Abstract

We have investigated the effects of nonnormal incident rays in calculating the refractive index profile of a dielectric sample using the reflectance measurement data obtained with a scanning confocal epimicroscope and also by solving three-dimensional vector wave equations for linearly polarized light. The numerically calculated reflection data of tightly focused Gaussian beams with different numerical apertures (NAs) on planar surfaces with various refractive indices confirm that the reflectance increases with an increase in the NA of a focusing objective lens. This is due to the nonnormal incident ray components of a Gaussian beam. We have found that the refractive index obtained with the assumption of a normal incident beam is far from the real value when the NA of a focusing lens becomes larger than 0.5, and thus the variation in the reflectance for different angular components in a Gaussian beam must be taken into consideration while using a larger NA lens. Errors in practical refractive index calculation for an optical fiber based on a normal incident beam in reflectance measurements can be as large as 1% in comparison to real values calculated by our three-dimensional vector wave equations.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. F. L Pedrotti, L. M. Pedrotti, and L. S. Pedrotti, Introduction to Optics (Academic, 1993).
  2. Y. Youk and D. Y. Kim, "Tightly focused epimicroscope technique for submicrometer-resolved highly sensitive refractive index measurement of an optical waveguide," Appl. Opt. 46, 2949-2953 (2007).
    [CrossRef] [PubMed]
  3. S. B. Cho, Y. Youk, and D. Y. Kim, "Stable system technique for measuring the refractive index profile of an optical fiber by modified fiber-type confocal microscope method," Proc. SPIE 6469, 646914 (2007).
  4. W. J. Stewart, "A new technique for measuring the refractive index profiles of graded optical fibers," in Technical Digest 1006 of the 1997 International Conference on Integrated Optics and Optical Fiber Communication (IECE, Japan, Tokyo, 1997), pp. 395-398.
  5. W. J. Stewart, "Optical fiber and perform profiling technology," IEEE J. Quantum Electron. QE-18, 1451-1466 (1982).
    [CrossRef]
  6. K. W. Raine, J. G. Baines, and R. J. King, "Comparison of refractive index measurements of optical fibres by three methods," IEE Proc. 135, 190-195 (1988).
  7. K. W. Raine, J. G. N. Baines, and D. E. Putland, "Refractive index profiling--state of the art," J. Lightwave Technol. 7, 1162-1169 (1989).
    [CrossRef]
  8. T. Wilson, J. N. Gannaway, and C. J. R. Sheppard, "Optical fibre profiling using a scanning optical microscope," Opt. Quantum Electron. 12, 341-345 (1980).
    [CrossRef]
  9. J.-X Cheng, A. Volkmer, and X. S. Xie, "Theoretical and experimental characterization of coherent anti-stokes raman scattering microscopy," J. Opt. Soc. Am. B 19, 1363-1375 (2002).
    [CrossRef]
  10. A. E. Siegman, Lasers (Academic, 1986).
  11. S. Hasegawa, N. Aoyama, A. Futamata, and T. Uchiyama, "Optical tunneling effect calculation of a solid immersion lens for use in optical disk memory," Appl. Opt. 38, 2297-2300 (1999).
    [CrossRef]
  12. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London , Ser. A 253, 358-379 (1959).
    [CrossRef]

2007 (1)

2002 (1)

1999 (1)

1989 (1)

K. W. Raine, J. G. N. Baines, and D. E. Putland, "Refractive index profiling--state of the art," J. Lightwave Technol. 7, 1162-1169 (1989).
[CrossRef]

1988 (1)

K. W. Raine, J. G. Baines, and R. J. King, "Comparison of refractive index measurements of optical fibres by three methods," IEE Proc. 135, 190-195 (1988).

1982 (1)

W. J. Stewart, "Optical fiber and perform profiling technology," IEEE J. Quantum Electron. QE-18, 1451-1466 (1982).
[CrossRef]

1980 (1)

T. Wilson, J. N. Gannaway, and C. J. R. Sheppard, "Optical fibre profiling using a scanning optical microscope," Opt. Quantum Electron. 12, 341-345 (1980).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London , Ser. A 253, 358-379 (1959).
[CrossRef]

Appl. Opt. (2)

IEE Proc. (1)

K. W. Raine, J. G. Baines, and R. J. King, "Comparison of refractive index measurements of optical fibres by three methods," IEE Proc. 135, 190-195 (1988).

IEEE J. Quantum Electron. (1)

W. J. Stewart, "Optical fiber and perform profiling technology," IEEE J. Quantum Electron. QE-18, 1451-1466 (1982).
[CrossRef]

J. Lightwave Technol. (1)

K. W. Raine, J. G. N. Baines, and D. E. Putland, "Refractive index profiling--state of the art," J. Lightwave Technol. 7, 1162-1169 (1989).
[CrossRef]

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

Opt. Quantum Electron. (1)

T. Wilson, J. N. Gannaway, and C. J. R. Sheppard, "Optical fibre profiling using a scanning optical microscope," Opt. Quantum Electron. 12, 341-345 (1980).
[CrossRef]

Proc. R. Soc. London (1)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London , Ser. A 253, 358-379 (1959).
[CrossRef]

Other (4)

S. B. Cho, Y. Youk, and D. Y. Kim, "Stable system technique for measuring the refractive index profile of an optical fiber by modified fiber-type confocal microscope method," Proc. SPIE 6469, 646914 (2007).

W. J. Stewart, "A new technique for measuring the refractive index profiles of graded optical fibers," in Technical Digest 1006 of the 1997 International Conference on Integrated Optics and Optical Fiber Communication (IECE, Japan, Tokyo, 1997), pp. 395-398.

F. L Pedrotti, L. M. Pedrotti, and L. S. Pedrotti, Introduction to Optics (Academic, 1993).

A. E. Siegman, Lasers (Academic, 1986).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

(Color online) Schematic of a linearly polarized collimated Gaussian beam focusing in a coordinate system (x, y, z). The direction of polarization of the electric field is along the x axis.

Fig. 2
Fig. 2

(Color online) (a), (c), and (e) are x, y, z components of the incident electric field in the focal plane. The wavelength of the beam is 675   nm , the focal length f is 2   mm , the beam waist of fundamental Gaussian profile, denoted by w 0 , is 1.05   mm , and the NA of the objective lens is 0.7.

Fig. 3
Fig. 3

(Color online) (a), (c), and (e) are x, y, z components of the reflected electric field in the focal plane. The wavelength of the beam is 675   nm , the focal length f is 2   mm , the beam waist of fundamental Gaussian profile, denoted by w 0 , is 1.05   mm , and the NA of the objective lens is 0.7.

Fig. 4
Fig. 4

(Color online) Refraction of focused light on the air–fiber interface.

Fig. 5
Fig. 5

Reflectance change according to an increase in NA. The reflectance increases according to the NA of a focused Gaussian beam. The focal length f is 2   mm . The beam waist, denoted by w 0 , is 1.05   mm .

Fig. 6
Fig. 6

(Color online) (a) Reflectance according to the refractive index with a fixed NA of the lens. The star shaped spot curve is the change in reflectance with a refractive index obtained by considering the normal incidence. The other spot curves are the changes in reflectance with refractive indices while considering the angular spectrum representation. (b) Refractive index error; n is the refractive index in the angular spectrum representation, and n pa is the refractive index in the normal incidence method.

Fig. 7
Fig. 7

(Color online) Schematic of modified fiber-type confocal scanning optical microscope. The wavelength of the beam is 675   nm , and the focal length f is 2   mm . The beam waist of fundamental Gaussian profile, denoted w 0 , is 1.05   mm , and the NA of the objective lens is 0.7.

Fig. 8
Fig. 8

(Color online) Relative refractive index difference of a commercial dispersion compensated fiber, where [a] is the refractive index difference profile calculated by Fresnel's equation using the angular spectrum representation and [b] is the refractive index difference profile calculated by Fresnel's equation at normal incidence using the paraxial approximation.

Equations (16)

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

R ( x , y ) = ( n 1 n + 1 ) 2 ,
n ( x , y ) = ( 1 + R ( x , y ) 1 R ( x , y ) ) .
E ( θ ) = E 0 exp [ ( f sin θ / w 0 ) 2 ] ,
E inc ( x , y , z ) = C 0 θ m 0 2 π ( ξ , η , ζ ) exp [ j k ( s x x + s y y + s z z ) ] × sin θ d θ d ϕ ,
s x = sin θ cos ϕ , s y = sin θ sin ϕ , s z = cos θ .
ξ = exp [ ( f sin θ / w 0 ) 2 ] ( cos θ ) 1 / 2 [ cos θ + sin 2 ϕ ( 1 cos θ ) ] ,
η = exp [ ( f sin θ / w 0 ) 2 ] ( cos θ ) 1 / 2 ( cos θ 1 ) sin ϕ cos ϕ ,
ζ = exp [ ( f sin θ / w 0 ) 2 ] ( cos θ ) 1 / 2 sin θ cos ϕ .
e pr = ( ξ cos ϕ + η sin ϕ ) r p ,
e sr = ( ξ sin ϕ η cos ϕ ) r s ,
r p = n 2 cos θ ( n 2 sin 2 θ ) 1 / 2 n 2 cos θ + ( n 2 sin 2 θ ) 1 / 2 ,
r s = cos θ ( n 2 sin 2 θ ) 1 / 2 cos θ + ( n 2 sin 2 θ ) 1 / 2 .
ξ r = e pr cos ϕ + e sr sin ϕ , η r = e pr sin ϕ + e sr cos ϕ ,
ζ r = ζ r normal .
R = P reflected d x d y P incident d x d y = ( | E x r H y r | + | E y r H x r | ) d x d y ( | E x inc H y inc | + | E y inc H x inc | ) d x d y ,
I ( n = n clad ) = R Vector ( n = n clad ) a 1 = R Paraxial ( n = n clad ) a 2 ,

Metrics