Abstract

Si-based total internal reflection (TIR) bio/chemical sensor presents an attractive alternative to Surface Plasmon Resonance (SPR) technology due to a relatively simple optical arrangement and technological implementation, as well as a relatively easy bio/chemical immobilization on Si/SiO2 surface with a number of novel attractive applications. This sensor is based on the control of phase difference between p- and s-polarized components of light reflected from Si/air or Si/water interface in TIR geometry and a high sensitivity of the sensor is granted by a high refractive index of Si (3.56 at 1200 nm). We study properties of TIR sensors in a configuration of spectral phase detection and identify conditions of maximal phase sensitive response. We also experimentally show that the detection limit of Si-based TIR sensor can be lowered down to a level of detection of commercially available SPR devices (10−6 Refractive Index Units, RIU) under the use of a proper low-noisy method of the phase control. The concept of Si-based TIR opens attractive prospects for the miniaturization of sensor devices, taking advantage of the advanced state of development of Si-based microfabrication technologies, while the proposed spectral phase detection scheme offers much easier packaging and calibration steps

© 2009 OSA

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References

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  1. P. N. Prasad, Introduction to Biophotonics, Wiley-Interscience (2003).
  2. “Handbook of Surface Plasmon Resonance”, Eds: R. B. M. Schasfoort, A. J. Tudos, Royal Society of Chemistry (2008).
  3. G. Harsanyi, Polymer Films In Sensor Applications (CRC Press 1995).
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    [Crossref]
  5. M. Espinosa Bosch, A. J. R. Sanchez, F. Sachez Rojas, and C. Bosch Ojeda, “Recent Development in Optical Fiber Biosensors,” Sensors 7, 797–859 ( 2007).
    [Crossref]
  6. B. Liedberg, C. Nylander, and I. Lundstrum, “Surface plasmon resonance for gas detection and biosensing,” Sens. Act. B Chem. 4(1), 299–304 ( 1983).
    [Crossref]
  7. B. Liedberg, C. Nylander, and I. Lundström, “Biosensing with surface plasmon resonance--how it all started,” Biosens. Bioelectron. 10(8), i–ix ( 1995).
    [Crossref] [PubMed]
  8. www.biacore.com
  9. A. V. Kabashin and P. I. Nikitin, “Interferometer based on a surface-plasmon resonance for sensor applications,” Quantum Electron. 27(7), 653–654 ( 1997).
    [Crossref]
  10. A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase Jumps and Interferometric Surface Plasmon Resonance Imaging,” Appl. Phys. Lett. 75(25), 3917–3919 ( 1999).
    [Crossref]
  11. S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29(20), 2378–2380 ( 2004).
    [Crossref] [PubMed]
  12. P. P. Markowicz, W. C. Law, A. Baev, P. Prasad, S. Patskovsky, and A. V. Kabashin, “Phase-sensitive time-modulated SPR polarimetry for wide dynamic range biosensing,” Opt. Express 15, 1745 ( 2007).
    [Crossref] [PubMed]
  13. S. Patskovsky, M. Maisonneuve, M. Meunier, and A. V. Kabashin, “Mechanical modulation method for ultrasensitive phase measurements in photonics biosensing,” Opt. Express 16(26), 21305–21314 ( 2008).
    [Crossref] [PubMed]
  14. S. Patskovsky, M. Meunier, and A. V. Kabashin, “Phase-sensitive silicon-based total internal reflection sensor,” Opt. Express 15(19), 12523–12528 ( 2007).
    [Crossref] [PubMed]
  15. M. Born, and E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  16. R. M. A. Azzam, “Differential reflection phase shift under conditions of attenuated internal reflection,” JOSA A Vol. 16 Iss. 7, (1999)
  17. R. M. A. Azzam, “Phase shifts that accompany total internal reflection at a dielectric-dielectric interface,” J. Opt. Soc. Am. A 21(8), 1559–1563 ( 2004).
    [Crossref]
  18. W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 ( 2007).
    [Crossref] [PubMed]
  19. M.W. Wang, F.H. Tsai, and Y.F. Chao “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455 –456 78–83 ( 2004).
    [Crossref]
  20. C. E. Stewart, I. R. Hooper, and J. R. Sambles, “Surface plasmon differential ellipsometry of aqueous solutions for bio-chemical sensing,” J. Phys. D Appl. Phys. 41(10), 105408–105415 ( 2008).
    [Crossref]
  21. www.luxpop.com
  22. S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. T. Luong, “Near-infrared surface plasmon resonance sensing on a silicon platform,” Sens. Actuators B Chem. 97(2-3), 409–414 ( 2004).
    [Crossref]
  23. S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. T. Luong, “Silicon-based surface plasmon resonance sensing with two surface plasmon polariton modes,” Appl. Opt. 42(34), 6905–6909 ( 2003).
    [Crossref] [PubMed]

2008 (2)

C. E. Stewart, I. R. Hooper, and J. R. Sambles, “Surface plasmon differential ellipsometry of aqueous solutions for bio-chemical sensing,” J. Phys. D Appl. Phys. 41(10), 105408–105415 ( 2008).
[Crossref]

S. Patskovsky, M. Maisonneuve, M. Meunier, and A. V. Kabashin, “Mechanical modulation method for ultrasensitive phase measurements in photonics biosensing,” Opt. Express 16(26), 21305–21314 ( 2008).
[Crossref] [PubMed]

2007 (4)

P. P. Markowicz, W. C. Law, A. Baev, P. Prasad, S. Patskovsky, and A. V. Kabashin, “Phase-sensitive time-modulated SPR polarimetry for wide dynamic range biosensing,” Opt. Express 15, 1745 ( 2007).
[Crossref] [PubMed]

S. Patskovsky, M. Meunier, and A. V. Kabashin, “Phase-sensitive silicon-based total internal reflection sensor,” Opt. Express 15(19), 12523–12528 ( 2007).
[Crossref] [PubMed]

M. Espinosa Bosch, A. J. R. Sanchez, F. Sachez Rojas, and C. Bosch Ojeda, “Recent Development in Optical Fiber Biosensors,” Sensors 7, 797–859 ( 2007).
[Crossref]

W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 ( 2007).
[Crossref] [PubMed]

2004 (4)

M.W. Wang, F.H. Tsai, and Y.F. Chao “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455 –456 78–83 ( 2004).
[Crossref]

S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. T. Luong, “Near-infrared surface plasmon resonance sensing on a silicon platform,” Sens. Actuators B Chem. 97(2-3), 409–414 ( 2004).
[Crossref]

R. M. A. Azzam, “Phase shifts that accompany total internal reflection at a dielectric-dielectric interface,” J. Opt. Soc. Am. A 21(8), 1559–1563 ( 2004).
[Crossref]

S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29(20), 2378–2380 ( 2004).
[Crossref] [PubMed]

2003 (1)

1999 (1)

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase Jumps and Interferometric Surface Plasmon Resonance Imaging,” Appl. Phys. Lett. 75(25), 3917–3919 ( 1999).
[Crossref]

1997 (1)

A. V. Kabashin and P. I. Nikitin, “Interferometer based on a surface-plasmon resonance for sensor applications,” Quantum Electron. 27(7), 653–654 ( 1997).
[Crossref]

1995 (2)

W. Lukosz, “Integrated optical direct chemical and biochemical sensors,” Sens. Act. B. 29(1-3), 37–50 ( 1995).
[Crossref]

B. Liedberg, C. Nylander, and I. Lundström, “Biosensing with surface plasmon resonance--how it all started,” Biosens. Bioelectron. 10(8), i–ix ( 1995).
[Crossref] [PubMed]

1983 (1)

B. Liedberg, C. Nylander, and I. Lundstrum, “Surface plasmon resonance for gas detection and biosensing,” Sens. Act. B Chem. 4(1), 299–304 ( 1983).
[Crossref]

Azzam, R. M. A.

Baev, A.

W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 ( 2007).
[Crossref] [PubMed]

P. P. Markowicz, W. C. Law, A. Baev, P. Prasad, S. Patskovsky, and A. V. Kabashin, “Phase-sensitive time-modulated SPR polarimetry for wide dynamic range biosensing,” Opt. Express 15, 1745 ( 2007).
[Crossref] [PubMed]

Bosch Ojeda, C.

M. Espinosa Bosch, A. J. R. Sanchez, F. Sachez Rojas, and C. Bosch Ojeda, “Recent Development in Optical Fiber Biosensors,” Sensors 7, 797–859 ( 2007).
[Crossref]

Chao, Y.F.

M.W. Wang, F.H. Tsai, and Y.F. Chao “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455 –456 78–83 ( 2004).
[Crossref]

Espinosa Bosch, M.

M. Espinosa Bosch, A. J. R. Sanchez, F. Sachez Rojas, and C. Bosch Ojeda, “Recent Development in Optical Fiber Biosensors,” Sensors 7, 797–859 ( 2007).
[Crossref]

Grigorenko, A. N.

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase Jumps and Interferometric Surface Plasmon Resonance Imaging,” Appl. Phys. Lett. 75(25), 3917–3919 ( 1999).
[Crossref]

Ho, H. P.

W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 ( 2007).
[Crossref] [PubMed]

S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29(20), 2378–2380 ( 2004).
[Crossref] [PubMed]

Hooper, I. R.

C. E. Stewart, I. R. Hooper, and J. R. Sambles, “Surface plasmon differential ellipsometry of aqueous solutions for bio-chemical sensing,” J. Phys. D Appl. Phys. 41(10), 105408–105415 ( 2008).
[Crossref]

Kabashin, A. V.

S. Patskovsky, M. Maisonneuve, M. Meunier, and A. V. Kabashin, “Mechanical modulation method for ultrasensitive phase measurements in photonics biosensing,” Opt. Express 16(26), 21305–21314 ( 2008).
[Crossref] [PubMed]

S. Patskovsky, M. Meunier, and A. V. Kabashin, “Phase-sensitive silicon-based total internal reflection sensor,” Opt. Express 15(19), 12523–12528 ( 2007).
[Crossref] [PubMed]

P. P. Markowicz, W. C. Law, A. Baev, P. Prasad, S. Patskovsky, and A. V. Kabashin, “Phase-sensitive time-modulated SPR polarimetry for wide dynamic range biosensing,” Opt. Express 15, 1745 ( 2007).
[Crossref] [PubMed]

W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 ( 2007).
[Crossref] [PubMed]

S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. T. Luong, “Near-infrared surface plasmon resonance sensing on a silicon platform,” Sens. Actuators B Chem. 97(2-3), 409–414 ( 2004).
[Crossref]

S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. T. Luong, “Silicon-based surface plasmon resonance sensing with two surface plasmon polariton modes,” Appl. Opt. 42(34), 6905–6909 ( 2003).
[Crossref] [PubMed]

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase Jumps and Interferometric Surface Plasmon Resonance Imaging,” Appl. Phys. Lett. 75(25), 3917–3919 ( 1999).
[Crossref]

A. V. Kabashin and P. I. Nikitin, “Interferometer based on a surface-plasmon resonance for sensor applications,” Quantum Electron. 27(7), 653–654 ( 1997).
[Crossref]

Kong, S. K.

Law, W. C.

Law, W.-C.

W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 ( 2007).
[Crossref] [PubMed]

Liedberg, B.

B. Liedberg, C. Nylander, and I. Lundström, “Biosensing with surface plasmon resonance--how it all started,” Biosens. Bioelectron. 10(8), i–ix ( 1995).
[Crossref] [PubMed]

B. Liedberg, C. Nylander, and I. Lundstrum, “Surface plasmon resonance for gas detection and biosensing,” Sens. Act. B Chem. 4(1), 299–304 ( 1983).
[Crossref]

Lin, C.

Lukosz, W.

W. Lukosz, “Integrated optical direct chemical and biochemical sensors,” Sens. Act. B. 29(1-3), 37–50 ( 1995).
[Crossref]

Lundström, I.

B. Liedberg, C. Nylander, and I. Lundström, “Biosensing with surface plasmon resonance--how it all started,” Biosens. Bioelectron. 10(8), i–ix ( 1995).
[Crossref] [PubMed]

Lundstrum, I.

B. Liedberg, C. Nylander, and I. Lundstrum, “Surface plasmon resonance for gas detection and biosensing,” Sens. Act. B Chem. 4(1), 299–304 ( 1983).
[Crossref]

Luong, J. H. T.

S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. T. Luong, “Near-infrared surface plasmon resonance sensing on a silicon platform,” Sens. Actuators B Chem. 97(2-3), 409–414 ( 2004).
[Crossref]

S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. T. Luong, “Silicon-based surface plasmon resonance sensing with two surface plasmon polariton modes,” Appl. Opt. 42(34), 6905–6909 ( 2003).
[Crossref] [PubMed]

Maisonneuve, M.

Markowicz, P.

W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 ( 2007).
[Crossref] [PubMed]

Markowicz, P. P.

Meunier, M.

Nikitin, P. I.

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase Jumps and Interferometric Surface Plasmon Resonance Imaging,” Appl. Phys. Lett. 75(25), 3917–3919 ( 1999).
[Crossref]

A. V. Kabashin and P. I. Nikitin, “Interferometer based on a surface-plasmon resonance for sensor applications,” Quantum Electron. 27(7), 653–654 ( 1997).
[Crossref]

Nylander, C.

B. Liedberg, C. Nylander, and I. Lundström, “Biosensing with surface plasmon resonance--how it all started,” Biosens. Bioelectron. 10(8), i–ix ( 1995).
[Crossref] [PubMed]

B. Liedberg, C. Nylander, and I. Lundstrum, “Surface plasmon resonance for gas detection and biosensing,” Sens. Act. B Chem. 4(1), 299–304 ( 1983).
[Crossref]

Patskovsky, S.

Prasad, P.

Prasad, P. N.

W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 ( 2007).
[Crossref] [PubMed]

Roy, I.

W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 ( 2007).
[Crossref] [PubMed]

Sachez Rojas, F.

M. Espinosa Bosch, A. J. R. Sanchez, F. Sachez Rojas, and C. Bosch Ojeda, “Recent Development in Optical Fiber Biosensors,” Sensors 7, 797–859 ( 2007).
[Crossref]

Sambles, J. R.

C. E. Stewart, I. R. Hooper, and J. R. Sambles, “Surface plasmon differential ellipsometry of aqueous solutions for bio-chemical sensing,” J. Phys. D Appl. Phys. 41(10), 105408–105415 ( 2008).
[Crossref]

Sanchez, A. J. R.

M. Espinosa Bosch, A. J. R. Sanchez, F. Sachez Rojas, and C. Bosch Ojeda, “Recent Development in Optical Fiber Biosensors,” Sensors 7, 797–859 ( 2007).
[Crossref]

Stewart, C. E.

C. E. Stewart, I. R. Hooper, and J. R. Sambles, “Surface plasmon differential ellipsometry of aqueous solutions for bio-chemical sensing,” J. Phys. D Appl. Phys. 41(10), 105408–105415 ( 2008).
[Crossref]

Tsai, F.H.

M.W. Wang, F.H. Tsai, and Y.F. Chao “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455 –456 78–83 ( 2004).
[Crossref]

Wang, M.W.

M.W. Wang, F.H. Tsai, and Y.F. Chao “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455 –456 78–83 ( 2004).
[Crossref]

Wu, S. Y.

Yong, K.-T.

W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 ( 2007).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase Jumps and Interferometric Surface Plasmon Resonance Imaging,” Appl. Phys. Lett. 75(25), 3917–3919 ( 1999).
[Crossref]

Biosens. Bioelectron. (2)

W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 ( 2007).
[Crossref] [PubMed]

B. Liedberg, C. Nylander, and I. Lundström, “Biosensing with surface plasmon resonance--how it all started,” Biosens. Bioelectron. 10(8), i–ix ( 1995).
[Crossref] [PubMed]

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

J. Phys. D Appl. Phys. (1)

C. E. Stewart, I. R. Hooper, and J. R. Sambles, “Surface plasmon differential ellipsometry of aqueous solutions for bio-chemical sensing,” J. Phys. D Appl. Phys. 41(10), 105408–105415 ( 2008).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Quantum Electron. (1)

A. V. Kabashin and P. I. Nikitin, “Interferometer based on a surface-plasmon resonance for sensor applications,” Quantum Electron. 27(7), 653–654 ( 1997).
[Crossref]

Sens. Act. B Chem. (1)

B. Liedberg, C. Nylander, and I. Lundstrum, “Surface plasmon resonance for gas detection and biosensing,” Sens. Act. B Chem. 4(1), 299–304 ( 1983).
[Crossref]

Sens. Act. B. (1)

W. Lukosz, “Integrated optical direct chemical and biochemical sensors,” Sens. Act. B. 29(1-3), 37–50 ( 1995).
[Crossref]

Sens. Actuators B Chem. (1)

S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. T. Luong, “Near-infrared surface plasmon resonance sensing on a silicon platform,” Sens. Actuators B Chem. 97(2-3), 409–414 ( 2004).
[Crossref]

Sensors (1)

M. Espinosa Bosch, A. J. R. Sanchez, F. Sachez Rojas, and C. Bosch Ojeda, “Recent Development in Optical Fiber Biosensors,” Sensors 7, 797–859 ( 2007).
[Crossref]

Thin Solid Films (1)

M.W. Wang, F.H. Tsai, and Y.F. Chao “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455 –456 78–83 ( 2004).
[Crossref]

Other (7)

M. Born, and E. Wolf, Principles of Optics (Pergamon, New York, 1975).

R. M. A. Azzam, “Differential reflection phase shift under conditions of attenuated internal reflection,” JOSA A Vol. 16 Iss. 7, (1999)

P. N. Prasad, Introduction to Biophotonics, Wiley-Interscience (2003).

“Handbook of Surface Plasmon Resonance”, Eds: R. B. M. Schasfoort, A. J. Tudos, Royal Society of Chemistry (2008).

G. Harsanyi, Polymer Films In Sensor Applications (CRC Press 1995).

www.biacore.com

www.luxpop.com

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic of Si-based TIR sensor; (b) Calculated spectral dependences for phases δp , δs and phase difference Δ in the sensor with air as a sample medium. The dependences are given for the incident angle of 16.6°

Fig. 2
Fig. 2

(a) Calculated spectral dependences for phases δp, δs and phase difference Δ in TIR Si-based sensor with water as a sample medium. The dependences are given for the incident angles of 22.4° (black curves) and 22.2° (red curves); (b) Calculated response of the phase difference Δ to variations of the refractive index

Fig. 3
Fig. 3

Spectral TIR phase difference as a function of the medium refractive index changes in air and in water

Fig. 4
Fig. 4

(a) Schematics of phase measurements in Si-based TIR geometry using PEM; (b) Response of the system under the replacement of pure Ar by pure N2

Equations (2)

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F 1 = A J 1 ( M ) cos ( φ ) F 2 = A J 2 ( M ) sin ( φ )
tan ( φ ) = F 2 J 1 ( M ) / F 1 J 2 ( M )

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