Aberration and the Strehl ratio

An exact expression is derived for the Strehl ratio as a function of the minimum root-mean-square aberration value that may produce it. The result is applicable to any aperture and is shown to imply a similar expression for the Strehl ratio as a function of the minimum amplitude range of the aberration.     

Nonaxial Strehl ratio of wavefront coding systems with a cubic phase mask

An approximate expression of the peak position of the point-spread function (PSF) of wavefront coding systems with a cubic phase mask (CPM) is derived and verified by simulation results. An approach called the nonaxial Strehl ratio (NASR) is used to evaluate the performance of wavefront coding systems with defocus aberrations. The characteristics of the NASR are investigated. The relationships between the NASR and the similarity of out-of-focus modulation transfer function (MTF) and recoverability of blurred encoding images are presented, and some useful guidelines are given for the optimization of phase mask parameters according to these relationships.     

Strehl ratio for focusing into biaxially birefringent media

The Strehl ratio for focusing into biaxially birefringent media with small birefringence is investigated. An analytical expression for the Strehl ratio as a function of the polarization and birefringence parameters is derived in the paraxial limit from a generalization of the scalar aberration theory. The expression agrees well with results of a numerical calculation.     

Strehl ratio versus defocus for noncentrally obscured pupils

We discuss from the viewpoint of the Strehl ratio versus defocus, or the normalized axial-irradiance distribution, the influence of decentering the dark mask of an annular pupil. Our treatment, which is valid for pupil apertures with any Fresnel number, permits us to infer that the axial behavior of a noncentrally obscured pupil is equivalent to that of an apodizer with continuous amplitude variations. Hence the Strehl ratio versus defocus of an optical system can be shaped by use of noncentered dark masks that act as continuous gray apodizers. Several numerically evaluated examples are presented.     

Dependence of Strehl ratio on f-number of optical system

Formulas for a minimum of wave aberration variance and a maximum of the Strehl ratio in the optimal image point are derived using the third- and fifth-order aberration theory. Moreover, relations for the calculation of the optimal value of f-number of the optical system were derived, which enabled us to theoretically analyze real optical systems and their image quality. The optimal f-number corresponds to such a value of f-number when the image quality of a real optical system is comparable to an aberration-free optical system. This value may also serve as an auxiliary criterion of the image quality of the optical system, for example, in photography.     

On- and off-axis statistical behavior of adaptive-optics-corrected short-exposure Strehl ratio

Statistical behavior of the adaptive-optics- (AO-) corrected short-exposure point-spread function (PSF) is derived assuming a perfect correction of the phase's low spatial frequencies. Analytical expressions of the Strehl ratio (SR) fluctuations of on- and off-axis short-exposure PSFs are obtained. A theoretical expression of the short SR angular correlation is proposed and used to derive a definition of an anisoplanatic angle for AO-corrected images. Several applications of the analytical expressions are proposed: AO performance characterization, postprocessing imaging, light coupling into fiber, and exoplanet detection from a ground-based telescope.     

Fundamental constraint on the maximum Strehl ratio of bifocal intraocular lenses

Both design experiences and experimental results of a bifocal intraocular lens (BIOL) show that the performances of a BIOL are always considerably worse than those of the diffraction limit. But this significant conclusion was neither quantified nor rigorously proved in previous literatures. To provide theoretical guidance for designing a BIOL, a fundamental constraint on the maximum Strehl ratio of both refractive BIOLs and diffractive BIOLs with arbitrary bifocal intensity ratio is quantified and rigorously proved in this paper. This fundamental constraint restricts the performances of a BIOL considerably worse than those of the diffraction limit.     

Calibration of a deformable mirror and Strehl ratio measurements by use of phase diversity

Calibration experiments with a bimorph mirror are presented. Phase-diversity wave-front sensing is used for measuring the control matrix, nulling wave-front errors in the optical setup, including the mirror, and measuring Strehl ratios and residual higher-order aberrations. The Strehl ratio of the calibrated system is measured to be 0.975, corresponding to 1/40 wave rms in the residual wave front. The conclusion is that a phase-diversity wave-front sensor is easier to install and use than interferometers and can replace them in optical setups for testing adaptive optics systems.     

Zernike representation and Strehl ratio of optical systems with variable numerical aperture

We consider optical systems with variable numerical aperture (NA) on the level of the Zernike coefficients of the correspondingly scalable pupil function. We thus present formulas for the Zernike coefficients and their first two derivatives as a function of the scaling factor ε ≤ 1, and we apply this to the Strehl ratio and its derivatives of NA-reduced optical systems. The formulas for the Zernike coefficients of NA-reduced optical systems are also useful for the forward calculation of point-spread functions and aberration retrieval within the Extended Nijboer–Zernike (ENZ) formalism for optical systems with reduced NA or systems that have a central obstruction. Thus, we retrieve a Gaussian, comatic pupil function on an annular set from the intensity point-spread function in the focal region under high-NA conditions.     

Topological phase in a nonparaxial Gaussian beam

An analysis is made of the structure and evolution of the singularities of a nonparaxial Gaussian beam. It is shown that a Gaussian beam may be represented by a family of straight lines lying on the surface of a hyperboloid and that the wavefront of this beam is a function of a point source situated at a point on the z axis with the imaginary coordinate iz ₀. The argument of this complex function is the topological phase of the beam which characterizes the rotation of the wavefront. The singularities of a nonparaxial Gaussian beam are located in the focal plane and are annular edge dislocations. Dislocation processes near the constriction of the Gaussian beam only occur as a result of aperture diffraction. Pis’ma Zh. Tekh. Fiz. 25, 14–20 (November 26, 1999)     

Determination of waist parameters of a Gaussian beam

A simple method is presented for determining the waist position and the waist size of a Gaussian beam from measured spot sizes. The method does not require any least-squares process, and substitution of measured spot sizes directly into the formulas gives the waist parameters. Only few data are required to determine the parameters accurately as long as measured spot sizes contain small errors.     

Diffraction of a Gaussian beam by a logarithmic axicon

We derive an explicit analytical relationship to describe the axial light intensity when a Gaussian beam is diffracted by the logarithmic axicon (LA). An evaluation formula for the effective radius of the diffraction pattern that we deduce shows the said radius to be in inverse proportion to the LA "force" parameter. The finite-difference time-domain-based simulation has shown that using the LA makes it possible to go beyond the diffraction limit: in the LA vicinity, the FWHM of the light beam can be as small as one fifth of the illumination wavelength.     

Transmission of a Gaussian beam through a Fabry-Perot interferometer

The transmission of a Gaussian beam through a Fabry-Perot interferometer (FPI) has been investigated. The equation for the electric field of the transmitted beam was derived and then the transmitted irradiance I(t) was numerically calculated for different selected parameters of both the FPI and the beam. The results show that the energy profile of the transmitted beam has been distorted to different degrees depending on the various parameters of the Gaussian beam and the FPI. Moreover the results show that the positions of the peaks of the transmitted beam are shifted, especially for intermediate waists for which the arctan term is nonlinear. The results also show that for nonnormal incidence successive transmitted beams are spatially separated and are not interfering appreciably with each other.     

Decentered elliptical Gaussian beam

A new kind of laser beam, called a decentered elliptical Gaussian beam (DEGB), is defined by a tensor method. The propagation formula for a DEGB passing through an axially nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be reduced to the formula for a fundamental elliptical Gaussian beam and a decentered Gaussian beam under certain conditions. As an example application of the derived formula, the propagation characteristics of a DEGB in free space are calculated and discussed. As another example we study the properties of a generalized laser beam array constructed by use of a DEGB as the fundamental mode.     

Atmospheric optical communication with a Gaussian Schell beam

We consider a wireless optical communication link in which the laser source is a Gaussian Schell beam. The effects of atmospheric turbulence strength and degree of source spatial coherence on aperture averaging and average bit error rate are examined. To accomplish this, we have derived analytic expressions for the spatial covariance of irradiance fluctuations and log-intensity variance for a Gaussian beam of any degree of coherence in the weak fluctuation regime. When spatial coherence of the transmitted source beam is reduced, intensity fluctuations (scintillations) decrease, leading to a significant reduction in the bit error rate of the optical communication link. We have also identified an enhanced aperture-averaging effect that occurs in tightly focused coherent Gaussian beams and in collimated and slightly divergent partially coherent beams. The expressions derived provide a useful design tool for selecting the optimal transmitter beam size, receiver aperture size, beam spatial coherence, transmitter focusing, etc., for the anticipated atmospheric channel conditions.     

Psychometrically appropriate assessment of afocal optics by measurement of the Strehl intensity ratio

Several different performance criteria have been proposed for assessing the quality of visual afocal sights. Earlier research by one of the authors (Haig) has shown that a high degree of correlation exists between a subjective assessment of performance and the Strehl intensity ratio of the optical system. We discuss some of the problems in choosing an objective performance criterion for visual sights and describe equipment that has been developed for measuring the line Strehl ratio of binoculars, both on and off axes. The equipment can be modified for testing other types of visual sight. It can also be used for measuring several additional performance parameters such as the modulation transfer function, transmission, and field curvature.     

Gaussian content as a laser beam quality parameter

We propose the Gaussian content (GC) as an optional quality parameter for the characterization of laser beams. It is defined as the overlap integral of a given field with an optimally defined Gaussian. The definition is especially suited for applications where coherence properties are targeted. Mathematical definitions and basic calculation procedures are given along with results for basic beam profiles. The coherent combination of an array of laser beams and the optimal coupling between a diode laser and a single-mode fiber are elaborated as application examples. The measurement of the GC and its conservation upon propagation are experimentally confirmed.