Abstract

Hollow-beam geometry, in conjunction with mode-selective detection, is of importance for the development of high-sensitivity devices for the measurement of dynamic light scattering in living tissues. Its application to scattering methods in the eye makes it possible to increase diagnostic ability for some diseases that alter the scattering parameters in the vitreous as well as in other transparent tissues of the eye. We present a thorough theoretical analysis of the hollow-beam geometry proposed recently for dynamic light scattering measurements in the human eye. The aims of the analysis are the determination of the excitation and the observation beam profiles at the focal plane and the evaluation of the volume under test in the measurement, which allow prediction of the intensity of the measured signal. The above is carried out with comparisons with the classical setup. From the theoretical point of view, the most appealing feature of the hollow-beam geometry is high collection efficiency combined with high stability. In the analysis performed, the concept of the characteristic length of a scattering system is introduced. With simple formalism, this parameter allows the calculation of the collection efficiency for general beam shaping and is extremely useful for the comparison of the performance of different systems.

© 1997 Optical Society of America

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References

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  1. B. Chu, Laser Light Scattering (Academic, New York, 1991); B. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  2. J. N. Weiss, S. E. Bursell, R. E. Gleasen, B. H. Eichold, “Photon correlation spectroscopy of in vivo human cornea,” Cornea 5, 19–24 (1986).
  3. S. E. Bursell, J. R. Serur, J. F. Haughton, “Cholesterol level assessed with photon correlation spectroscopy,” in Lasers in Medicine, S. N. Joffe, J. A. Parrish, R. Scott, eds., Proc. SPIE712, 175–181 (1986).
    [Crossref]
  4. G. Benedeck, L. Chylack, T. Libondi, P. Magnante, M. Pennet, “Quantitative detection of the molecular changes associated with early cataractogenesis in the living human lens using quasi-elastic light scattering,” Curr. Eye Res. 6, 1421–1432 (1987).
    [Crossref]
  5. L. Rovati, F. Fankhauser, J. Ricka, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
    [Crossref]
  6. F. Fankhauser, L. Rovati, J. Ricka, “In-vivo dynamic light scattering changes of the vitreous in diabetes mellitus,” Invest. Ophthalmol. Visual Sci. 37, S973 (1996).
  7. J. Ricka, “Dynamic light scattering with single-mode and multimode receivers,” Appl. Opt. 32, 2860–2875 (1993).
    [Crossref] [PubMed]
  8. T. Gisler, H. Rüger, S. Egelhaaf, J. Tschumi, P. Schurtenberger, J. Ricka, “Mode-selective dynamic light scattering: theory vs. experimental realization,” Appl. Opt. 34, 3546–3554 (1995).
    [Crossref] [PubMed]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 8.
  10. I. S. Shteyn, I. M. Rizhik, Table of Integrals, Series and Products (Academic, New York, 1984), Chap. 6.
  11. F. Könz, J. Ricka, M. Frenz, F. Fankhauser, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
    [Crossref]
  12. E. J. Nijhof, W. S. J. Uijttewaal, R. M. Heethaar, “A laser-Doppler system for measuring distributions of blood particles in narrow flow channels,” IEEE Trans. Instrum. Meas. 43, 430–435 (1994).
    [Crossref]

1996 (2)

L. Rovati, F. Fankhauser, J. Ricka, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
[Crossref]

F. Fankhauser, L. Rovati, J. Ricka, “In-vivo dynamic light scattering changes of the vitreous in diabetes mellitus,” Invest. Ophthalmol. Visual Sci. 37, S973 (1996).

1995 (2)

T. Gisler, H. Rüger, S. Egelhaaf, J. Tschumi, P. Schurtenberger, J. Ricka, “Mode-selective dynamic light scattering: theory vs. experimental realization,” Appl. Opt. 34, 3546–3554 (1995).
[Crossref] [PubMed]

F. Könz, J. Ricka, M. Frenz, F. Fankhauser, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
[Crossref]

1994 (1)

E. J. Nijhof, W. S. J. Uijttewaal, R. M. Heethaar, “A laser-Doppler system for measuring distributions of blood particles in narrow flow channels,” IEEE Trans. Instrum. Meas. 43, 430–435 (1994).
[Crossref]

1993 (1)

1987 (1)

G. Benedeck, L. Chylack, T. Libondi, P. Magnante, M. Pennet, “Quantitative detection of the molecular changes associated with early cataractogenesis in the living human lens using quasi-elastic light scattering,” Curr. Eye Res. 6, 1421–1432 (1987).
[Crossref]

1986 (1)

J. N. Weiss, S. E. Bursell, R. E. Gleasen, B. H. Eichold, “Photon correlation spectroscopy of in vivo human cornea,” Cornea 5, 19–24 (1986).

Benedeck, G.

G. Benedeck, L. Chylack, T. Libondi, P. Magnante, M. Pennet, “Quantitative detection of the molecular changes associated with early cataractogenesis in the living human lens using quasi-elastic light scattering,” Curr. Eye Res. 6, 1421–1432 (1987).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 8.

Bursell, S. E.

J. N. Weiss, S. E. Bursell, R. E. Gleasen, B. H. Eichold, “Photon correlation spectroscopy of in vivo human cornea,” Cornea 5, 19–24 (1986).

S. E. Bursell, J. R. Serur, J. F. Haughton, “Cholesterol level assessed with photon correlation spectroscopy,” in Lasers in Medicine, S. N. Joffe, J. A. Parrish, R. Scott, eds., Proc. SPIE712, 175–181 (1986).
[Crossref]

Chu, B.

B. Chu, Laser Light Scattering (Academic, New York, 1991); B. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Chylack, L.

G. Benedeck, L. Chylack, T. Libondi, P. Magnante, M. Pennet, “Quantitative detection of the molecular changes associated with early cataractogenesis in the living human lens using quasi-elastic light scattering,” Curr. Eye Res. 6, 1421–1432 (1987).
[Crossref]

Egelhaaf, S.

Eichold, B. H.

J. N. Weiss, S. E. Bursell, R. E. Gleasen, B. H. Eichold, “Photon correlation spectroscopy of in vivo human cornea,” Cornea 5, 19–24 (1986).

Fankhauser, F.

L. Rovati, F. Fankhauser, J. Ricka, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
[Crossref]

F. Fankhauser, L. Rovati, J. Ricka, “In-vivo dynamic light scattering changes of the vitreous in diabetes mellitus,” Invest. Ophthalmol. Visual Sci. 37, S973 (1996).

F. Könz, J. Ricka, M. Frenz, F. Fankhauser, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
[Crossref]

Frenz, M.

F. Könz, J. Ricka, M. Frenz, F. Fankhauser, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
[Crossref]

Gisler, T.

Gleasen, R. E.

J. N. Weiss, S. E. Bursell, R. E. Gleasen, B. H. Eichold, “Photon correlation spectroscopy of in vivo human cornea,” Cornea 5, 19–24 (1986).

Haughton, J. F.

S. E. Bursell, J. R. Serur, J. F. Haughton, “Cholesterol level assessed with photon correlation spectroscopy,” in Lasers in Medicine, S. N. Joffe, J. A. Parrish, R. Scott, eds., Proc. SPIE712, 175–181 (1986).
[Crossref]

Heethaar, R. M.

E. J. Nijhof, W. S. J. Uijttewaal, R. M. Heethaar, “A laser-Doppler system for measuring distributions of blood particles in narrow flow channels,” IEEE Trans. Instrum. Meas. 43, 430–435 (1994).
[Crossref]

Könz, F.

F. Könz, J. Ricka, M. Frenz, F. Fankhauser, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
[Crossref]

Libondi, T.

G. Benedeck, L. Chylack, T. Libondi, P. Magnante, M. Pennet, “Quantitative detection of the molecular changes associated with early cataractogenesis in the living human lens using quasi-elastic light scattering,” Curr. Eye Res. 6, 1421–1432 (1987).
[Crossref]

Magnante, P.

G. Benedeck, L. Chylack, T. Libondi, P. Magnante, M. Pennet, “Quantitative detection of the molecular changes associated with early cataractogenesis in the living human lens using quasi-elastic light scattering,” Curr. Eye Res. 6, 1421–1432 (1987).
[Crossref]

Nijhof, E. J.

E. J. Nijhof, W. S. J. Uijttewaal, R. M. Heethaar, “A laser-Doppler system for measuring distributions of blood particles in narrow flow channels,” IEEE Trans. Instrum. Meas. 43, 430–435 (1994).
[Crossref]

Pennet, M.

G. Benedeck, L. Chylack, T. Libondi, P. Magnante, M. Pennet, “Quantitative detection of the molecular changes associated with early cataractogenesis in the living human lens using quasi-elastic light scattering,” Curr. Eye Res. 6, 1421–1432 (1987).
[Crossref]

Ricka, J.

L. Rovati, F. Fankhauser, J. Ricka, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
[Crossref]

F. Fankhauser, L. Rovati, J. Ricka, “In-vivo dynamic light scattering changes of the vitreous in diabetes mellitus,” Invest. Ophthalmol. Visual Sci. 37, S973 (1996).

T. Gisler, H. Rüger, S. Egelhaaf, J. Tschumi, P. Schurtenberger, J. Ricka, “Mode-selective dynamic light scattering: theory vs. experimental realization,” Appl. Opt. 34, 3546–3554 (1995).
[Crossref] [PubMed]

F. Könz, J. Ricka, M. Frenz, F. Fankhauser, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
[Crossref]

J. Ricka, “Dynamic light scattering with single-mode and multimode receivers,” Appl. Opt. 32, 2860–2875 (1993).
[Crossref] [PubMed]

Rizhik, I. M.

I. S. Shteyn, I. M. Rizhik, Table of Integrals, Series and Products (Academic, New York, 1984), Chap. 6.

Rovati, L.

F. Fankhauser, L. Rovati, J. Ricka, “In-vivo dynamic light scattering changes of the vitreous in diabetes mellitus,” Invest. Ophthalmol. Visual Sci. 37, S973 (1996).

L. Rovati, F. Fankhauser, J. Ricka, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
[Crossref]

Rüger, H.

Schurtenberger, P.

Serur, J. R.

S. E. Bursell, J. R. Serur, J. F. Haughton, “Cholesterol level assessed with photon correlation spectroscopy,” in Lasers in Medicine, S. N. Joffe, J. A. Parrish, R. Scott, eds., Proc. SPIE712, 175–181 (1986).
[Crossref]

Shteyn, I. S.

I. S. Shteyn, I. M. Rizhik, Table of Integrals, Series and Products (Academic, New York, 1984), Chap. 6.

Tschumi, J.

Uijttewaal, W. S. J.

E. J. Nijhof, W. S. J. Uijttewaal, R. M. Heethaar, “A laser-Doppler system for measuring distributions of blood particles in narrow flow channels,” IEEE Trans. Instrum. Meas. 43, 430–435 (1994).
[Crossref]

Weiss, J. N.

J. N. Weiss, S. E. Bursell, R. E. Gleasen, B. H. Eichold, “Photon correlation spectroscopy of in vivo human cornea,” Cornea 5, 19–24 (1986).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 8.

Appl. Opt. (2)

Cornea (1)

J. N. Weiss, S. E. Bursell, R. E. Gleasen, B. H. Eichold, “Photon correlation spectroscopy of in vivo human cornea,” Cornea 5, 19–24 (1986).

Curr. Eye Res. (1)

G. Benedeck, L. Chylack, T. Libondi, P. Magnante, M. Pennet, “Quantitative detection of the molecular changes associated with early cataractogenesis in the living human lens using quasi-elastic light scattering,” Curr. Eye Res. 6, 1421–1432 (1987).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

E. J. Nijhof, W. S. J. Uijttewaal, R. M. Heethaar, “A laser-Doppler system for measuring distributions of blood particles in narrow flow channels,” IEEE Trans. Instrum. Meas. 43, 430–435 (1994).
[Crossref]

Invest. Ophthalmol. Visual Sci. (1)

F. Fankhauser, L. Rovati, J. Ricka, “In-vivo dynamic light scattering changes of the vitreous in diabetes mellitus,” Invest. Ophthalmol. Visual Sci. 37, S973 (1996).

Opt. Eng. (1)

F. Könz, J. Ricka, M. Frenz, F. Fankhauser, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
[Crossref]

Rev. Sci. Instrum. (1)

L. Rovati, F. Fankhauser, J. Ricka, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
[Crossref]

Other (4)

S. E. Bursell, J. R. Serur, J. F. Haughton, “Cholesterol level assessed with photon correlation spectroscopy,” in Lasers in Medicine, S. N. Joffe, J. A. Parrish, R. Scott, eds., Proc. SPIE712, 175–181 (1986).
[Crossref]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 8.

I. S. Shteyn, I. M. Rizhik, Table of Integrals, Series and Products (Academic, New York, 1984), Chap. 6.

B. Chu, Laser Light Scattering (Academic, New York, 1991); B. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

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Figures (10)

Fig. 1
Fig. 1

Schematic representation of the hollow-beam geometry. The excitation beam is assumed to be Gaussian and is hollowed through removal of its central part; the scattering light from the VUT is collected coaxially. Single lens L is used to focus the excitation beam and collect the scattered light.

Fig. 2
Fig. 2

Normalized intensity distribution of the excitation beam, impinging on focusing lens L: d, cavity radius; ae, effective Gaussian beam radius.

Fig. 3
Fig. 3

Diffraction of a focused Gaussian spherical wave at a circular obstacle. The emerging wave from the lens converges at the axial focal point O. The vectors Rp and Rq identify the generic observation point P and source point Q, respectively.

Fig. 4
Fig. 4

Normalized intensity distribution of the excitation beam in the neighborhood of the focal point O (me = 1.92): u, v, two adimensional cylindrical coordinates defined by Eq. (5).

Fig. 5
Fig. 5

Normalized intensity distribution of the excitation beam at the focal plane (u = 0) with and without the cavity in the beam (me = 0 and me = 1.92). The first zero of the intensity distribution of the hollow beam occurs for v = 2.41.

Fig. 6
Fig. 6

Relative error of the intensity distribution of the excitation beam obtained by Eq. (3) with respect to the numeric evaluation of the Fresnel integrals. Note the excellent agreement between the result obtained with analytical form and numerical evaluation. It is obvious that the worst result can be observed on the curve where the worst convergence of the series is located, i.e., |αe| = |v|.

Fig. 7
Fig. 7

Normalized intensity distribution of the observation beam in the neighborhood of the focal point O (mo= 192): u, v, two adimensional cylindrical coordinates defined by Eq. (5).

Fig. 8
Fig. 8

(a) Characteristic length and (b) scattering volume as a function of the cavity radius d. The predetection signal is inversely proportional to the characteristic length, and therefore the maximum collection efficiency is achieved in correspondence with the cavity radius d = 0.686ae. The ratio between ae and ao was set to 10 and the scattering angle to 174°.

Fig. 9
Fig. 9

Normalized integrals of |Io(0, v)|2 and |Ie(0, v)|2 as a function of 1/me. The observation beam intensity integral is independent of d and therefore the integral of |Io(0, v)|2 is inversely proportional to me = d2/ae2 because |Uo(0, v)|2d2|Io(0, v)|2. The integral of |Ie(0, v)|2 exhibits a nonlinear dependence of the ratio me.

Fig. 10
Fig. 10

Scattering volume as a function of the distance d for the classical setup with scattering angle equal to 174°. The excitation beam radius is set to be 10 times bigger than the observation beam radius.

Equations (36)

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Eer=CeArexp-ik·rexp-iωt,  Eor=CoBrexp-ik·rexp-iωt,
Uer=CeArexp-ik·r,  Uor=CoBrexp-ik·r
UeQ=Cef exp-ikfexp-ξ2-η22ae2.
UeP=-iλCefW-D exp-ikfexp-ξ2-η22ae2dS,
ξ=dρ sin ϑη=dρ cos ϑ  x=r sin φy=r cos φ  u=2πλd2f2zv=2πλdfr  me=d2ae2αe=me+iu.
s-f=-RpRqf,  ζ=-f2-d2ρ21/2-f+12d2ρ2f,  ks-f=-kRpRqf=-kxξ+yη+zζf-vρ cosθ-φ+f2d2u-12uρ2.
UeP=-iλd2f2Ce expif2d2u×02π1expivρ cosϑ-φexp-α2ρ2ρdρdϑ,
J0x=12π02πexpix cosαdα, UeP=-12Aiπλd2f2 expif2d2uexp-α2ρ2×J0vρρdρ=-Aiπλd2f2 expif2d2uIeu, v.
Ieu, v=21exp-α2ρ2J0vρρdρ=20exp-α2ρ2J0vρρdρ-201exp-α2ρ2J0vρρdρ.
20exp-α2ρ2J0vρρdρ=2αe exp-v22αe,
αev<1
ddxxn+1Jn+1x=xn+1Jnx,
Ieu, v=2αe exp-v22αe-2αe exp-αe2n=1αenvnJnv.
αev>1
ddxx-nJnx=-x-nJn+1x,  limx0Jnxxn=12nn!,
Ieu, v=2αe exp-αe2n=0vnαenJnv.
Ueu, v2=Aπλd2f222αe exp-v22αe-2αe exp-αe2n=1αenvnJnv22αe exp-αe2n=0vnαenJnv2αevαev<1>1.
Iou, v=20exp-αo2ρ2J0vρρdρ=2αo exp-v22αo,
αo=mo+iu,  mo=d2/ao2.
J=IeRλ, ϑΩoVsϑ,
Ie=Je2π0Ue0, v2rdr.
Je=Je exp-d2ae2;
Ωo=λ22π0Uo0, v2rdr;
Vsϑ=2π-0Uou, vUeu, v2rdrdz.
J=JeRλ, ϑλ2-0Uou, vUeu, v2rdrdz0Uo0, v2rdr0Ue0, v2rdr,
J=JeRλ, ϑλ2λ2π-0Uou, vUeu, v2vdvdu0Uo0, v2vdv0Ue0, v2vdv=JeRλ, ϑλ2Kϑ.
Kϑ=1k0Uo0, v2vdv 0Ue0, v2vdv-0Uou, vUeu, v2vdvdu.
Vsϑ=2π-0Uou, vUeu, v2rdrdz=λ32π2f4d4-0Uou, vUeu, v2vdvduλ32π2π-ϑ4×0Uo0, v2vdv 0Ue0, v2vdvKϑk.
J=JeRλ, ϑλ2sin ϑπae2+a021/2,
aeλf2πaee,  aoλf2πao.
Kϑ=sin ϑπae2-ao21/2dfπae2-ao21/2,
Kϑ1kπd2ae2+d2ao21/2.
d5ao+5ae2=ao552.
Kϑkmindaoπ=48.73.
12dae20Ue0, v2vdv,  12dao20Uo0, v2vdv,
Vsϑ=λ32π2π-ϑ4d44ao2ae21Kϑkλ32π4π-ϑ4π3/2aoaed2ao2d2+ae2d21/2.

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