 = y r x o x o + y r x o x o + y r x i x i + y r x i x i x r x o x o + x r x o x o + x r x i x i + x r x i x i =tan( α r ),
2 x i ( θ ) θ 2 | θ= θ 0 = ( tan( α r ) x r x o y r x o ) x o +( tan( α r ) x r x o y r x o ) x o +( tan( α r ) y r x i ) x r x i x i ( y r x i tan( α r ) x r x i ) .
x i ( θ 0 +dθ )= x i ( θ 0 )+dθ x i ( θ ) θ | θ= θ 0 + d θ 2 2 2 x i ( θ ) θ 2 | θ= θ 0 x i ( θ ) θ | θ= θ 0 +dθ = x i ( θ ) θ | θ= θ 0 +dθ 2 x i ( θ ) θ 2 | θ= θ 0 .
ΔT( θ )= 1 c ( x i ( θ )+ x o ( θ ) L i ( θ ) L o ( θ )( x i ( 0 )+ x o ( 0 ) L i ( 0 ) L o ( 0 ) ) )=βθ,
{ y r ( θ ) / θ x r ( θ ) / θ tan( α r ( θ ) )=0 ΔT( θ ) θ β=0 ,
( x 1 y 1 )=( cos( δθ ) sin( δθ ) sin( δθ ) cos( δθ ) )( x b ( θ ) R i )( x b ( θ )+ R i δθ R i x b ( θ )δθ )+O( δ θ 2 ).
tan( α b )= lim δθ0 y 2 y 1 x 2 x 1 = x b ( θ ) x b ( θ ) / θ R i .

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