Wave Theory for a Simple Lens

A new wave theory describing image formation by a simple lens is formulated in the angular spectrum representation. It is closely related to Luneburg's theory of instrumental optics but is free of certain geometrical approximations made in his theory. The validity of Luneburg's essentially geometrical treatment of the lens aperture is discussed and the approximations involved are found to be justified for isoplanatic optical systems with small numerical aperture. Some concepts usually found in lens theories based on geometrical optics are seen to have analogues in the present wave theory. In particular, a connection is made between homogeneous plane waves in an angular spectrum expansion of the field and the light rays belonging to a family of rays that pass through the lens. The fundamental relations assumed in Fourier optics are shown to follow from this theory when they are applied to the special case of isoplanatic optical systems with small numerical aperture. The image field of a scalar dipole formed by a diffraction-limited lens is calculated using these results.     

Measurement of the orbital angular momentum at photon level via the spatial probability distribution

We demonstrate that by diffracting light at single photon level with orbital angular momentum (OAM) by an equilateral triangular aperture, it is possible to determine their OAM amount by simply counting the number of maxima in the side of a generated triangular shaped hexagonal lattice in the spatial photon probability distribution. The sign of the OAM is obtained by the orientation of the latticed triangle. We also show that by changing the aperture size it is possible to discriminate OAM state superpositions.     

On the Study of Partial Coherence via Diffraction Experiments

The radiation pattern of a circular aperture illuminated both uniformly and non-uniformly by partially coherent light is calculated in the diffraction limit. Furthermore, the angular resolution of a circular lens is examined for four reasonable correlation functions. Also the effects of the parameters—partial coherence, misalignment, and non-uniform illumination—on the diffraction pattern of a double slit are considered for a particular mathematical model of the radiation field. The results of this analysis are used to discuss the the use of diffraction experiments in the study of the coherence properties of light.     

Analysis of a simple method for the reduction of phonon peak broadening in surface Brillouin light scattering

We present an investigation of the effect of the collecting lens aperture on the line shape of phonon peaks observed in surface Brillouin light scattering (SBLS) from surfaces of opaque materials and transparent thin films. In general, the broadening that is due to the aperture is asymmetric and can be as large as 60% of the peak frequency shift in the case of a f/1.4 aperture with an angle of incidence theta(i) = 30 degrees . We calculated SBLS spectra accounting for the spread in scattering wave vectors across the collecting lens aperture, the polarization and angular dependence of the scattering, and the spectrometer instrumental function. By performing a detailed comparison between measured and calculated SBLS spectra for Si(001), we identified a set of simple rules for the placement of a rectangular slit in the collecting lens aperture to reduce the effects of aperture broadening. By use of a slit, the peak linewidths can be reduced substantially, without reducing the peak heights significantly, while eliminating false shifts in the measured frequency values.     

Orbital angular momentum density and spiral spectrum of Lorentz–Gauss vortex beams diffracted by a rectangular aperture

Based on the expansions of the Lorentz distribution and the aperture function, an analytical expression of Lorentz–Gauss vortex beams with a topological charge of ±1 diffracted by a rectangular aperture is derived. One can judge the sign of the topological charge from the normalized intensity and the phase distributions. The effects of the rectangular aperture on the orbital angular momentum density and the spiral spectrum are investigated, respectively. When the length and the width of the rectangular aperture are not equal, the orbital angular momentum density distribution becomes twisted and tilted. When the size of the rectangular aperture increases, the magnitude of the orbital angular momentum density and the weight coefficient of the dominant spectrum both increase, while the weight coefficients of other minor spectra decrease. In addition, the expansion of the spiral spectrum in the case of rectangular aperture is smaller than that in the case of the single slit. The difference between the adjacent spectra in the case of the rectangular aperture is four, which is twice the difference in the case of the single slit. Moreover, the weight coefficient of the dominant spectrum in the case of the rectangular aperture is relatively larger. To measure the topological charge of the diffracted Lorentz–Gauss vortex beam, this research denotes that the rectangular aperture is superior to the single slit.     

Optical properties of a retroreflecting sheet

The power reflection and polarization properties of a close-packed array of retroreflectors are modeled, and a commercially available sheet is measured to verify the predictions. The modeling technique is conceptually simple and applicable to a wide range of structures of this type. The close-packed sheet retroreflects over a range of angles of incidence of approximately -40 to 40 deg in both directions and returns the polarization that illuminates it largely unchanged. Predictions of returned power are within 10% for light incident within 15 deg of normal and within 20% for angles less than 20 deg. Angles of polarization rotation are predicted to within 10 deg over a similar range of input angles. The model predicts the angular aperture of the sheet and the major features of the angular response. Future research will focus on design of structures with wider angles of acceptance and responses optimized for specific applications.     

Free space optical communications through clouds: analysis of signal characteristics

Free space optical communications (FSOC) is a method by which one transmits a modulated beam of light through the atmosphere for broadband applications. Fundamental limitations of FSOC arise from the environment through which light propagates. This work addresses transmitted light beam dispersion (spatial, angular, and temporal dispersion) in FSOC operating as a ground-to-air link when clouds exist along the communications channel. Light signals (photons) transmitted through clouds will interact with the cloud particles. Photon-particle interaction causes dispersion of light signals, which has significant effects on signal attenuation and pulse spread. The correlation between spatial and angular dispersion is investigated as well, which plays an important role on the receiver design. Moreover, the paper indicates that temporal dispersion (pulse spread) and energy loss strongly depend on the aperture size of the receiver, the field-of-view (FOV), and the position of the receiver relative to the optical axis of the transmitter.     

Predictions of Rayleigh's diffraction theory for the effect of focal shift in high-aperture systems

In their work on diffraction [J. Opt. Soc. Am.51, 1050 1961], Osterberg and Smith have computed in an exact manner from the Rayleigh-Sommerfeld diffraction integral of the first kind the irradiance distribution along the axis of a converging spherical wave, and they found that in a scalar optical system of high relative aperture and finite value of Fresnel number, the central peak value of the axial irradiance may occur inside, at, or outside the geometrical focal point as the angular semiaperture of the system is less than, equal to, or greater than, respectively, a particular angle that falls near 70 degrees . These findings are now reexamined using a different assumption that takes into account diffraction at the edge of the aperture. Different results are obtained that agree well with the predictions of other theories of diffraction of light and give confidence to the common conclusions drawn by investigators of the effect of focal shift, that the point of the principal maximum of axial irradiance is not at the geometrical focus but shifted toward the aperture in systems of different relative aperture and finite value of Fresnel number.     

Three-dimensional analysis of mode discrimination in vertical-cavity surface-emitting lasers

Optical mode discrimination in vertical-cavity surface-emitting lasers that contain distributed Bragg reflectors (DBRs) and a spatially limited gain medium is analyzed numerically. It is assumed that the output field is linearly polarized owing to gain selectivity. The analysis employs a three-dimensional model and an angular spectrum of plane-wave decomposition with the proper polarizations. Two types of round aperture are considered, namely, a Gaussian aperture and a ring-peak aperture that represents gain saturation. Coupled with the DBRs, the former aperture yields nearly Laguerre-Gaussian modes, whereas the latter aperture significantly distorts the mode shapes. In both cases, narrowband DBRs provide the best mode discrimination.     

Channel modeling of light signals propagating through a battlefield environment: analysis of channel spatial, angular, and temporal dispersion

Free-space optical communication (FSOC) is used to transmit a modulated beam of light through the atmosphere for broadband applications. Fundamental limitations of FSOC arise from the environment through which light propagates. We address transmitted light signal dispersion (spatial, angular, and temporal dispersion) in FSOC that operates in the battlefield environment. Light signals (photons) transmitted through the battlefield environment will interact with particles of man-made smoke such as fog oil, along the propagation path. Photon-particle interaction causes dispersion of light signals, which has significant effects on signal attenuation and pulse spread. We show that physical properties of battlefield particles play important roles in determining dispersion of received light signals. The correlation between spatial and angular dispersion is investigated as well, which has significant effects on receiver design issues. Moreover, our research indicates that temporal dispersion (delay spread) and the received power strongly depend on the receiver aperture size, field of view (FOV), and the position of the receiver relative to the optical axis of the transmitter. The results describe only specific scenarios for given types of battlefield particles. Generalization of the results requires additional work. Based on properties of the correlation, a sensitive receiver with a small FOV is needed that can find the line-of-sight photons and work with them.     

Optical Function of the Convexiclivate Fovea with Particular Regard to Notosudid Deep-sea Teleosts

The optical function of the deep symmetrical convexiclivate purecone fovea of deep-sea notosudids has been examined within the limits of geometrical optics, assuming, inter alia, that the full lens aperture is available for formation of foveal images, that the spherical aberration of the lens is negligible, and that the refractive index of the retina is slightly higher than that of the vitreous. Refraction at the vitreo-retinal boundary in the deep part of the fovea distorts a decentred image of a point source into an elongated spot, almost a double spot, while light from an extended source (residual daylight) will produce uneven illumination within the fovea, with a ring-shaped area of least illumination concentric with the foveal centre. It is suggested that the main optical function of the notosudid fovea is to break the camouflage of mesopelagic animals using small photophores for countershading. The theory implies the angular distribution of the photophore light to differ from that of the residual daylight. The fovea can transform such a difference in angular distribution of light from a small photophore into a difference in illumination of the two parts of the double spot produced by decentred foveal images of point sources.     

Detection of biological particles by the use of circular dichroism measurements improved by scattering theory

Light scattered from optically active spheres was theoretically analyzed for biodetection. The circularly polarized signal of near-forward scattering from circularly dichroic spheres was calculated. Both remote and point biodetection were considered. The analysis included the effect of a circular aperture and beam block at the detector. If the incident light is linearly polarized, a false signal would limit the sensitivity of the biodetector. If the incident light is randomly polarized, shot noise would limit the sensitivity. Suggested improvements to current techniques include a beam block, precise angular measurements, randomly polarized light, index-matching fluid, and larger apertures for large particles.     

Collection efficiency of a single optical fiber in turbid media

If a single optical fiber is used for both delivery and collection of light, two major factors affect the measurement of collected light: (1) the light transport in the medium that describes the amount of light that returns to the fiber and (2) the light coupling to the optical fiber that depends on the angular distribution of photons entering the fiber. We focus on the importance of the latter factor and describe how the efficiency of the coupling depends on the optical properties of the medium. For highly scattering tissues, the efficiency is well predicted by the numerical aperture (NA) of the fiber. For lower scattering, such as in soft tissues, photons arrive at the fiber from deeper depths, and the coupling efficiency could increase twofold to threefold above that predicted by the NA.     

All-sky camera with a concave mirror

The characteristics of an all-sky camera with a concave mirror are analyzed. A differential equation for a concave aspheric mirror with constant angular magnification is derived for the general dependence of the camera image height on the camera field angle. This equation is solved in parametric form for the case of a concave mirror with a constant angular magnification. The explicit equations for the shape of the aspheric mirror are given for some particular values of the angular magnification. Parametric equations of the surface shape for sevenfold angular magnification are developed into a power series that is used to analyze the imaging performance of such a mirror. The performance of the concave aspheric mirror is compared with that of a spherical mirror. The minimal camera-to-mirror distance is determined as a function of the blur allowed and the camera lens aperture. Some characteristics of convex mirrors are also presented for comparison.     

Propagation and backpropagation for ultrasonic wavefront design

Wave backpropagation is a concept that can be used to calculate the excitation signals for an array with programmable transmit waveforms to produce a specified field that has no significant evanescent wave components. This concept can also be used to find the field at a distance away from an aperture based on measurements made in the aperture. For a uniform medium, three methods exist for the calculation of wave propagation and backpropagation: the diffraction integral method, the angular spectrum method, and the shift-and-add method. The boundary conditions that are usually implicitly assumed by these methods are analyzed, and the relationship between these methods are explored. The application of the angular spectrum method to other kinds of boundary conditions is discussed, as is the relationship between wave backpropagation, phase conjugation, and the time-reversal mirror. Wave backpropagation is used, as an example, to calculate the excitation signals for a ring transducer to produce a specified pulsatile plane wave with a limited spatial extent.     

Efficient three-dimensional imaging from a small cylindrical aperture

Small-diameter cylindrical imaging platforms, such as those being considered in the development of in vivo ultrasonic microprobes, pose unique image formation challenges. The curved apertures they provide are incompatible with many of the commonly used frequency-domain synthetic aperture imaging algorithms. At the same time, their frequently small diameters place limits on the available aperture and the angular resolution that may be achieved. We obtain a three-dimensional, frequency-domain imaging algorithm for this geometry by making suitable approximations to the point spread function for wave propagation in cylindrical coordinates and obtaining its Fourier transform by analogy with the equivalent problem in Cartesian coordinates. For the most effective use of aperture, we propose using a focused transducer to place a virtual source a short distance from the probe. The focus is treated as a diverging source by the imaging algorithm, which then forms images on deeper cylindrical shells. This approach retains the simplicity and potential angular resolution of a single element, yet permits full use of the available probe aperture and a higher energy output. Computer simulations and experimental results using wire targets show that this imaging technique attains the resolution limit dictated by the operating wavelength and the transducer characteristics     

Two-dimensional synthetic aperture laser optical feedback imaging using galvanometric scanning

We have improved the resolution of our laser optical feedback imaging (LOFI) setup by using a synthetic aperture (SA) process. We report a two-dimensional (2D) SA LOFI experiment where the unprocessed image (i.e., the classical LOFI image) is obtained point by point, line after line using full 2D galvanometric scanning. The 2D superresolved image is then obtained by successively computing two angular SA operations while a one-dimensional angular synthesis is preceded by a frequency synthesis to obtain a 2D superresolved image conventionally in the synthetic aperture radar (SAR) method and their corresponding laser method called synthetic aperture ladar. The numerical and experimental results are compared.     

Resolution improvement in digital holography by angular and polarization multiplexing

Angular and polarization multiplexing techniques are utilized in both object and reference arms in the digital holographic microscopy system to improve its resolution. The angular multiplexing provides on-axis and off-axis illumination and reference beams with different carrier frequencies. Polarization multiplexing prohibits the occurrence of interference between low and high object spatial frequencies and reference beams. The proposed system does not require special light sources or filtering masks. Experimental results show that the resolution of the synthesized image exceeds the resolution determined by the numerical aperture of the imaging microscope objective.     

Instrumentation to measure the backscattering coefficient b(b) for arbitrary phase functions

A single detector instrument concept that collects scattered light over the full range of backscattering angles is described. Its light collection aperture is designed so as to introduce a sin θ factor into the collection probability. Hence, the instrument is exactly a b(b) meter; it directly measures b(b), not a proxy for it. For an infinitesimal aperture to the detector, the instrument would give b(b) exactly; for a finite aperture (e.g., 1.26 cm(2)), it would typically give b(b) to an accuracy of a few tenths of 1%. The instrumentation itself is as simple as that of the well-known fixed-angle meters-it projects a beam of light into the medium and collects backscattered light with a single detector; the differences are the position of the detector and the shape/orientation of the entrance aperture to the detector.     

Fundamental Parameters Line Profile Fitting in Laboratory Diffractometers

The fundamental parameters approach to line profile fitting uses physically based models to generate the line profile shapes. Fundamental parameters profile fitting (FPPF) has been used to synthesize and fit data from both parallel beam and divergent beam diffractometers. The refined parameters are determined by the diffractometer configuration. In a divergent beam diffractometer these include the angular aperture of the divergence slit, the width and axial length of the receiving slit, the angular apertures of the axial Soller slits, the length and projected width of the x-ray source, the absorption coefficient and axial length of the sample. In a parallel beam system the principal parameters are the angular aperture of the equatorial analyser/Soller slits and the angular apertures of the axial Soller slits. The presence of a monochromator in the beam path is normally accommodated by modifying the wavelength spectrum and/or by changing one or more of the axial divergence parameters. Flat analyzer crystals have been incorporated into FPPF as a Lorentzian shaped angular acceptance function. One of the intrinsic benefits of the fundamental parameters approach is its adaptability any laboratory diffractometer. Good fits can normally be obtained over the whole 20 range without refinement using the known properties of the diffractometer, such as the slit sizes and diffractometer radius, and emission profile.