Abstract

We present a new (to our knowledge) technique for introducing phase shifts between the laterally sheared emergent beam components of a cyclic path optical configuration (CPOC). The phase shifts are introduced by applying a small change in the angle of incidence of the incident beam due to the small angular rotation of the CPOC setup. Phase-shifting interferometry has been applied along with this phase-shifting technique for a CPOC with lateral shear to find the surface slope/profile of curved optical surfaces. Results for a spherical optical surface have been discussed. An optical setup for measurement of the surface profile of toroidal beam line mirrors of synchrotron radiation sources is proposed.

© 2011 Optical Society of America

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References

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  1. M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 140–142.
  2. S. Chatterjee and Y. P. Kumar, “Long trace profiler with cyclic optical configuration,” Appl. Opt. 41, 5857–5859 (2002)
    [CrossRef] [PubMed]
  3. S. Chatterjee and Y. P. Kumar, “Measurement of residual wedge angle of a high optical quality transparent parallel plate with a multi-pass optical configuration,” Opt. Eng. 49, 053605 (2010).
    [CrossRef]
  4. S. Chatterjee and Y. P. Kumar, “Simple technique for the generation of white light Young’s fringes with a cyclic optical configuration,” Appl. Opt. 47, 2956–2960 (2008).
    [CrossRef] [PubMed]
  5. S. Chatterjee and Y. P. Kumar, “Simple technique for the generation of white-light Haidinger fringes with cyclic optical configuration,” Opt. Lett. 34, 1291–1293 (2009)
    [CrossRef] [PubMed]
  6. S. Chatterjee and Y. P. Kumar, “External measurement of dihedral right angles with cyclic optical configuration,” Appl. Opt. 48, 1598–1605 (2009).
    [CrossRef] [PubMed]
  7. S. Chatterjee and Y. P. Kumar, “Measurement of centering error of a lens with cyclic optical configuration,” Opt. Eng. 49, 043601 (2010).
    [CrossRef]
  8. Y. P. Kumar and S. Chatterjee, “Simultaneous measurement of refractive index and wedge angle of optical windows using Fizeau interferometry and a cyclic path optical configuration,” Appl. Opt. 48, 4756–4761 (2009).
    [CrossRef] [PubMed]
  9. Y. P. Kumar and S. Chatterjee, “Application of Newton’s method to determine the focal length of lenses using a lateral shearing interferometer and cyclic path optical configuration set up,” Opt. Eng. 49, 053604 (2010).
    [CrossRef]
  10. C. C. Cheng and M. G. Raymer, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
    [CrossRef]
  11. J. S. Darlin, M. P. Kothiyal, and R. S. Sirohi, “Self-referencing cyclic shearing interferometer for collimation testing,” J. Mod. Opt. 44, 929–939 (1997).
    [CrossRef]
  12. P. Harharan and M. Roy, “A geometric-phase interferometer,” J. Mod. Opt. 391811–1815 (1992).
    [CrossRef]
  13. S. Chatterjee and Y. P. Kumar, Opt. Eng. 48, 115601(2009).
    [CrossRef]
  14. J A. Ferrari and E. Garbasi, “Phase shifting (Sagnac) interferometer with external phase control,” Appl. Opt. 44, 4510–4512 (2005).
    [CrossRef] [PubMed]
  15. Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase shifting, lateral shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–41 (2005).
    [CrossRef]
  16. N. I. T. Arellano, A. M. Garcia, G. R. Zurita, J. A. R. Alvarez, and A. M. Perez, “Slope measurement of a phase object using a polarizing phase shifting high frequency Ronchi grating interferometer,” Appl. Opt. 49, 6402–6408 (2010).
    [CrossRef]
  17. M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–444 (1985).
    [CrossRef] [PubMed]
  18. D. Malacara, “Radial, rotational and reversal shear interferometers,” Optical Shop Testing, 3rd ed., D.Malacara, ed. 3rd ed. (Wiley, 2007), pp. 189–197.
  19. K. Yoshihara, “On the triangular path interferometer,” Jpn. J. Appl. Phys. 7, 529–535 (1968).
    [CrossRef]
  20. M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, D.Malacara, ed. (Wiley2007), pp. 122–184.
    [CrossRef]
  21. P. Hariharan, “Phase shifting interferometery: minimization of systematic errors,” Opt. Eng. 39, 967–969 (2000).
    [CrossRef]
  22. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase shifting interferometery: a simple error compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  23. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, Vol.  xxviii. E.Wolf, ed. (North-Holland, 1988), pp. 349–393.
    [CrossRef]
  24. D. Malacara, S. Malacara, and Z. Malacara, “Some methods to introduce phase shifting,” in Interferogram Analysis for Optical Testing (Marcel Dekker, 1998), pp. 248–255.
  25. S. Chatterjee and Y. P. Kumar, “Measurement of surface profile of a long-radius optical surface with wedge phase shifting lateral shear interferometer,” Opt. Eng. 49, 103602(2010).
    [CrossRef]

2010 (5)

S. Chatterjee and Y. P. Kumar, “Measurement of residual wedge angle of a high optical quality transparent parallel plate with a multi-pass optical configuration,” Opt. Eng. 49, 053605 (2010).
[CrossRef]

Y. P. Kumar and S. Chatterjee, “Application of Newton’s method to determine the focal length of lenses using a lateral shearing interferometer and cyclic path optical configuration set up,” Opt. Eng. 49, 053604 (2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “Measurement of centering error of a lens with cyclic optical configuration,” Opt. Eng. 49, 043601 (2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “Measurement of surface profile of a long-radius optical surface with wedge phase shifting lateral shear interferometer,” Opt. Eng. 49, 103602(2010).
[CrossRef]

N. I. T. Arellano, A. M. Garcia, G. R. Zurita, J. A. R. Alvarez, and A. M. Perez, “Slope measurement of a phase object using a polarizing phase shifting high frequency Ronchi grating interferometer,” Appl. Opt. 49, 6402–6408 (2010).
[CrossRef]

2009 (4)

2008 (1)

2005 (2)

J A. Ferrari and E. Garbasi, “Phase shifting (Sagnac) interferometer with external phase control,” Appl. Opt. 44, 4510–4512 (2005).
[CrossRef] [PubMed]

Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase shifting, lateral shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–41 (2005).
[CrossRef]

2002 (1)

2000 (2)

P. Hariharan, “Phase shifting interferometery: minimization of systematic errors,” Opt. Eng. 39, 967–969 (2000).
[CrossRef]

C. C. Cheng and M. G. Raymer, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[CrossRef]

1997 (1)

J. S. Darlin, M. P. Kothiyal, and R. S. Sirohi, “Self-referencing cyclic shearing interferometer for collimation testing,” J. Mod. Opt. 44, 929–939 (1997).
[CrossRef]

1992 (1)

P. Harharan and M. Roy, “A geometric-phase interferometer,” J. Mod. Opt. 391811–1815 (1992).
[CrossRef]

1987 (1)

1985 (1)

1968 (1)

K. Yoshihara, “On the triangular path interferometer,” Jpn. J. Appl. Phys. 7, 529–535 (1968).
[CrossRef]

Alvarez, J. A. R.

Arellano, N. I. T.

Barnes, T. H.

Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase shifting, lateral shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–41 (2005).
[CrossRef]

Chatterjee, S.

S. Chatterjee and Y. P. Kumar, “Measurement of surface profile of a long-radius optical surface with wedge phase shifting lateral shear interferometer,” Opt. Eng. 49, 103602(2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “Measurement of residual wedge angle of a high optical quality transparent parallel plate with a multi-pass optical configuration,” Opt. Eng. 49, 053605 (2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “Measurement of centering error of a lens with cyclic optical configuration,” Opt. Eng. 49, 043601 (2010).
[CrossRef]

Y. P. Kumar and S. Chatterjee, “Application of Newton’s method to determine the focal length of lenses using a lateral shearing interferometer and cyclic path optical configuration set up,” Opt. Eng. 49, 053604 (2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, Opt. Eng. 48, 115601(2009).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “External measurement of dihedral right angles with cyclic optical configuration,” Appl. Opt. 48, 1598–1605 (2009).
[CrossRef] [PubMed]

Y. P. Kumar and S. Chatterjee, “Simultaneous measurement of refractive index and wedge angle of optical windows using Fizeau interferometry and a cyclic path optical configuration,” Appl. Opt. 48, 4756–4761 (2009).
[CrossRef] [PubMed]

S. Chatterjee and Y. P. Kumar, “Simple technique for the generation of white-light Haidinger fringes with cyclic optical configuration,” Opt. Lett. 34, 1291–1293 (2009)
[CrossRef] [PubMed]

S. Chatterjee and Y. P. Kumar, “Simple technique for the generation of white light Young’s fringes with a cyclic optical configuration,” Appl. Opt. 47, 2956–2960 (2008).
[CrossRef] [PubMed]

S. Chatterjee and Y. P. Kumar, “Long trace profiler with cyclic optical configuration,” Appl. Opt. 41, 5857–5859 (2002)
[CrossRef] [PubMed]

Cheng, C. C.

C. C. Cheng and M. G. Raymer, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[CrossRef]

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, Vol.  xxviii. E.Wolf, ed. (North-Holland, 1988), pp. 349–393.
[CrossRef]

Darlin, J. S.

J. S. Darlin, M. P. Kothiyal, and R. S. Sirohi, “Self-referencing cyclic shearing interferometer for collimation testing,” J. Mod. Opt. 44, 929–939 (1997).
[CrossRef]

Delisle, C.

Eiju, T.

Ferrari, J A.

Garbasi, E.

Garcia, A. M.

Harharan, P.

P. Harharan and M. Roy, “A geometric-phase interferometer,” J. Mod. Opt. 391811–1815 (1992).
[CrossRef]

Hariharan, P.

Kothiyal, M. P.

J. S. Darlin, M. P. Kothiyal, and R. S. Sirohi, “Self-referencing cyclic shearing interferometer for collimation testing,” J. Mod. Opt. 44, 929–939 (1997).
[CrossRef]

M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–444 (1985).
[CrossRef] [PubMed]

Kumar, Y. P.

Y. P. Kumar and S. Chatterjee, “Application of Newton’s method to determine the focal length of lenses using a lateral shearing interferometer and cyclic path optical configuration set up,” Opt. Eng. 49, 053604 (2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “Measurement of surface profile of a long-radius optical surface with wedge phase shifting lateral shear interferometer,” Opt. Eng. 49, 103602(2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “Measurement of centering error of a lens with cyclic optical configuration,” Opt. Eng. 49, 043601 (2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “Measurement of residual wedge angle of a high optical quality transparent parallel plate with a multi-pass optical configuration,” Opt. Eng. 49, 053605 (2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “Simple technique for the generation of white-light Haidinger fringes with cyclic optical configuration,” Opt. Lett. 34, 1291–1293 (2009)
[CrossRef] [PubMed]

S. Chatterjee and Y. P. Kumar, Opt. Eng. 48, 115601(2009).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “External measurement of dihedral right angles with cyclic optical configuration,” Appl. Opt. 48, 1598–1605 (2009).
[CrossRef] [PubMed]

Y. P. Kumar and S. Chatterjee, “Simultaneous measurement of refractive index and wedge angle of optical windows using Fizeau interferometry and a cyclic path optical configuration,” Appl. Opt. 48, 4756–4761 (2009).
[CrossRef] [PubMed]

S. Chatterjee and Y. P. Kumar, “Simple technique for the generation of white light Young’s fringes with a cyclic optical configuration,” Appl. Opt. 47, 2956–2960 (2008).
[CrossRef] [PubMed]

S. Chatterjee and Y. P. Kumar, “Long trace profiler with cyclic optical configuration,” Appl. Opt. 41, 5857–5859 (2002)
[CrossRef] [PubMed]

Lo, Y. H.

Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase shifting, lateral shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–41 (2005).
[CrossRef]

Malacara, D.

D. Malacara, “Radial, rotational and reversal shear interferometers,” Optical Shop Testing, 3rd ed., D.Malacara, ed. 3rd ed. (Wiley, 2007), pp. 189–197.

D. Malacara, S. Malacara, and Z. Malacara, “Some methods to introduce phase shifting,” in Interferogram Analysis for Optical Testing (Marcel Dekker, 1998), pp. 248–255.

Malacara, S.

D. Malacara, S. Malacara, and Z. Malacara, “Some methods to introduce phase shifting,” in Interferogram Analysis for Optical Testing (Marcel Dekker, 1998), pp. 248–255.

Malacara, Z.

D. Malacara, S. Malacara, and Z. Malacara, “Some methods to introduce phase shifting,” in Interferogram Analysis for Optical Testing (Marcel Dekker, 1998), pp. 248–255.

Mantravadi, M.

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 140–142.

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, D.Malacara, ed. (Wiley2007), pp. 122–184.
[CrossRef]

Oreb, B. F.

Paez, G.

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 140–142.

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, D.Malacara, ed. (Wiley2007), pp. 122–184.
[CrossRef]

Perez, A. M.

Raymer, M. G.

C. C. Cheng and M. G. Raymer, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[CrossRef]

Roy, M.

P. Harharan and M. Roy, “A geometric-phase interferometer,” J. Mod. Opt. 391811–1815 (1992).
[CrossRef]

Sirohi, R. S.

J. S. Darlin, M. P. Kothiyal, and R. S. Sirohi, “Self-referencing cyclic shearing interferometer for collimation testing,” J. Mod. Opt. 44, 929–939 (1997).
[CrossRef]

Somervell, A. R. D.

Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase shifting, lateral shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–41 (2005).
[CrossRef]

Strojnik, M.

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 140–142.

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, D.Malacara, ed. (Wiley2007), pp. 122–184.
[CrossRef]

Yoshihara, K.

K. Yoshihara, “On the triangular path interferometer,” Jpn. J. Appl. Phys. 7, 529–535 (1968).
[CrossRef]

Zurita, G. R.

Appl. Opt. (8)

S. Chatterjee and Y. P. Kumar, “Long trace profiler with cyclic optical configuration,” Appl. Opt. 41, 5857–5859 (2002)
[CrossRef] [PubMed]

S. Chatterjee and Y. P. Kumar, “Simple technique for the generation of white light Young’s fringes with a cyclic optical configuration,” Appl. Opt. 47, 2956–2960 (2008).
[CrossRef] [PubMed]

S. Chatterjee and Y. P. Kumar, “External measurement of dihedral right angles with cyclic optical configuration,” Appl. Opt. 48, 1598–1605 (2009).
[CrossRef] [PubMed]

Y. P. Kumar and S. Chatterjee, “Simultaneous measurement of refractive index and wedge angle of optical windows using Fizeau interferometry and a cyclic path optical configuration,” Appl. Opt. 48, 4756–4761 (2009).
[CrossRef] [PubMed]

J A. Ferrari and E. Garbasi, “Phase shifting (Sagnac) interferometer with external phase control,” Appl. Opt. 44, 4510–4512 (2005).
[CrossRef] [PubMed]

N. I. T. Arellano, A. M. Garcia, G. R. Zurita, J. A. R. Alvarez, and A. M. Perez, “Slope measurement of a phase object using a polarizing phase shifting high frequency Ronchi grating interferometer,” Appl. Opt. 49, 6402–6408 (2010).
[CrossRef]

M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–444 (1985).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase shifting interferometery: a simple error compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

J. Mod. Opt. (3)

C. C. Cheng and M. G. Raymer, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[CrossRef]

J. S. Darlin, M. P. Kothiyal, and R. S. Sirohi, “Self-referencing cyclic shearing interferometer for collimation testing,” J. Mod. Opt. 44, 929–939 (1997).
[CrossRef]

P. Harharan and M. Roy, “A geometric-phase interferometer,” J. Mod. Opt. 391811–1815 (1992).
[CrossRef]

Jpn. J. Appl. Phys. (1)

K. Yoshihara, “On the triangular path interferometer,” Jpn. J. Appl. Phys. 7, 529–535 (1968).
[CrossRef]

Opt. Eng. (6)

P. Hariharan, “Phase shifting interferometery: minimization of systematic errors,” Opt. Eng. 39, 967–969 (2000).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “Measurement of surface profile of a long-radius optical surface with wedge phase shifting lateral shear interferometer,” Opt. Eng. 49, 103602(2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, Opt. Eng. 48, 115601(2009).
[CrossRef]

Y. P. Kumar and S. Chatterjee, “Application of Newton’s method to determine the focal length of lenses using a lateral shearing interferometer and cyclic path optical configuration set up,” Opt. Eng. 49, 053604 (2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “Measurement of centering error of a lens with cyclic optical configuration,” Opt. Eng. 49, 043601 (2010).
[CrossRef]

S. Chatterjee and Y. P. Kumar, “Measurement of residual wedge angle of a high optical quality transparent parallel plate with a multi-pass optical configuration,” Opt. Eng. 49, 053605 (2010).
[CrossRef]

Opt. Laser Eng. (1)

Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase shifting, lateral shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–41 (2005).
[CrossRef]

Opt. Lett. (1)

Other (5)

D. Malacara, “Radial, rotational and reversal shear interferometers,” Optical Shop Testing, 3rd ed., D.Malacara, ed. 3rd ed. (Wiley, 2007), pp. 189–197.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, Vol.  xxviii. E.Wolf, ed. (North-Holland, 1988), pp. 349–393.
[CrossRef]

D. Malacara, S. Malacara, and Z. Malacara, “Some methods to introduce phase shifting,” in Interferogram Analysis for Optical Testing (Marcel Dekker, 1998), pp. 248–255.

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 140–142.

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, D.Malacara, ed. (Wiley2007), pp. 122–184.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Optical schematic of the CPOC.

Fig. 2
Fig. 2

Geometric construction of the ray paths of the emergent rays, from the output port of the CPOC, for a tilt in the direction of the incident ray.

Fig. 3
Fig. 3

Ray paths for calculation of the OPD between tilted emergent rays.

Fig. 4
Fig. 4

Optical schematic of the CPOC-based PLSI setup.

Fig. 5
Fig. 5

Lateral shearing interferogram obtained by using the setup shown in Fig. 4.

Fig. 6
Fig. 6

Unwrapped OPD values ( Δ W ) obtained along a central tangential section and the corresponding fitted line.

Fig. 7
Fig. 7

Variation of slope (a) along the section of the TM.

Fig. 8
Fig. 8

Variation of slope error ( Δ a ) along the section of the TM.

Fig. 9
Fig. 9

Height profile along the section of the TM.

Fig. 10
Fig. 10

Error in height profile along the section of the TM.

Fig. 11
Fig. 11

Optical schematic of the setup for the measurement of slope errors of the BM.

Equations (10)

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

Δ = S sin δ i ,
Δ W = W [ ( X S ) , Z ] W ( X , Z ) ,
Δ = S ( δ i ) + ( δ i ) S .
I ( x , y ) = I 0 ( x , y ) { 1 + V ( x , y ) cos [ ϕ ( x , y ) + α j ] } ,
ϕ = arctan [ 2 ( I 4 I 2 ) / ( I 1 + I 5 2 I 3 ) ] .
Δ ϕ = ( ε 2 / 4 ) sin 2 ϕ ,
δ = ( λ / 2 π ) ϕ ,
d = { [ ( R / 2 ) D ] λ / S } ,
[ ( R / 2 ) D ] / S = ( d / λ ) = [ 1 / ( λ / d ) ] = [ 1 / θ ] ,
R = 2 [ ( S / θ ) + D ] .

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