Abstract

Retinal image light distributions in a standard optical model of a diffraction-limited eye with round pupils are presented for several patterns of amplitude and phase modulation of the light admitted into the eye. Of special interest are circularly symmetrical configurations of truncated Bessel amplitude transmission functions, and of light subjected to axicon deviation. It is shown by several examples that this kind of beam shaping allows generation of retinal imagery, which can be more robust to defocus while maintaining minimal image degradation, and it points to situations of two separate zones simultaneously in sharp focus, several diopters apart.

© 2019 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  3. G. C. Steward, The Symmetrical Optical System (Cambridge University, 1928).
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    [Crossref]
  5. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
    [Crossref]
  6. T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17, 15558–15570 (2009).
    [Crossref]
  7. A. J. Lambert, E. M. Daly, and J. C. Dainty, “Improved fixation quality provided by a Bessel beacon in an adaptive optics system,” Ophthalmic Physiolog. Opt. 33, 403–411 (2013).
    [Crossref]
  8. D. Bhattarai, M. Suheimat, A. J. Lambert, and D. A. Atchison, “Fixation stability with Bessel beams,” Optom. Vis. Sci 96, 95–102 (2019).
    [Crossref]
  9. P. Artal, “Calculations of two-dimensional foveal retinal images in real eyes,” J. Opt. Soc. Am. A 7, 1374–1381 (1990).
    [Crossref]
  10. G. Bickel, G. Häusler, and M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–978 (1985).
    [Crossref]
  11. B. Dossier, “Recherches sur l’apodisation des images optiques,” Revue d’Optique 33, 57–267 (1954).
  12. K. Strehl, “Aplanatische und fehlerhaft Abbildung im Fernrohr,” Zeitschrift für Instrumentenkunde 15, 362–370 (1895).
  13. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass, 1963).
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  15. N. Charman, Handbook of Optics, M. Bass, ed. (McGraw-Hill, 2010), Vol. 3, pp. 1.1–1.65.

2019 (1)

D. Bhattarai, M. Suheimat, A. J. Lambert, and D. A. Atchison, “Fixation stability with Bessel beams,” Optom. Vis. Sci 96, 95–102 (2019).
[Crossref]

2013 (1)

A. J. Lambert, E. M. Daly, and J. C. Dainty, “Improved fixation quality provided by a Bessel beacon in an adaptive optics system,” Ophthalmic Physiolog. Opt. 33, 403–411 (2013).
[Crossref]

2009 (1)

2005 (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

1993 (1)

1990 (1)

1985 (1)

G. Bickel, G. Häusler, and M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–978 (1985).
[Crossref]

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[Crossref]

1954 (2)

J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
[Crossref]

B. Dossier, “Recherches sur l’apodisation des images optiques,” Revue d’Optique 33, 57–267 (1954).

1895 (1)

K. Strehl, “Aplanatische und fehlerhaft Abbildung im Fernrohr,” Zeitschrift für Instrumentenkunde 15, 362–370 (1895).

Artal, P.

Atchison, D. A.

D. Bhattarai, M. Suheimat, A. J. Lambert, and D. A. Atchison, “Fixation stability with Bessel beams,” Optom. Vis. Sci 96, 95–102 (2019).
[Crossref]

Bhattarai, D.

D. Bhattarai, M. Suheimat, A. J. Lambert, and D. A. Atchison, “Fixation stability with Bessel beams,” Optom. Vis. Sci 96, 95–102 (2019).
[Crossref]

Bickel, G.

G. Bickel, G. Häusler, and M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–978 (1985).
[Crossref]

Charman, N.

N. Charman, Handbook of Optics, M. Bass, ed. (McGraw-Hill, 2010), Vol. 3, pp. 1.1–1.65.

Cižmár, T.

Dainty, J. C.

A. J. Lambert, E. M. Daly, and J. C. Dainty, “Improved fixation quality provided by a Bessel beacon in an adaptive optics system,” Ophthalmic Physiolog. Opt. 33, 403–411 (2013).
[Crossref]

Daly, E. M.

A. J. Lambert, E. M. Daly, and J. C. Dainty, “Improved fixation quality provided by a Bessel beacon in an adaptive optics system,” Ophthalmic Physiolog. Opt. 33, 403–411 (2013).
[Crossref]

Dholakia, K.

T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17, 15558–15570 (2009).
[Crossref]

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

Dossier, B.

B. Dossier, “Recherches sur l’apodisation des images optiques,” Revue d’Optique 33, 57–267 (1954).

Goodman, J. W.

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

Häusler, G.

G. Bickel, G. Häusler, and M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–978 (1985).
[Crossref]

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[Crossref]

Lambert, A. J.

D. Bhattarai, M. Suheimat, A. J. Lambert, and D. A. Atchison, “Fixation stability with Bessel beams,” Optom. Vis. Sci 96, 95–102 (2019).
[Crossref]

A. J. Lambert, E. M. Daly, and J. C. Dainty, “Improved fixation quality provided by a Bessel beacon in an adaptive optics system,” Ophthalmic Physiolog. Opt. 33, 403–411 (2013).
[Crossref]

Liang, J.

Maul, M.

G. Bickel, G. Häusler, and M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–978 (1985).
[Crossref]

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

McLeod, J. H.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass, 1963).

Steward, G. C.

G. C. Steward, The Symmetrical Optical System (Cambridge University, 1928).

Strehl, K.

K. Strehl, “Aplanatische und fehlerhaft Abbildung im Fernrohr,” Zeitschrift für Instrumentenkunde 15, 362–370 (1895).

Suheimat, M.

D. Bhattarai, M. Suheimat, A. J. Lambert, and D. A. Atchison, “Fixation stability with Bessel beams,” Optom. Vis. Sci 96, 95–102 (2019).
[Crossref]

Westheimer, G.

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

J. Opt. Soc. Am. (1)

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

Ophthalmic Physiolog. Opt. (1)

A. J. Lambert, E. M. Daly, and J. C. Dainty, “Improved fixation quality provided by a Bessel beacon in an adaptive optics system,” Ophthalmic Physiolog. Opt. 33, 403–411 (2013).
[Crossref]

Opt. Eng. (1)

G. Bickel, G. Häusler, and M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–978 (1985).
[Crossref]

Opt. Express (1)

Optom. Vis. Sci (1)

D. Bhattarai, M. Suheimat, A. J. Lambert, and D. A. Atchison, “Fixation stability with Bessel beams,” Optom. Vis. Sci 96, 95–102 (2019).
[Crossref]

Proc. R. Soc. London Ser. A (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[Crossref]

Revue d’Optique (1)

B. Dossier, “Recherches sur l’apodisation des images optiques,” Revue d’Optique 33, 57–267 (1954).

Zeitschrift für Instrumentenkunde (1)

K. Strehl, “Aplanatische und fehlerhaft Abbildung im Fernrohr,” Zeitschrift für Instrumentenkunde 15, 362–370 (1895).

Other (4)

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass, 1963).

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

N. Charman, Handbook of Optics, M. Bass, ed. (McGraw-Hill, 2010), Vol. 3, pp. 1.1–1.65.

G. C. Steward, The Symmetrical Optical System (Cambridge University, 1928).

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Figures (8)

Fig. 1.
Fig. 1. Cross-sectional profile of Bessel beam ${J_o}(r)$ showing the first few lobes. Solid line: amplitude of a true Bessel beam. Dotted line: transmissivity of an opacity conforming to a Bessel function with the same parameter. The first few zeros occur at 2.4, 5.5. 8.6, 11.8 … Inset: face-on appearance of a Bessel intensity function.
Fig. 2.
Fig. 2. Rays as deviated by a purely prismatic axicon with apical angle $\chi $. All rays are deviated towards (or away from) the axis though an angle $\delta =(n - 1).\chi$. At a distance $r^{\prime}$ from the axis, the optical path between the incident plane and the point of interception by the receiving plane is changed by a phase factor ${2}\pi \delta .r^{\prime}/\lambda $.
Fig. 3.
Fig. 3. Retinal imagery with Bessel beam illumination. In each panel, top: amplitude distribution radially across the eye’s 2 mm radius entrance pupil; (below) cross section of the retinal illuminance through the center of the circularly symmetrical point-spread function in a range of image planes in ½ diopter steps from 2.5 D myopic to 2.5 D hyperopic defocus.
Fig. 4.
Fig. 4. Retinal image with small-angle axicon pupil modulation. Top: path difference phase deviations from sphericity of the wavefront admitted into the eye’s pupil (phase deviations) and (bottom) cross section of retinal illuminance through the center of the point-spread function in a range of image planes in ½ diopter steps from 2.5 D myopic to 2.5 D hyperopic defocus in three conditions. Left: normal viewing, in focus. Middle: with a coaxial axicon, deviating angle 0.3 milliradians, in or near the eye’s entrance pupil; phase increases linearly with radial distance in the pupil. Right: normal viewing with 1D defocus, when there is a quadratic phase deviation of the wavefront. Comparison with equivalent panels in Fig. 3 shows that the effect of a small-angle axicon in or near the pupil is akin to changing focus.
Fig. 5.
Fig. 5. Axicon with larger deviating angle placed in or near the eye’s entrance pupil ${E^\prime}$ deviates meridional sheets of rays away from Gaussian focus ${O^\prime}$ to peripheral locations, forming a ring image. Solid lines: axicon imagery; dotted lines: normal path of light from lens to retina.
Fig. 6.
Fig. 6. Rays from a parallel beam traced through a purely prismatic axicon $A$. When the pupil of a viewing eye is placed at $P$, deviated beams from all half-meridians around the clock fill the pupil, all participating in the generation of the interference pattern that is admitted into the eye’s pupil. There is a zone in the axicon’s image space in which the cross-sectional light distribution is that of a Bessel or quasi-Bessel beam [6].
Fig. 7.
Fig. 7. Retinal image in an actual eye (data from Artal [9]). Left: cross-section of point-spread function through its center; right: optical transfer function. Two conditions: solid lines, normal in-focus viewing; dashed lines, within 1.5 diopter defocus with truncated Bessel function pupil modulation (normalized). The figure shows the expected effect of suitable pupil transmission modulation functions on ocular imagery in a real eye.
Fig. 8.
Fig. 8. Left: light distributions in the retinal image of an edge in focus (middle of the seven ogives) and in three equally spaced states of defocus both in the myopic and hyperopic directions. Solid lines: normal viewing—plane wavefront; dashed lines: with a pupil admittance of the form of a truncated Bessel beam. The latter condition gives an image in two separate states of defocus that is almost as sharp as that in the in-focus condition with normal viewing. Right: a measure of image degradation—the reciprocal of the edge gradient at the middle of the ogive light distribution of a sharp target edge as seen on the left of the figure—as a function of state of focus for both normal viewing (solid curve) and Bessel beam imagery (dashed curve). The graph tellingly illustrates the dual-focus property of this type of Bessel beam.

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A ( r ) e i ϕ ( r ) .

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