Abstract

A gradient-index axicon with its initial focus offset from the back surface was designed with the thin-lens approximation. Two samples were fabricated by means of the time-varying boundary condition diffusion method, which is based on the modified quasi-chemical diffusion model. Intensity profile measurements were taken along the focal region of the axicons. The samples produced extended line foci. From the intensity measurements, the central spot widths and back focal lengths were determined. The peak widths matched theoretical predictions made with the diffraction theory for the samples and showed good agreement with the predicted widths for a pseudo-Bessel beam, showing that the axicon produced a pseudo-diffractionless beam.

© 2000 Optical Society of America

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References

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1998 (2)

1996 (1)

1995 (1)

J. M. Inman, J. L. Bentley, S. N. Houde-Walter, “Modeling ion-exchanged glass photonics: the modified quasi-chemical diffusion coefficient,” J. Non-Cryst. Solids 191, 1–2 (1995).
[CrossRef]

1994 (1)

1992 (1)

1991 (2)

R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991).
[CrossRef]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1990 (1)

1988 (2)

1987 (2)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

D. DeBeer, S. R. Hartmann, R. Friedberg, J. Durnin, J. J. Miceli, J. H. Eberly, “Comment on ‘diffraction-free beams’ (with reply),” Phys. Rev. Lett. 59, 2611–2612 (1987).
[CrossRef]

1983 (1)

1982 (1)

W. N. Charman, “Theoretical aspects of concentric varifocal lenses,” Ophthalmic Physiol. Opt. 2, 75–86 (1982).
[CrossRef] [PubMed]

1962 (1)

1960 (1)

Bara, S.

Bentley, J. L.

T. H. Tomkinson, J. L. Bentley, M. K. Crawford, C. J. Harkrider, D. T. Moore, J. L. Rouke, “Rigid endoscopic relay systems: a comparative study,” Appl. Opt. 35, 6674–6683 (1996).
[CrossRef] [PubMed]

J. M. Inman, J. L. Bentley, S. N. Houde-Walter, “Modeling ion-exchanged glass photonics: the modified quasi-chemical diffusion coefficient,” J. Non-Cryst. Solids 191, 1–2 (1995).
[CrossRef]

J. L. Bentley, “Integration of the design and manufacture of gradient-index optical systems,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1995).

Bowen, J. P.

Caldwell, J. B.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Charman, W. N.

W. N. Charman, “Theoretical aspects of concentric varifocal lenses,” Ophthalmic Physiol. Opt. 2, 75–86 (1982).
[CrossRef] [PubMed]

Crawford, M. K.

DeBeer, D.

D. DeBeer, S. R. Hartmann, R. Friedberg, J. Durnin, J. J. Miceli, J. H. Eberly, “Comment on ‘diffraction-free beams’ (with reply),” Phys. Rev. Lett. 59, 2611–2612 (1987).
[CrossRef]

Deguchi, M.

M. Deguchi, D. T. Moore, D. S. Kindred, “Zoom lens design using gradient-index lenses,” in Image Acquisition and Scientific Imaging Systems, H. C. Titus, A. Waks, eds., Proc. SPIE2173, 161–168 (1994).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

D. DeBeer, S. R. Hartmann, R. Friedberg, J. Durnin, J. J. Miceli, J. H. Eberly, “Comment on ‘diffraction-free beams’ (with reply),” Phys. Rev. Lett. 59, 2611–2612 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

D. DeBeer, S. R. Hartmann, R. Friedberg, J. Durnin, J. J. Miceli, J. H. Eberly, “Comment on ‘diffraction-free beams’ (with reply),” Phys. Rev. Lett. 59, 2611–2612 (1987).
[CrossRef]

Fantone, S. D.

Fischer, D. J.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Friedberg, R.

D. DeBeer, S. R. Hartmann, R. Friedberg, J. Durnin, J. J. Miceli, J. H. Eberly, “Comment on ‘diffraction-free beams’ (with reply),” Phys. Rev. Lett. 59, 2611–2612 (1987).
[CrossRef]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fujiwara, S.

Gardner, L. R.

Gomez-Reino, C.

Gonzalez, R. M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Harkrider, C. J.

Hartmann, S. R.

D. DeBeer, S. R. Hartmann, R. Friedberg, J. Durnin, J. J. Miceli, J. H. Eberly, “Comment on ‘diffraction-free beams’ (with reply),” Phys. Rev. Lett. 59, 2611–2612 (1987).
[CrossRef]

Haun, N.

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Herman, R. M.

Houde-Walter, S. N.

J. M. Inman, J. L. Bentley, S. N. Houde-Walter, “Modeling ion-exchanged glass photonics: the modified quasi-chemical diffusion coefficient,” J. Non-Cryst. Solids 191, 1–2 (1995).
[CrossRef]

Houk, M. T.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Inman, J. M.

J. M. Inman, J. L. Bentley, S. N. Houde-Walter, “Modeling ion-exchanged glass photonics: the modified quasi-chemical diffusion coefficient,” J. Non-Cryst. Solids 191, 1–2 (1995).
[CrossRef]

Jaroszewicz, Z.

Kindred, D. S.

Kolodziejczyk, A.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Linares, J.

Marchand, E. W.

McLeod, J. H.

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

D. DeBeer, S. R. Hartmann, R. Friedberg, J. Durnin, J. J. Miceli, J. H. Eberly, “Comment on ‘diffraction-free beams’ (with reply),” Phys. Rev. Lett. 59, 2611–2612 (1987).
[CrossRef]

Moore, D. T.

Morales, T.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Rouke, J. L.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Shiba, M.

Sochacki, J.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Tomkinson, T. H.

Wang, D. Y. H.

Wiggins, T. A.

Appl. Opt. (8)

J. Non-Cryst. Solids (1)

J. M. Inman, J. L. Bentley, S. N. Houde-Walter, “Modeling ion-exchanged glass photonics: the modified quasi-chemical diffusion coefficient,” J. Non-Cryst. Solids 191, 1–2 (1995).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Ophthalmic Physiol. Opt. (1)

W. N. Charman, “Theoretical aspects of concentric varifocal lenses,” Ophthalmic Physiol. Opt. 2, 75–86 (1982).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

D. DeBeer, S. R. Hartmann, R. Friedberg, J. Durnin, J. J. Miceli, J. H. Eberly, “Comment on ‘diffraction-free beams’ (with reply),” Phys. Rev. Lett. 59, 2611–2612 (1987).
[CrossRef]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Other (4)

M. Deguchi, D. T. Moore, D. S. Kindred, “Zoom lens design using gradient-index lenses,” in Image Acquisition and Scientific Imaging Systems, H. C. Titus, A. Waks, eds., Proc. SPIE2173, 161–168 (1994).
[CrossRef]

J. L. Bentley, “Integration of the design and manufacture of gradient-index optical systems,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1995).

C. J. Harkrider, D. T. Moore, “Time varying boundary condition diffusion for gradient-index design,” in International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 780–788 (1998).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

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Figures (10)

Fig. 1
Fig. 1

Parameters used for deriving the index profile of a GRIN axicon.

Fig. 2
Fig. 2

Simulated performance of axicon does not match desired, because design goals violate thin-lens assumption in the index-equation derivation. The difference in back focal lengths is not constant.

Fig. 3
Fig. 3

When sample thickness is accounted for, the necessary index of refraction has a slightly smaller index change than predicted by means of the thin-lens index equation.

Fig. 4
Fig. 4

Parameters used to determine ray-intercept height at the start of the back focus for use in the optical design program’s optimization process.

Fig. 5
Fig. 5

Measured index profile (from the fabricated sample) compared with the desired index profile. The wings at the edges of the measured data are from chips in the sample edges.

Fig. 6
Fig. 6

Theoretical performance of an ideal axicon compared with the polynomial approximation of an index. Even a higher-(sixteenth) order polynomial fails to predict the axicon’s focal behavior accurately.

Fig. 7
Fig. 7

Analytical index formula was fit to the measured index profile over 1.65 mm of the 2.0-mm semiaperture.

Fig. 8
Fig. 8

Experimental setup for measuring the axicon focus and example images from the axicon 2 beam. (a) Initial focus. (b) Appearance of first two full rings. (c) and (d) Images taken further along focus to show full pseudo-Bessel beam.

Fig. 9
Fig. 9

Measured peak widths for axicon 1 compared with prediction and pseudo-Bessel beam theory. Axicon 1 positions are shifted +0.5 mm for best agreement.

Fig. 10
Fig. 10

Measured peak widths for axicon 2 compared with prediction and pseudo-Bessel beam theory. Axicon 2 positions are shifted +0.75 mm for best agreement.

Tables (4)

Tables Icon

Table 1 MQC Coefficients for Glass CHGL-25 at 560 °Ca

Tables Icon

Table 2 Index Profile Fit Parameters (mm)

Tables Icon

Table 3 Fabricated Axicon Dimensions

Tables Icon

Table 4 Comparison of Measured Focal Lengths to Geometric Predictions with the Index-Fit Results and the Measured Axicon Lengths

Equations (16)

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

fr=f1+r-r1r2-r1 Δf,
tanθ=-dzdr-rfr,  where zr  fr,
 xa+bxdx=xb-ab2 lna+bx,
zr=r2-r1Δf r-f1-r1r2-r1 Δfr2-r1Δf2×lnf1+Δfr2-r1r-r1r1r.
n0t=nrt+zr  nr=n0-zrt,
nr=no-r2-r1tΔf2r-r1Δf-r2-r1f1-r1Δfln1+r-r1r2-r1Δff1,
nr=no-r2tΔf2rΔf-r2f1 ln1+rr2Δff1.
limΔf0nr=n0-r22tf1.
limf10nr, r1=0=n0-r2tΔf r.
nr=no-12tf1 r2+Δf3r2tf12 r3-Δf24r22tf13 r4+.
Dχ=c2χβχ=1+1-χβχ=0β-1+1×DB1-χα,  β=1-4χ-χ01-χ-χ0[1-expρ1/2,
ri=ri-rif1+Δf ri-r1r2-r1 f1.
0.1Ti2O+0.03ZrO2+0.02Al2O3+0.05Na2O+0.2Li2O+0.6SiO2.
nχ=1.6234-0.03025χ-0.01098χ2.
Jokr sinθz2,
wc=2 2.408k sinθz.

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