Abstract

A new procedure is presented for testing long, radial-gradient-index rod lenses in a phase-stepping Twyman–Green interferometer. In addition, a method for converting the measured optical phase data to the rod’s index of refraction profile is reviewed. This procedure was used to measure refractive-index profile dispersions over the wavelength range of 441 to 1064 nm. These measurements provide the first data for refractive-index profiles of rod lenses in the infrared and show the existence of positive dispersion, negative dispersion, and near-infrared dispersion reversal in some samples. The dispersion results suggest the need for modifications to the current glass model.

© 1999 Optical Society of America

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References

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    [CrossRef] [PubMed]
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  17. J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
    [CrossRef]
  18. Nippon Sheet Glass America, Inc., 27 World’s Fair Drive, Somerset, N.J. 08873.
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    [CrossRef] [PubMed]
  20. Gradient Lens Corporation, 207 Tremont Street, Rochester, N.Y. 14608.
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  23. S. D. Fantone, “Refractive index and spectral models for gradient-index materials,” Appl. Opt. 22, 432–440 (1983).
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  24. C. E. Saxer, “Interferometric characterization of the chromatic dispersion of gradient-index glasses,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1998).

1993

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

1992

H. Ohta, Y. Kohgo, Y. Goto, “The near-infrared electronic endoscope for diagnosis of esophageal varices,” Gastrointest. Endosc. 38, 330–335 (1992).
[CrossRef] [PubMed]

1990

H. Kohso, Y. Tatsumi, H. Fujino, K. Tokita, T. Kodama, K. Kashima, K. Kawai, “An investigation of an infrared ray electronic endoscope with a laser diode light source,” Endoscopy 22, 217–220 (1990).
[CrossRef] [PubMed]

1985

1983

1982

1973

E. G. Rawson, R. G. Murray, “Interferometric measurement of SELFOC dielectric constant coefficients to sixth order,” IEEE J. Quantum Electron. QE-9, 1114–1118 (1973).
[CrossRef]

1970

1965

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

1942

1940

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1991), pp. 121–122.

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients, (Dover, N.Y., 1968).

Burow, R.

Chen, S.-S.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Cline, T.

Davis, D. O.

Elssner, K. E.

Fantone, S. D.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, England, 1992), pp. 350–354.

Fujino, H.

H. Kohso, Y. Tatsumi, H. Fujino, K. Tokita, T. Kodama, K. Kashima, K. Kawai, “An investigation of an infrared ray electronic endoscope with a laser diode light source,” Endoscopy 22, 217–220 (1990).
[CrossRef] [PubMed]

Fusamoto, H.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Ghatak, A. K.

Goto, Y.

H. Ohta, Y. Kohgo, Y. Goto, “The near-infrared electronic endoscope for diagnosis of esophageal varices,” Gastrointest. Endosc. 38, 330–335 (1992).
[CrossRef] [PubMed]

Grzanna, J.

Hayashi, N.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Huggins, M. L.

Iga, K.

Jander, R.

Kamada, T.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Kashima, K.

H. Kohso, Y. Tatsumi, H. Fujino, K. Tokita, T. Kodama, K. Kashima, K. Kawai, “An investigation of an infrared ray electronic endoscope with a laser diode light source,” Endoscopy 22, 217–220 (1990).
[CrossRef] [PubMed]

Kashiwagi, T.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Kawai, K.

H. Kohso, Y. Tatsumi, H. Fujino, K. Tokita, T. Kodama, K. Kashima, K. Kawai, “An investigation of an infrared ray electronic endoscope with a laser diode light source,” Endoscopy 22, 217–220 (1990).
[CrossRef] [PubMed]

Kawano, S.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Kodama, T.

H. Kohso, Y. Tatsumi, H. Fujino, K. Tokita, T. Kodama, K. Kashima, K. Kawai, “An investigation of an infrared ray electronic endoscope with a laser diode light source,” Endoscopy 22, 217–220 (1990).
[CrossRef] [PubMed]

Kohgo, Y.

H. Ohta, Y. Kohgo, Y. Goto, “The near-infrared electronic endoscope for diagnosis of esophageal varices,” Gastrointest. Endosc. 38, 330–335 (1992).
[CrossRef] [PubMed]

Kohso, H.

H. Kohso, Y. Tatsumi, H. Fujino, K. Tokita, T. Kodama, K. Kashima, K. Kawai, “An investigation of an infrared ray electronic endoscope with a laser diode light source,” Endoscopy 22, 217–220 (1990).
[CrossRef] [PubMed]

Kumar, D. V.

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, N.Y., 1978), pp. 67–74.
[CrossRef]

Mead, R.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Mercado, R. I.

Merkel, K.

Murray, R. G.

E. G. Rawson, R. G. Murray, “Interferometric measurement of SELFOC dielectric constant coefficients to sixth order,” IEEE J. Quantum Electron. QE-9, 1114–1118 (1973).
[CrossRef]

Nagano, K.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Nelder, J. A.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Ohta, H.

H. Ohta, Y. Kohgo, Y. Goto, “The near-infrared electronic endoscope for diagnosis of esophageal varices,” Gastrointest. Endosc. 38, 330–335 (1992).
[CrossRef] [PubMed]

Peng, H. B.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, England, 1992), pp. 350–354.

Rawson, E. G.

E. G. Rawson, R. G. Murray, “Interferometric measurement of SELFOC dielectric constant coefficients to sixth order,” IEEE J. Quantum Electron. QE-9, 1114–1118 (1973).
[CrossRef]

Robb, P. N.

Sands, P. J.

Sato, N.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Saxer, C. E.

C. E. Saxer, “Interferometric characterization of the chromatic dispersion of gradient-index glasses,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1998).

Scarborough, J. B.

J. B. Scarborough, Numerical Mathematical Analysis (Johns Hopkins U. Press, Baltimore, 1966), Chap. 13, Art. 116.

Schwider, J.

Sharma, A.

Spolaczyk, R.

Sun, K. H.

Takei, Y.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Tatsumi, Y.

H. Kohso, Y. Tatsumi, H. Fujino, K. Tokita, T. Kodama, K. Kashima, K. Kawai, “An investigation of an infrared ray electronic endoscope with a laser diode light source,” Endoscopy 22, 217–220 (1990).
[CrossRef] [PubMed]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, England, 1992), pp. 350–354.

Tokita, K.

H. Kohso, Y. Tatsumi, H. Fujino, K. Tokita, T. Kodama, K. Kashima, K. Kawai, “An investigation of an infrared ray electronic endoscope with a laser diode light source,” Endoscopy 22, 217–220 (1990).
[CrossRef] [PubMed]

Tsuji, S.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Tsujii, M.

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, England, 1992), pp. 350–354.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1991), pp. 121–122.

Wyant, J. C.

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18 (5) , 65–71 (May1982).

Appl. Opt.

Comput. J.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Endoscopy

H. Kohso, Y. Tatsumi, H. Fujino, K. Tokita, T. Kodama, K. Kashima, K. Kawai, “An investigation of an infrared ray electronic endoscope with a laser diode light source,” Endoscopy 22, 217–220 (1990).
[CrossRef] [PubMed]

S. Tsuji, S. Kawano, N. Sato, N. Hayashi, H. B. Peng, M. Tsujii, K. Nagano, Y. Takei, S.-S. Chen, T. Kashiwagi, H. Fusamoto, T. Kamada, “Comparison of infrared electronic endoscopy using reflection and transmission,” Endoscopy 25, 278–281 (1993).
[CrossRef] [PubMed]

Gastrointest. Endosc.

H. Ohta, Y. Kohgo, Y. Goto, “The near-infrared electronic endoscope for diagnosis of esophageal varices,” Gastrointest. Endosc. 38, 330–335 (1992).
[CrossRef] [PubMed]

IEEE J. Quantum Electron.

E. G. Rawson, R. G. Murray, “Interferometric measurement of SELFOC dielectric constant coefficients to sixth order,” IEEE J. Quantum Electron. QE-9, 1114–1118 (1973).
[CrossRef]

J. Opt. Soc. Am.

Laser Focus

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18 (5) , 65–71 (May1982).

Other

E. W. Marchand, Gradient Index Optics (Academic, N.Y., 1978), pp. 67–74.
[CrossRef]

H. A. Buchdahl, Optical Aberration Coefficients, (Dover, N.Y., 1968).

J. B. Scarborough, Numerical Mathematical Analysis (Johns Hopkins U. Press, Baltimore, 1966), Chap. 13, Art. 116.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1991), pp. 121–122.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, England, 1992), pp. 350–354.

Nippon Sheet Glass America, Inc., 27 World’s Fair Drive, Somerset, N.J. 08873.

Gradient Lens Corporation, 207 Tremont Street, Rochester, N.Y. 14608.

C. E. Saxer, “Interferometric characterization of the chromatic dispersion of gradient-index glasses,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1998).

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Figures (7)

Fig. 1
Fig. 1

Multiwavelength Twyman–Green interferometer used to measure long GRIN rods. The sample shown is slightly less than 1/4 pitch in length, so that light is focused on the test-arm mirror.

Fig. 2
Fig. 2

(a) Interferometer’s test arm, unfolded about the test-arm mirror. (b) Example wavefront profile. (c) Corresponding OPD map profile.

Fig. 3
Fig. 3

Dispersion measurements for SRL-200-200 GRIN rod. (a) N 0 index interpolation and extrapolation, (b) N 1 dispersion, (c) pitch-length dispersion, (d) N 2 dispersion, (e) N 3 dispersion, (f) comparison of N 2 and N 3 dispersions.

Fig. 4
Fig. 4

Surface and transmission OPD map profiles for the SRL-200-200 GRIN rod. All OPD map profiles have been vertically shifted so that the OPD value on axis is zero. Tilt and defocus have not been removed in this plot. The vertical dashed lines window the curve-fitting region.

Fig. 5
Fig. 5

Dispersion measurements for Gradient Lens Corporation GRIN rod. (a) N 0 index interpolation and extrapolation, (b) N 1 dispersion, (c) pitch-length dispersion, (d) N 2 dispersion, (e) N 3 dispersion, (f) comparison of N 2 and N 3 dispersions.

Fig. 6
Fig. 6

Surface and transmission OPD map profiles for the EG27 GRIN rod. All OPD map profiles have been vertically shifted, so that the OPD value on axis is zero. Tilt and defocus have not been removed in this plot. The vertical dashed lines window the curve-fitting region.

Fig. 7
Fig. 7

Effects of a 0.002 error in N 0 on the higher-order index coefficients for the SRL-200-200 GRIN rod. (a) Original N 0 and N 0 with 0.002 error, (b) N 1 shifts to maintain pitch length, (c) N 2 is relatively unaffected, (d) N 3 is also relatively unaffected.

Tables (2)

Tables Icon

Table 1 Laser Sources and Wavelengths Used for Chromatic Dispersion Measurements in the Interferometer

Tables Icon

Table 2 Base Refractive-Index Dispersion for Sample EG27

Equations (31)

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

OPDr, λ=λ2π ϕr, λ.
Nr, λ=i=0 Niλr2ii=0P Niλr2i.
Wr=i=0M W2irr2i.
OPDr=i=0M a2irr2i.
Wr=-OPDr
ynu=TGynu,
TG=cosαˆsinαˆαN0-N0α sinαˆcosαˆ,
αˆ=αt,
α=-2N1/N01/2.
p=2π/α.
TH=12dNa01,
Tsys=cos2αˆ-dN0α sin2αˆNasin2αˆαN0+2d cos2 αˆNa2dN02α2 sin2 αˆNa-N0α sin2αˆcos2αˆ-dN0α sin2αˆNa.
C=-2a2r.
a2r=2dN02α2 sin2 αˆ-N0Naα sin2αˆ2Na cos2αˆ-dN0α sin2αˆ.
BFD=Na/N0α tan αˆ.
σ1=-1nKuaKj=1K a1j+j=1K a1j*-1nKuaK a¯1,
a1j=1/2N0k0-1yaua2ua+uaj,
k0j=N0j/N0j,
a1j*=12N0yaua3-N0yaua3j+ 4N2ya4+2N1ya2ua2-12 N0ua4jdz,
εy|SA=σ1ρ3 cos θ=1nKuaKWρy,
a¯1=4a4ryex4,
N2=4a4ryex4yent4-t1-t2N0-t3N1t4,
yexyent=cos2 αˆ-sin2 αˆ-2 N0Na αd sin αˆ cos αˆ.
N0=1.5845+7.64×10-3/λ2,
ωλ=λ-λ01+αλ-λ0,
N0ω=N0|λ0+ν1ω+ν2ω2+ν3ω3.
t1N0, N1, t, d=α4cβ+c3dN04s4Na3+α3N03s32Na2β-c-β3c2-3β2c3-3βc4-c5+β3s2+3β2cs2+3βc2s2+c3s2,
t2N0, N1, t, d=164-12α3β3s-20α3βs3-12α4t-12α4β4t-24α4β2s2t-12α4s4t+16α3βsβ2+s2cos2αˆ+4α3βs-β2+s2cos4αˆ+8α31+β4-s4sin2αˆ+α3-1-β4+6β2s2-s4sin4αˆ,
t3N0, N1, t, d=1164αβ3s-4αβs3+4α2t+4α2β4t+8α2β2s2t+4α2s4t+4αβs-β2+s2cos4αˆ+α-1-β4+6β2s2-s4sin4αˆ,
t4N0, N1, t, d=18α-20β3s-12βs3+12αt+12αβ4t+24αβ2s2t+12αs4t+16βsβ2+s2cos2αˆ+4βsβ2-s2cos4αˆ+81+β4-s4sin2αˆ+1+β4-6β2s2+s4sin4αˆ,
ccos αˆ,  ssin αˆ,  βc-2N0/Naαds,

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