Abstract

We present a theoretical study of a new total-internal-reflection fluorescence microscope for the detection of fluorescence at a water-glass interface. The system is designed for confocal imaging and spectroscopy of nanoparticles and single molecules. Focusing and fluorescence collection through standard glass coverslips is accomplished by a parabolic mirror lens. The large aperture of the element is used to excite fluorescence within the evanescent field of a diffraction-limited focus and to collect focal emission with high efficiency. Tight focusing and supercritical excitation reduce the detection volume for fluorescent analyte molecules well below that of an attoliter (10-18 L), which can be advantageous for monitoring surface binding of single molecules without interference from fluorescence of the unbound bulk. Calculations of the electric fields in the focus region and simulated confocal imaging demonstrate the performance of the system.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Axelrod, “Total internal reflection fluorescence microscopy,” in Methods in Cellular Imaging, A. Periasamy, ed. (Oxford U. Press, New York, 2001), pp. 362–380.
    [CrossRef]
  2. A. D. Stout, D. Axelrod, “Evanescent field excitation of fluorescence by epi-illumination microscopy,” Appl. Opt. 28, 5237–5242 (1989).
    [CrossRef]
  3. W. P. Ambrose, T. Basché, W. E. Moerner, “Detection and spectroscopy of single pentacene molecules in a p-terphenyl crystal by means of fluorescence excitation,” J. Chem. Phys. 95, 7150–7163 (1991).
    [CrossRef]
  4. L. Fleury, P. Tamarat, B. Lounis, J. Bernard, M. Orrit, “Fluorescence spectra of single pentacene molecules in p-terphenyl at 1.7 K,” Chem. Phys. Lett. 236, 87–95 (1995).
    [CrossRef]
  5. H. van der Meer, J. A. J. M. Disselhorst, J. Koehler, A. C. J. Brouwer, E. J. J. Groenen, J. Schmidt, “An insert for single-molecule magnetic-resonance spectroscopy in an external magnetic field,” Rev. Sci. Instrum. 66, 4853–4856 (1995).
    [CrossRef]
  6. T. Ruckstuhl, S. Jung, J. Enderlein, S. Seeger, “Forbidden light detection from single molecules,” Anal. Chem. 72, 2117–2123 (2000).
    [CrossRef] [PubMed]
  7. A. Drechsler, M. A. Lieb, C. Debus, A. J. Meixner, A. Tarrach, “Confocal microscopy with a high numerical aperture parabolic mirror,” Opt. Express 9, 637–644 (2001), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  8. W. Lukosz, R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total irradiated power,” J. Opt. Soc. Am. 67, 1607–1615 (1977).
    [CrossRef]
  9. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  10. P. Varga, P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17, 2081–2089 (2000).
    [CrossRef]
  11. P. Varga, P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A 17, 2090–2095 (2000).
    [CrossRef]
  12. M. A. Lieb, A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8, 458–474 (2000), http://www.opticsexpress.org .
    [CrossRef]
  13. H. Ling, S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  14. S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
    [CrossRef]
  15. P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  16. P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I.,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  17. P. Török, C. J. R. Sheppard, P. Varga, “Study of evanescent waves for transmission near-field optical microscopy,” J. Mod. Opt. 43, 1167–1183 (1996).
    [CrossRef]
  18. L. Novotny, Lecture Notes on Nano-optics (U. Rochester Press, Rochester, N.Y., 2000).
  19. M. A. Lieb, “Mikroskopie mit Parabolspiegeloptik. Theorie, Aufbau und Charakterisierung eines kombinierten konfokalen und nahfeld-optischen Mikroskops für die Einzelmolekül-Spektroskopie bei tiefen Temperaturen,” Ph.D dissertation (University of Siegen, Siegen, Germany, 2001), ISBN 3-8311-3424-3 (2002).
  20. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  21. W. Lukosz, R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 67, 195–199 (1977).
    [CrossRef]
  22. W. Lukosz, R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. III. Radiation patterns of dipoles with arbitrary orientation,” J. Opt. Soc. Am. 69, 1495–1503 (1979).
    [CrossRef]
  23. J. Enderlein, T. Ruckstuhl, S. Seeger, “Highly efficient optical detection of surface-generated fluorescence,” Appl. Opt. 38, 724–732 (1999).
    [CrossRef]

2001 (1)

2000 (4)

1999 (1)

1996 (1)

P. Török, C. J. R. Sheppard, P. Varga, “Study of evanescent waves for transmission near-field optical microscopy,” J. Mod. Opt. 43, 1167–1183 (1996).
[CrossRef]

1995 (4)

L. Fleury, P. Tamarat, B. Lounis, J. Bernard, M. Orrit, “Fluorescence spectra of single pentacene molecules in p-terphenyl at 1.7 K,” Chem. Phys. Lett. 236, 87–95 (1995).
[CrossRef]

H. van der Meer, J. A. J. M. Disselhorst, J. Koehler, A. C. J. Brouwer, E. J. J. Groenen, J. Schmidt, “An insert for single-molecule magnetic-resonance spectroscopy in an external magnetic field,” Rev. Sci. Instrum. 66, 4853–4856 (1995).
[CrossRef]

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I.,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

1993 (1)

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

1991 (1)

W. P. Ambrose, T. Basché, W. E. Moerner, “Detection and spectroscopy of single pentacene molecules in a p-terphenyl crystal by means of fluorescence excitation,” J. Chem. Phys. 95, 7150–7163 (1991).
[CrossRef]

1989 (1)

1984 (1)

1979 (1)

1977 (2)

W. Lukosz, R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total irradiated power,” J. Opt. Soc. Am. 67, 1607–1615 (1977).
[CrossRef]

W. Lukosz, R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 67, 195–199 (1977).
[CrossRef]

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Ambrose, W. P.

W. P. Ambrose, T. Basché, W. E. Moerner, “Detection and spectroscopy of single pentacene molecules in a p-terphenyl crystal by means of fluorescence excitation,” J. Chem. Phys. 95, 7150–7163 (1991).
[CrossRef]

Axelrod, D.

A. D. Stout, D. Axelrod, “Evanescent field excitation of fluorescence by epi-illumination microscopy,” Appl. Opt. 28, 5237–5242 (1989).
[CrossRef]

D. Axelrod, “Total internal reflection fluorescence microscopy,” in Methods in Cellular Imaging, A. Periasamy, ed. (Oxford U. Press, New York, 2001), pp. 362–380.
[CrossRef]

Basché, T.

W. P. Ambrose, T. Basché, W. E. Moerner, “Detection and spectroscopy of single pentacene molecules in a p-terphenyl crystal by means of fluorescence excitation,” J. Chem. Phys. 95, 7150–7163 (1991).
[CrossRef]

Bernard, J.

L. Fleury, P. Tamarat, B. Lounis, J. Bernard, M. Orrit, “Fluorescence spectra of single pentacene molecules in p-terphenyl at 1.7 K,” Chem. Phys. Lett. 236, 87–95 (1995).
[CrossRef]

Booker, G. R.

Brouwer, A. C. J.

H. van der Meer, J. A. J. M. Disselhorst, J. Koehler, A. C. J. Brouwer, E. J. J. Groenen, J. Schmidt, “An insert for single-molecule magnetic-resonance spectroscopy in an external magnetic field,” Rev. Sci. Instrum. 66, 4853–4856 (1995).
[CrossRef]

Cremer, C.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Debus, C.

Disselhorst, J. A. J. M.

H. van der Meer, J. A. J. M. Disselhorst, J. Koehler, A. C. J. Brouwer, E. J. J. Groenen, J. Schmidt, “An insert for single-molecule magnetic-resonance spectroscopy in an external magnetic field,” Rev. Sci. Instrum. 66, 4853–4856 (1995).
[CrossRef]

Drechsler, A.

Enderlein, J.

T. Ruckstuhl, S. Jung, J. Enderlein, S. Seeger, “Forbidden light detection from single molecules,” Anal. Chem. 72, 2117–2123 (2000).
[CrossRef] [PubMed]

J. Enderlein, T. Ruckstuhl, S. Seeger, “Highly efficient optical detection of surface-generated fluorescence,” Appl. Opt. 38, 724–732 (1999).
[CrossRef]

Fleury, L.

L. Fleury, P. Tamarat, B. Lounis, J. Bernard, M. Orrit, “Fluorescence spectra of single pentacene molecules in p-terphenyl at 1.7 K,” Chem. Phys. Lett. 236, 87–95 (1995).
[CrossRef]

Groenen, E. J. J.

H. van der Meer, J. A. J. M. Disselhorst, J. Koehler, A. C. J. Brouwer, E. J. J. Groenen, J. Schmidt, “An insert for single-molecule magnetic-resonance spectroscopy in an external magnetic field,” Rev. Sci. Instrum. 66, 4853–4856 (1995).
[CrossRef]

Hell, S.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Jung, S.

T. Ruckstuhl, S. Jung, J. Enderlein, S. Seeger, “Forbidden light detection from single molecules,” Anal. Chem. 72, 2117–2123 (2000).
[CrossRef] [PubMed]

Koehler, J.

H. van der Meer, J. A. J. M. Disselhorst, J. Koehler, A. C. J. Brouwer, E. J. J. Groenen, J. Schmidt, “An insert for single-molecule magnetic-resonance spectroscopy in an external magnetic field,” Rev. Sci. Instrum. 66, 4853–4856 (1995).
[CrossRef]

Kunz, R. E.

Laczik, Z.

Lee, S. W.

Lieb, M. A.

A. Drechsler, M. A. Lieb, C. Debus, A. J. Meixner, A. Tarrach, “Confocal microscopy with a high numerical aperture parabolic mirror,” Opt. Express 9, 637–644 (2001), http://www.opticsexpress.org .
[CrossRef] [PubMed]

M. A. Lieb, A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8, 458–474 (2000), http://www.opticsexpress.org .
[CrossRef]

M. A. Lieb, “Mikroskopie mit Parabolspiegeloptik. Theorie, Aufbau und Charakterisierung eines kombinierten konfokalen und nahfeld-optischen Mikroskops für die Einzelmolekül-Spektroskopie bei tiefen Temperaturen,” Ph.D dissertation (University of Siegen, Siegen, Germany, 2001), ISBN 3-8311-3424-3 (2002).

Ling, H.

Lounis, B.

L. Fleury, P. Tamarat, B. Lounis, J. Bernard, M. Orrit, “Fluorescence spectra of single pentacene molecules in p-terphenyl at 1.7 K,” Chem. Phys. Lett. 236, 87–95 (1995).
[CrossRef]

Lukosz, W.

Meixner, A. J.

Moerner, W. E.

W. P. Ambrose, T. Basché, W. E. Moerner, “Detection and spectroscopy of single pentacene molecules in a p-terphenyl crystal by means of fluorescence excitation,” J. Chem. Phys. 95, 7150–7163 (1991).
[CrossRef]

Novotny, L.

L. Novotny, Lecture Notes on Nano-optics (U. Rochester Press, Rochester, N.Y., 2000).

Orrit, M.

L. Fleury, P. Tamarat, B. Lounis, J. Bernard, M. Orrit, “Fluorescence spectra of single pentacene molecules in p-terphenyl at 1.7 K,” Chem. Phys. Lett. 236, 87–95 (1995).
[CrossRef]

Reiner, G.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Ruckstuhl, T.

T. Ruckstuhl, S. Jung, J. Enderlein, S. Seeger, “Forbidden light detection from single molecules,” Anal. Chem. 72, 2117–2123 (2000).
[CrossRef] [PubMed]

J. Enderlein, T. Ruckstuhl, S. Seeger, “Highly efficient optical detection of surface-generated fluorescence,” Appl. Opt. 38, 724–732 (1999).
[CrossRef]

Schmidt, J.

H. van der Meer, J. A. J. M. Disselhorst, J. Koehler, A. C. J. Brouwer, E. J. J. Groenen, J. Schmidt, “An insert for single-molecule magnetic-resonance spectroscopy in an external magnetic field,” Rev. Sci. Instrum. 66, 4853–4856 (1995).
[CrossRef]

Seeger, S.

T. Ruckstuhl, S. Jung, J. Enderlein, S. Seeger, “Forbidden light detection from single molecules,” Anal. Chem. 72, 2117–2123 (2000).
[CrossRef] [PubMed]

J. Enderlein, T. Ruckstuhl, S. Seeger, “Highly efficient optical detection of surface-generated fluorescence,” Appl. Opt. 38, 724–732 (1999).
[CrossRef]

Sheppard, C. J. R.

P. Török, C. J. R. Sheppard, P. Varga, “Study of evanescent waves for transmission near-field optical microscopy,” J. Mod. Opt. 43, 1167–1183 (1996).
[CrossRef]

Stelzer, E. H. K.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Stout, A. D.

Tamarat, P.

L. Fleury, P. Tamarat, B. Lounis, J. Bernard, M. Orrit, “Fluorescence spectra of single pentacene molecules in p-terphenyl at 1.7 K,” Chem. Phys. Lett. 236, 87–95 (1995).
[CrossRef]

Tarrach, A.

Török, P.

van der Meer, H.

H. van der Meer, J. A. J. M. Disselhorst, J. Koehler, A. C. J. Brouwer, E. J. J. Groenen, J. Schmidt, “An insert for single-molecule magnetic-resonance spectroscopy in an external magnetic field,” Rev. Sci. Instrum. 66, 4853–4856 (1995).
[CrossRef]

Varga, P.

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Anal. Chem. (1)

T. Ruckstuhl, S. Jung, J. Enderlein, S. Seeger, “Forbidden light detection from single molecules,” Anal. Chem. 72, 2117–2123 (2000).
[CrossRef] [PubMed]

Appl. Opt. (2)

Chem. Phys. Lett. (1)

L. Fleury, P. Tamarat, B. Lounis, J. Bernard, M. Orrit, “Fluorescence spectra of single pentacene molecules in p-terphenyl at 1.7 K,” Chem. Phys. Lett. 236, 87–95 (1995).
[CrossRef]

J. Chem. Phys. (1)

W. P. Ambrose, T. Basché, W. E. Moerner, “Detection and spectroscopy of single pentacene molecules in a p-terphenyl crystal by means of fluorescence excitation,” J. Chem. Phys. 95, 7150–7163 (1991).
[CrossRef]

J. Microsc. (1)

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

J. Mod. Opt. (1)

P. Török, C. J. R. Sheppard, P. Varga, “Study of evanescent waves for transmission near-field optical microscopy,” J. Mod. Opt. 43, 1167–1183 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

W. Lukosz, R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 67, 195–199 (1977).
[CrossRef]

Opt. Express (2)

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

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Rev. Sci. Instrum. (1)

H. van der Meer, J. A. J. M. Disselhorst, J. Koehler, A. C. J. Brouwer, E. J. J. Groenen, J. Schmidt, “An insert for single-molecule magnetic-resonance spectroscopy in an external magnetic field,” Rev. Sci. Instrum. 66, 4853–4856 (1995).
[CrossRef]

Other (4)

D. Axelrod, “Total internal reflection fluorescence microscopy,” in Methods in Cellular Imaging, A. Periasamy, ed. (Oxford U. Press, New York, 2001), pp. 362–380.
[CrossRef]

L. Novotny, Lecture Notes on Nano-optics (U. Rochester Press, Rochester, N.Y., 2000).

M. A. Lieb, “Mikroskopie mit Parabolspiegeloptik. Theorie, Aufbau und Charakterisierung eines kombinierten konfokalen und nahfeld-optischen Mikroskops für die Einzelmolekül-Spektroskopie bei tiefen Temperaturen,” Ph.D dissertation (University of Siegen, Siegen, Germany, 2001), ISBN 3-8311-3424-3 (2002).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Schematic of the confocal TIRF microscope.

Fig. 2
Fig. 2

Contour plots of constants |E| 2, |E x |2, |E y |2, and |E z |2 in the xy, xz, and yz planes. Scale bars, one vacuum laser wavelength(1 λvac). Dashed lines, interface positions. The numbers indicate multiplication factors of the intensity images.

Fig. 3
Fig. 3

Cross sections of constant |E| 2. The intensity is normalized on the surface value at the optical axis. (a) Relative intensity along the x axis (solid curve) and along the y axis (dashed curve). (b) Relative intensity along the z axis. λvac is the laser wavelength in vacuum.

Fig. 4
Fig. 4

Intensity distribution in the detection planes of dipoles positioned in the primary geometrical focus with (from the left) x, y, and z orientation. Circles correspond to a pinhole with diameter 80 λem, which will be used for the computation of the CEF.

Fig. 5
Fig. 5

Series of x-dipole images in the detection plane with increasing offset from the optical axis: rm=n λem812, 12, 0, n=08.

Fig. 6
Fig. 6

Contour plots of the CEFs of principal dipole orientations. The maximum is achieved by p = (0, 0, 1) because of the slightly higher concentration of light at the detection plane. Scale bars, 1 λvac.

Fig. 7
Fig. 7

Polar dipole emission distribution averaged over all azimuth angles φ: Solid curves, parallel-oriented dipoles; dashed curves, perpendicularly oriented dipoles.

Fig. 8
Fig. 8

Influence of surface distance on dipole emission intensity into 47° ≤ θ ≤ 75°.

Fig. 9
Fig. 9

Contour plot of the CEF in the z direction. The scale bar corresponds to 1 λvac.

Fig. 10
Fig. 10

Confocal images of surface-bound molecules with principal dipole orientations. The scale bars correspond to 1 λvac.

Fig. 11
Fig. 11

Detection volume of the confocal TIRF microscope for randomly orientated emitters.

Fig. 12
Fig. 12

Ray-tracing calculations of the primary geometrical focus simulating spherical aberration. Top, identical refractive indices for the coverslip (thickness, 150 μm; n glass = 1.5230) and the immersion oil. Middle and bottom, index of the oil increased to 1.5231 and 1.5232, respectively.

Equations (14)

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

θc=arcsinnwater/nglass,
θmin,max=arctanr1,214fp r1,22-fp-1,
Ex=ikfp2I0+I2 cos 2φp, Ey=ikfp2 I2 sin 2φp, Ez=-kfpI1 cos φp,
I0=θcθmax l0θ2 sin θ1+cos θ1+cos θJ0krp sin θ sin θp×exp-ikrp cos θ cos θpdθ, I1=θcθmax l0θ2 sin2 θ1+cos θ J1krp sin θ sin θp×exp-ikrp cos θ cos θpdθ, I2=θcθmax l0θ2 sin θ1+cos θ1-cos θ×J2krp sin θ sin θp×exp-ikrp cos θ cos θpdθ,
l0θ=exp-r2θw02, rθ=4fp1+1+tan2 θ2 tan θ,
I0,glass=θcθmax l0θ2 sin θ1+cos θ1+cos θexp-iΦ11+rs-rp cos θexpiΦ11J0Φ12dθ, I1,glass=θcθmax l0θ2 sin2 θ1+cos θexp-iΦ11+rp expiΦ11J1Φ12dθ, I2,glass=θcθmax l0θ2 sin θ1+cos θ1-cos θexp-iΦ11+rs+rp cos θexpiΦ11J2Φ12dθ,
I0,water=θcθmax l0θ2 sin θ1+cos θts+tp cos θtexpiΦ21J0Φ22dθ, I1,water=θcθmax l0θ2 sin θ1+cos θ tp sin θt×expiΦ21J1Φ22dθ, I2,water=θcθmax l0θ2 sin θ1+cos θts-tp cos θtexpiΦ21J2Φ22dθ,
Φ11=kglassrp cos θ cos θp, Φ12=kglassrp sin θ sin θp, Φ21=kwaterrp cosθt cos θp, Φ22=kwaterrp sin θt sin θp,
sin θt=nglassnwatersin θ, cos θt=1-nglassnwatersin θ21/2.
Erd=-ik3f016π3ε0αminαmax02π1+cos θm2fmcos θd×p·em,ed-p·ee×expikrd·sd-nglassrm·sm×sin θddθddφ,
θm=2 arctanf02fpsin θd,
αmin,max=arctanr2,1f0.
e=-sin φ, cos φ, 0, em,=cos θm cos φ, cos θm sin φ, sin θm, ed,=cos θd cos φ, cos θd sin φ, sin θd, sm=sin θm cos φ, sin θm sin φ, -cos θm, sd=sin θd cos φ, sin θd sin φ, -cos θd.
rm=n λem812, 12, 0, n=08.

Metrics