Optical Wavefronts and Their Aberrations

If we consider a pencil of rays issuing from a point on the axis of a symmetrical optical system ( i. e ., a system of refracting spherical surfaces, the centres of which lie on a straight line called the axis of the system), it is well known that, if the pencil be a thin one, of which the mean ray is along the axis, the first approximation to the emergent pencil is another punctual pencil, of which the rays pass through an image point, also situated on the axis. The general method of treatment of such image points, which are usually referred to as “geometrical” images, is due to Gauss, and is developed in any text book of Geometrical Optics. When, however, the pencil considered is one of finite aperture, the outlying rays do not, after emergence, pass through the Gaussian image point, nor do they have the inclination assigned to them by the Gaussian calculation. The emergent rays lying in any one axial plane touch an envelope or caustic, which has one cusp at the Gaussian image, with the axis as proper tangent. The intercepts of any given emergent ray upon the axis and the image plane, measured from the Gaussian image, are known as the longitudinal and transverse spherical aberrations of that ray.

As we know, wave aberration is the difference between the ideal wave and the real wave. One of the important characteristics of ordinary optical system images is the condition of equal optical path distance. If all light from a point A passing through the optical system focuses at A', we can say that point A' is the ideal image of object point A and that the optical path distances of all rays from A to A' are the same. When there is aberration in the optical system, wave aberration is the difference of geometric path distances multiplied by the index. In figure 1 wave front P from point B passing through the optical system forms a new wave surface pp'. Generally we use the intersection point ' of the principal ray and the ideal imaginary surface as the center of the ideal reference spherical surface so PP is the wave aberration

We present a complete and precise quantitative characterization of the different process steps used in an elastomeric inverse moulding and vacuum casting technique. We use the latter replication technique to fabricate concave replicas from an array of convex thermal reflow microlenses. During the inverse elastomeric moulding we obtain a secondary silicone mould of the original silicone mould in which the master component is embedded. Using vacuum casting, we are then able to cast out of the second mould several optical transparent poly-urethane arrays of concave refractive microlenses. We select ten particular representative microlenses on the original, the silicone moulds and replica sample and quantitatively characterize and statistically compare them during the various fabrication steps. For this purpose, we use several state-of-the-art and ultra-precise characterization tools such as a stereo microscope, a stylus surface profilometer, a non-contact optical profilometer, a Mach–Zehnder interferometer, a Twyman–Green interferometer and an atomic force microscope to compare various microlens parameters such as the lens height, the diameter, the paraxial focal length, the radius of curvature, the Strehl ratio, the peak-to-valley and the root-mean-square wave aberrations and the surface roughness. When appropriate, the microlens parameter under test is measured with several different measuring tools to check for consistency in the measurement data. Although none of the lens samples shows diffraction-limited performance, we prove that the obtained replicated arrays of concave microlenses exhibit sufficiently low surface roughness and sufficiently high lens quality for various imaging applications.

In this paper, a simple and practical method is proposed. The fringe patterns are generated on a computer display (for example, a LCD or TFT monitor) and the images refracted via the tested optical component are recorded by a CCD camera. The intensity patterns in these images are distorted because of the tested object. Using the well-known phase-shifting algorithms, the alterant phase caused by the tested object can be precisely obtained. And the aberration of each refracted ray with the reference sphere can be obtained by use of geometry principle. Then the wave front aberration is restored with Zernike polynomial fitting algorithm according to the relation between wave aberration and ray aberration. The technique allows the measurement for high aberrated wave fronts, owing to its ability to measure large ray slope changes, and the use of a computer display generation leads flexible adjustment of period and direction of patterns and accurate phase shift. Compared with other techniques, this technique is simpler, cheaper and more flexible.

Applications for concave mirrors that are off axis portions of surfaces of revolution are increasing due to continued improvements in fabrication techniques and an accompanying reduction in cost. For imaging applications, it is desirable to know the effect that object position has on the resulting image position and quality and how these image characteristics depend on the surface geometry. We have addressed these topics for the specific case of an ellipsoid of revolution. We analyze the imaging properties of the ellipsoid by considering that a point source is located at an arbitrary position in the vicinity of one focus. The point source is modeled as emitting a cone of rays symmetrically distributed about a specified direction. The intersection of this diverging cone with the ellipsoidal mirror defines the region of the ellipsoid that images the point source. Ray aberrations are evaluated by considering the transverse ray intercepts at a target plane that is oriented normal to the converging cone of rays. Wave aberrations are determined in the traditional manner by considering the optical path difference between the convergent wavefront and a spherical reference wave centered at the paraxial image location. We will show how aberrations vary as different portions of the ellipsoid are involved in the imaging process and also illustrate the effect on the geometrical image produced by a tilt of the target plane.

Spot diagrams are the intersection coordinate points in the image plane of the incoming rays from a point object. When the point object is imagined to be at infinity, the spot diagram can be considered as a representation of the point spread function (PSF) of the imaging lens. The performance of an imaging system for various applications can be analyzed with the help of the spot diagrams. In this paper, we will present our work using exact ray tracing that can be employed to compute the spot diagrams of both scalar and vector beams such as radially polarized beam, azimuthally polarized beam, etc.

In optical design, we trace rays from a point object through a system to determine the aberrations of the wavefront at its exit pupil. In optical testing, we determine the aberrations of a system or an element interferometrically. In both cases, we obtain aberration numbers at an array of points. We can calculate the PSF or other associated image quality measures from these numbers. We can also calculate the aberration variance, which, in turn, gives some idea of the image quality. However, such measures do not shed light on the content of the aberration function. To understand the nature of this function, we want to know the amount of certain familiar aberrations discussed in Chapter 2 that are present, so that perhaps something can be done about them in improving the design or the system under test. A straightforward approach to determine the content of an aberration function is to decompose it into a set of orthogonal polynomials that represent balanced classical aberrations and include wavefront defocus and tilt. The Zernike circle polynomials are in widespread use for this purpose for systems with circular pupils. These polynomials are unique in the sense that they are not only orthogonal across a unit circle, but they also represent balanced aberrations yielding minimum variance, as we shall see in Chapter 4. In this chapter, we discuss the basic properties of the orthogonal polynomials. We also describe the Gram–Schmidt orthogonalization process for obtaining orthogonal polynomials over one domain from those that are orthogonal over another domain, e.g., obtaining polynomials that are orthogonal over an annular pupil from the circle polynomials. We emphasize the use of orthonormal polynomials so that their coefficients represent the standard deviations of the corresponding polynomial aberration terms.

According to sparrow criterion, two point sources are said to be resolved when the second derivative of the total distribution of irradiance in the diffracted image of the two- points, vanishes at a point midway between the respective Gaussian image points. When this condition is satisfied, the distance between the two object points gives the sparrow limit of resolution. In the resultant irradiance distribution curve, the central dip just vanishes when the resolution begins. Initially the dip is at the middle point between the two Gaussian image points when the separation between two point object is larger than the critical limit. As the distance between the points is reduced, the dip reduces in its upward concavity and it just vanishes at a particular separation of the two point objects.

So far, we have considered refracting imaging systems: a spherical refracting surface in Chapter 1, a thin lens in Chapter 2, and a plane-parallel plate in Chapter 3. Now we consider the imaging properties of a spherical reflecting surface, i.e., a spherical mirror. These properties can be obtained in a manner similar to that for a spherical refracting surface. However, the geometry of the problem is different since now a ray incident on the surface is reflected back into the same medium containing the incident ray, instead of being refracted into another medium. Accordingly, it is instructive to draw object and image rays and not blindly use the imaging and aberration relations appropriate for a reflecting surface. In this chapter, we give the relations describing the primary aberrations of a spherical mirror for an arbitrary position of the aperture stop. These relations are applied to specific cases, one when the aperture stop is located at the mirror and the other when it is located at its center of curvature. It is shown that, in the first case, field curvature and distortion are zero. In the second case, coma, astigmatism, and distortion are zero. A numerical problem illustrates these results. <strong>4.2 PRIMARY ABERRATION FUNCTION</strong> Consider an imaging system consisting of a spherical mirror of radius of curvature <i>R</i> and focal length <i>f</i>&#x2032;. Let the aperture stop and the corresponding exit pupil of the system be located as indicated in Figure 4-1. The line joining the center of curvature <i>C</i> of the mirror and the center of the aperture stop (and, therefore, the center <i>O</i> of the exit pupil) defines the optical axis of the system. Consider an object lying at a distance <i>S</i> from the vertex <i>V</i>₀ of the mirror. Let the height of an object point <i>P</i> from the optical axis be <i>h</i>. The distance <i>S</i>&#x2032; and the height <i>h</i>&#x2032; of its Gaussian image <i>P</i>&#x2032; are given by (4-1) and (4-2a) (4-2b) respectively, where <i>M</i> is the magnification of the image.

The departures from sphericity of a wavefront can be specified by either the wave aberration function or the transversal ray aberrations. Another possibility of specifying the departures from sphericity is by considering the envelope of a set of rays. Here we indicate that the analytical form of the caustic can be obtained readily from the wave aberration function or from the transversal ray aberrations. We show that the analytical form of the caustic and the parametric expressions for ray tracing are the particular solution and the general solution, respectively, of Clairaut's nonlinear differential equation.

Based on the Extended Nijboer-Zernike theory,the effect of different amplitudes for exit pupils on the image intensity in the focal plane was analyzed.A novel approach was applied to testing the wavefront according to the actual condition of the amplitude in the exit pupil,which can help eliminating the error caused by the nonuniformity illuminated pupil and the Fast Fourier Transform in the original phase retrieval algorithms.A testing experiment was performed on an imaging optical system,and obtained results show that the tested wave fronts in the exit pupil of a camera lens are 0.196 5λ in PV and 0.022 4λ in RMS(the testing wavelength λ is 632.8 nm).The aberrations in the wavefront are mainly astigmatism,coma and high order astigmatism.Furthermore,the approach can also be used to analyze the amplitude in the exit pupil of a camera lens and calculate the light intensity distribution on other focal planes.The experiment proves this approach available.

Various authors have presented the aberration function of an optical system as a power series expansion with respect to the ray coordinates in the exit pupil and the coordinates of the intersection point with the image field of the optical system. In practical applications, for reasons of efficiency and accuracy, an expansion with the aid of orthogonal polynomials is preferred for which, since the 1980s, orthogonal Zernike polynomials have become the reference. In the literature, some conversion schemes of power series coefficients to coefficients for the corresponding Zernike polynomial expansion have been given. In this paper we present an analytic solution for the conversion problem from a power series expansion in three or four dimensions to a double Zernike polynomial expansion. The solution pertains to a general optical system with four independent pupil and field coordinates and to a system with rotational symmetry in which case three independent coordinate combinations have to be considered. The conversion of the coefficients is analytically in closed form and the result is independent of a specific sampling scheme or sampling density as this is the case for the commonly used least squares fitting techniques. Computation schemes are given that allow the evaluation of coefficients of arbitrarily high order in pupil and field coordinates.

A new device used for tilted wavefront correction in adaptive optical cavity of CO2 laser was presented. The devices arecomposed of a pair of microlens arrays in the Galilean telescopic geometry, which is designed using binary opticaltechnology. The microlens arrays pair is used for correction one order aberration of the laser resonator. The analyses showthat the device has the advantages.Keywords: adaptive optics, laser resonator, tilted wavefront, aberration correction, binary optics

Taking a certain type of photoelectric system of armored vehicle as an example, the influence of angular vibration on the optical system image quality is analyzed. Firstly, the paper introduces the characteristics of photovoltaic system and the basic theory of imaging; Then it uses the ZEMAX to build optical system model; finally, it simulates the MTF and Spot Diagram as the change of lens angular. By making quantitative analysis of the photoelectric system influence on imaging quality of lens deflection angular, we get the relationship between geometry maximum radius and mean root square radius with the lens deflection angular, and evaluate the effects of angular vibration on image quality synthetically, which has important reference value to the design of photoelectric system.

In this paper, a new method used to calculate the Zernike coefficients of wave aberration from ray aberration and from finite shearing of wavefront is proposed. The simulated results are compared with the least squares fitting method, and indicate that the Zernike coefficients of wave aberration can be calculated as accurately with the new method as with the least squares fitting method, but with less computation and operation.© (1994) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.