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Fig. 8. (a) Example of an experimental 512 x 512 8-bits hologram; (b) a z00m, corresponding to the dashed square of (a), where the interference fringes appear more visible; (c) the Fourier spectrum of (a) containing the zero order (zo), the real image (ri), the virtual image (vi) and also parasitic interferences spatial frequencies (p); (d) Fourier spectrum after application of the bandpass filter.

Figure 8 (a) Example of an experimental 512 x 512 8-bits hologram; (b) a z00m, corresponding to the dashed square of (a), where the interference fringes appear more visible; (c) the Fourier spectrum of (a) containing the zero order (zo), the real image (ri), the virtual image (vi) and also parasitic interferences spatial frequencies (p); (d) Fourier spectrum after application of the bandpass filter.

Related Figures (11)

Fig. 1. Scalar Debye theory: focusing of a spherical wave through a lens of focal f, half-aperture a, maximum subtended half angle a. The Debye theory combines in this way the geometrical and the wave optics, because all the individual plane waves
Fig. 1. Scalar Debye theory: focusing of a spherical wave through a lens of focal f, half-aperture a, maximum subtended half angle a. The Debye theory combines in this way the geometrical and the wave optics, because all the individual plane waves
We have considered here the case of an incident field linearly polarized in the x direction, but the expressions could be generalized for any arbitrary polarization state (Mansuripur, 2002). Even if the incident field had a component only along the x direction, the field at the focal plane will have components along all the three directions x (unit vector i), y (unit vector j) and z (unit vector k) (Fig. 2) and for a specific position can be calculated as follows (Gu, 2000): Fig. 2. Vectorial model: focusing ofa linearly polarized (x direction) beam through a lens of focal length f, half-aperture a, maximum subtended half angle a.
We have considered here the case of an incident field linearly polarized in the x direction, but the expressions could be generalized for any arbitrary polarization state (Mansuripur, 2002). Even if the incident field had a component only along the x direction, the field at the focal plane will have components along all the three directions x (unit vector i), y (unit vector j) and z (unit vector k) (Fig. 2) and for a specific position can be calculated as follows (Gu, 2000): Fig. 2. Vectorial model: focusing ofa linearly polarized (x direction) beam through a lens of focal length f, half-aperture a, maximum subtended half angle a.
Fig. 3. The x-, y- and z-components of the vectorial transverse APSF (xy) for a x 63/0.85 MO. ly: ly: 1;, are in proportion of, respectively, 1: 0.0032: 0.1290. Calculations were performed on a 4 x 4 mm? surface in the xy plane for all the component. The intensity distributions are normalized, the phase distributions are coded between —z = black and +z = white.
Fig. 3. The x-, y- and z-components of the vectorial transverse APSF (xy) for a x 63/0.85 MO. ly: ly: 1;, are in proportion of, respectively, 1: 0.0032: 0.1290. Calculations were performed on a 4 x 4 mm? surface in the xy plane for all the component. The intensity distributions are normalized, the phase distributions are coded between —z = black and +z = white.
Fig. 4. APSF computed in the axial direction with the vectorial model for a x 63/0.85 MO (a) and a x 100/1.3 MO (b). The intensity distributions are enhanced by a nonlinear distribution of the grey levels, the phase distributions are coded in 8 bits between —z = black and +z = white.
Fig. 4. APSF computed in the axial direction with the vectorial model for a x 63/0.85 MO (a) and a x 100/1.3 MO (b). The intensity distributions are enhanced by a nonlinear distribution of the grey levels, the phase distributions are coded in 8 bits between —z = black and +z = white.
Fig. 5. Axial APSF (xz) examples when the cover slip and the immersion oil refractive index are varied according to Gibson model: ideal case ng = 1.525, ni = 1.518 (a), cover slip refractive index is varied ng = 1.530 (b), immersion oil refractive index is varied ni = 1.514 (c). The MO considered here was x 100/1.30. The intensity distributions are enhanced by a nonlinear distribution of the grey levels, the phase distributions are coded in eight bits between —z = black and +z = white. Aberrations are present in most optical imaging systems: lenses, lenses assembly and MO. They are generally a consequence of the fabrication process; spherical aberrations in particular are due to grinding and polishing process of the lens which naturally tends to produce spherical surfaces. These aberrations are usually compensated by the recourse to the assembly of several lenses having complementary geometrical and dielectriccharacteristics. Other types ofaberrations appear when the focused beam crosses one or several dielectric layers, for which the MO has not been designed. To analyze the aberrations appearing when a MO is used under inappropriate conditions, Gibson (Gibson & Lanni, 1991) proposed a simple approach, based on the scalar diffraction theory and geometrical optics calculations. The aberration function is obtained through a calculation involving the ideal design parameters of the MO (cover slip refractive index, cover slip thickness, immersion oil refractive index) and their effective value. The simulations presented here were obtained by implementing the approach suggested by Haeberlé (2003), who combined the Gibson and the vectorial T6r6k models. Figure 5 summarizes the results of the simulations of the APSF of a x100 MO with 1.3 NA, with nonpolarized light, used under different conditions. If nonpolarized light is diffracted by the MO, the birefringence caused by the high NA MO would mix the cross-polarized components of the beam, while keeping a nonzero correlation among the cross-polarized components of the beam. Therefore, the cross-polarized components of the outgoing beam would cancel out for statistical reason, because the cross-polarized component of the emitted field would be uncorrelated. Therefore, the use of nonpolarized light ensures
Fig. 5. Axial APSF (xz) examples when the cover slip and the immersion oil refractive index are varied according to Gibson model: ideal case ng = 1.525, ni = 1.518 (a), cover slip refractive index is varied ng = 1.530 (b), immersion oil refractive index is varied ni = 1.514 (c). The MO considered here was x 100/1.30. The intensity distributions are enhanced by a nonlinear distribution of the grey levels, the phase distributions are coded in eight bits between —z = black and +z = white. Aberrations are present in most optical imaging systems: lenses, lenses assembly and MO. They are generally a consequence of the fabrication process; spherical aberrations in particular are due to grinding and polishing process of the lens which naturally tends to produce spherical surfaces. These aberrations are usually compensated by the recourse to the assembly of several lenses having complementary geometrical and dielectriccharacteristics. Other types ofaberrations appear when the focused beam crosses one or several dielectric layers, for which the MO has not been designed. To analyze the aberrations appearing when a MO is used under inappropriate conditions, Gibson (Gibson & Lanni, 1991) proposed a simple approach, based on the scalar diffraction theory and geometrical optics calculations. The aberration function is obtained through a calculation involving the ideal design parameters of the MO (cover slip refractive index, cover slip thickness, immersion oil refractive index) and their effective value. The simulations presented here were obtained by implementing the approach suggested by Haeberlé (2003), who combined the Gibson and the vectorial T6r6k models. Figure 5 summarizes the results of the simulations of the APSF of a x100 MO with 1.3 NA, with nonpolarized light, used under different conditions. If nonpolarized light is diffracted by the MO, the birefringence caused by the high NA MO would mix the cross-polarized components of the beam, while keeping a nonzero correlation among the cross-polarized components of the beam. Therefore, the cross-polarized components of the outgoing beam would cancel out for statistical reason, because the cross-polarized component of the emitted field would be uncorrelated. Therefore, the use of nonpolarized light ensures
Fig. 6. Axial profiles through the xz section of the IPSF and PPSF for the x 100/1.30 MO, without (a) and with (b) spherical aberrations induced by the use of nondesigned parameters (cover slip, immersion oil). A linear phase corresponding to the displacement along the optical axis has been subtracted in both (a) and (b) to enhance the anomalous phase behaviour in presence of aberration.
Fig. 6. Axial profiles through the xz section of the IPSF and PPSF for the x 100/1.30 MO, without (a) and with (b) spherical aberrations induced by the use of nondesigned parameters (cover slip, immersion oil). A linear phase corresponding to the displacement along the optical axis has been subtracted in both (a) and (b) to enhance the anomalous phase behaviour in presence of aberration.
Fig. 7. Experimental setup for the APSF measurement: BS beam-splitter, BE beam expander, NF neutral density filter, 4/2 half-wave plate, M mirror, FC fibre coupling lens, PS piezo system, MS micrometric stage, MO microscope objective, O object wave, R reference wave. Inset: a detail showing the off-axis geometry at the incidence on the CCD. Our setup is based on a Mach-Zehnder interferometer configuration. In the object arm, a point-source object is imaged through the MO or through the lens to characterize. (Fig. 7). As a point source we use a near field scanning optical microscopy (NSOM) fibre, with a 60-nm-diameter emitting tip. The light source is a A = 532 nm laser (frequency-doubled Nd: YAG) with adjustable power up to 100 mW and the laser is coupled in the optical fibre by a lens. The MO is mounted on micrometric xyz platforms combined with tilt facilities, allowing for proper alignment of the MO to avoid aberrations coming from the setup misalignment. Fine fibre movements, are achieved by a piezoelectric xyz stage, which permits nanometric displacements (1 step = 1.22 nm, within arange of 80 wm). ACCD camera (CCD1) is positioned at a large distance of about 1500 mm to create a sufficiently high magnification (about 1000x for a 100x MO) image of the point object,
Fig. 7. Experimental setup for the APSF measurement: BS beam-splitter, BE beam expander, NF neutral density filter, 4/2 half-wave plate, M mirror, FC fibre coupling lens, PS piezo system, MS micrometric stage, MO microscope objective, O object wave, R reference wave. Inset: a detail showing the off-axis geometry at the incidence on the CCD. Our setup is based on a Mach-Zehnder interferometer configuration. In the object arm, a point-source object is imaged through the MO or through the lens to characterize. (Fig. 7). As a point source we use a near field scanning optical microscopy (NSOM) fibre, with a 60-nm-diameter emitting tip. The light source is a A = 532 nm laser (frequency-doubled Nd: YAG) with adjustable power up to 100 mW and the laser is coupled in the optical fibre by a lens. The MO is mounted on micrometric xyz platforms combined with tilt facilities, allowing for proper alignment of the MO to avoid aberrations coming from the setup misalignment. Fine fibre movements, are achieved by a piezoelectric xyz stage, which permits nanometric displacements (1 step = 1.22 nm, within arange of 80 wm). ACCD camera (CCD1) is positioned at a large distance of about 1500 mm to create a sufficiently high magnification (about 1000x for a 100x MO) image of the point object,
Fig. 9. Temporal phase fluctuations at CCD1, CCD2, and the phase difference between the to phases (1000 holograms recorded in 40 s).
Fig. 9. Temporal phase fluctuations at CCD1, CCD2, and the phase difference between the to phases (1000 holograms recorded in 40 s).
Fig. 10. Axial APSF (amplitude and phase) for different cover slip thickness compensation in an adjustable collar x20 0.4 NA microscope objective: collar at position O (a), 0.5 (b) and 1 (c). The insets enhance the phase singularities appearing at the zero intensity points. The intensity distributions are enhanced by a nonlinear distribution of the grey levels, the phase distributions are coded in eight bits between —z = black and z = white.
Fig. 10. Axial APSF (amplitude and phase) for different cover slip thickness compensation in an adjustable collar x20 0.4 NA microscope objective: collar at position O (a), 0.5 (b) and 1 (c). The insets enhance the phase singularities appearing at the zero intensity points. The intensity distributions are enhanced by a nonlinear distribution of the grey levels, the phase distributions are coded in eight bits between —z = black and z = white.
Fig. 11. Axial (a) and radial (b) comparisons in amplitude and phase between experimental APSF measurement (up) and calculated APSF with the Gibson and Lanni model [10] for a x 100 1.3 NA microscope objective. Measurements performed in oil (n = 1.518) without cover slip. The intensity distributions are enhanced by a nonlinear distribution of the grey levels, the phase distributions are coded in 8 bits between —z = black and z = white.
Fig. 11. Axial (a) and radial (b) comparisons in amplitude and phase between experimental APSF measurement (up) and calculated APSF with the Gibson and Lanni model [10] for a x 100 1.3 NA microscope objective. Measurements performed in oil (n = 1.518) without cover slip. The intensity distributions are enhanced by a nonlinear distribution of the grey levels, the phase distributions are coded in 8 bits between —z = black and z = white.
Fig. 12. (a) Measured xz sections of the IPSF and PPSF for the x 100/1.30 MO used under nondesigned conditions (no cover slip, no immersion oil); (b) simulated and measured phase profile behaviour — from which a linear phase corresponding to the displacement along the optical axis has been subtracted — along the optical axis.
Fig. 12. (a) Measured xz sections of the IPSF and PPSF for the x 100/1.30 MO used under nondesigned conditions (no cover slip, no immersion oil); (b) simulated and measured phase profile behaviour — from which a linear phase corresponding to the displacement along the optical axis has been subtracted — along the optical axis.
Related topics:
Materials EngineeringCondensed Matter PhysicsDigital HolographyThree Dimensional ImagingPhase measurementSpherical AberrationPoint Spread FunctionBiochemistry and cell biology
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