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We developed the pupil astigmatism criteria for correcting the quadratic field-dependent aberrations. These criteria provide an elegant way to determine and correct aberrations that have quadratic field dependence and arbitrary pupil dependence in the same way that the Abbe sine condition is used for aberrations with linear field dependence. Like t...
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... the tangential radius of curvature is When the indices of refraction and the ray directions before and after an optical surface are known, the surface normal therefore the slope can be calculated by the vector form of Snell's law. 4 In Fig. 5 a ray with ray vector n i A ˆ is incident on an optical surface, and the refracted ray has a ray vector n r B ˆ ; Snell's law ...
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Citations
... Braat and Greve [3] designed a system with two aspherical surfaces using the Wasserman-Wolf method [4]. Zhao and Burge [5] developed pupil astigmatism criteria for correcting quadratic field-dependent aberrations and demonstrated their use in the design of two optical systems with three and four aspherical surfaces, respectively. In designing their aspherical surfaces, it was assumed that the slope of the current segment was aligned along the tangent of the preceding segment {line 13 after Eq. (7) of [5]}. ...
... Zhao and Burge [5] developed pupil astigmatism criteria for correcting quadratic field-dependent aberrations and demonstrated their use in the design of two optical systems with three and four aspherical surfaces, respectively. In designing their aspherical surfaces, it was assumed that the slope of the current segment was aligned along the tangent of the preceding segment {line 13 after Eq. (7) of [5]}. In practice, however, this assumption does not always hold, and thus the actual optical performance of the system is therefore lower than the predicted performance. ...
... The most notable benefit of aspherical refractive surfaces is their ability to correct spherical aberration [5]. As shown in Fig. 1, the design of a refractive aspherical surface requires a knowledge of the incidence angle θ i and refraction angle θ i corresponding to the unit directional vectors of the incoming and outgoing rays, i.e., ℓ i−1 and ℓ i , respectively. ...
Snell’s law describes the relationship between the incidence angle and reflection (or refraction) angle of a light ray impinging on the interface between two different isotropic media. In this paper, Snell’s law is used to derive the unit normal vectors of an aspherical surface given a knowledge of the unit directional vectors of the incoming and outgoing rays. The proposed method has important applications in the design and fabrication of aspherical surfaces since the surface normal vectors determine not only the optical performance of the surface but also the cutting tool angles required to machine the surfaces.
... Such systems of axial symmetry have been designed using the Wasserman-Wolf method. 10 A similar approach can be used to design the planesymmetric version of such systems. As the sine conditions for axisymmetric systems make it possible to design rather unconventional aplanatic systems, 11 so the generalized conditions developed here may be used in similar ways to design unconventional well-corrected planesymmetric systems. ...
The Abbe sine condition and the recently developed pupil astigmatism conditions provide a powerful set of relationships for describing imaging systems that are free from aberrations that have linear and quadratic dependence on field, to all orders in the pupil. We have proved both of these conditions and applied them to axisymmetric imaging systems. We now extend our approach to plane-symmetric systems. Still using Hamilton's characteristic functions, we derive the general sine conditions and the pupil astigmatism conditions that describe plane-symmetric systems that are free of all aberrations with linear and quadratic field dependence.
... These criteria can be used together with Fermat's Principle and the Abbe sine condition to design a high-N.A. system that performs perfectly over a moderate field of view. 8 After the first manuscript of this paper was submitted to this journal, a reviewer pointed out that a similar set of criteria was derived by H. Boegehold and M. Herzberger 9 more than 70 years ago. They took a different approach and obtained a more general set of criteria for correction of all the aberrations that are quadratic in field. ...
Aberrations of imaging systems can be described by using a polynomial expansion of the dependence on field position, or the off-axis distance of a point object. On-axis, or zero-order, aberrations can be calculated directly. It is well-known that aberrations with linear field dependence can be calculated and controlled by using the Abbe sine condition, which evaluates only on-axis behavior. We present a new set of relationships that fully describe the aberrations that depend on the second power of the field. A simple set of equations is derived by using Hamilton's characteristic functions and simplified by evaluating astigmatism in the pupil. The equations, which we call the pupil astigmatism criteria, use on-axis behavior to evaluate and control all aberrations with quadratic dependence on the field and arbitrary dependence on the pupil. These relations are explained and are validated by using several specific optical designs.
This study applies a skew ray tracing approach based on a 4×4 homogeneous coordinate transformation matrix and Snell's law to analyze the errors of a ray light path as it passes through a series of optical elements in an asymmetrical optical system. The proposed error analysis methodology considers two principal sources of a light path error, namely: i) the translational errors and the rotational errors which determine the deviation of the light path at each boundary surface, and ii) the differential changes induced in the incident point position and unit directional vector of the refracted/reflected ray as a result of differential changes in the position and unit directional vector of the light source. The validity of the proposed methodology is verified by analyzing the effects of optical errors in a corner cube.
We have discussed the proper gimbal point position in conformal optical system based on the pupil astigmatism criteria. Then, we derive the best position of the aberration correction gimbal point by using aberration correction criteria in rotationally symmetrical system. The result proves the application potential of the aberration correction criteria in conformal optical system design and the dome's image quality has been improved greatly by properly choosing the position of the gimbal point.
The revolution in surface machining, testing and alignment of optical systems open new horizons for lens design. Imaging systems that are generally non axi-symmetric can be made. Freeform optical surfaces can be machined. In this paper we propose a method based on the generalized Sine condition for the design of aplanatic optical systems that are generally made of free form optics. As a proof of concept we present the design of a two mirrors fast aplanatic telescope. We also show the possibility of designing plane symmetric optical systems.
We present numeric methods for designing an high NA axi-symmetric imaging optical system that is corrected for all orders of spherical aberrations, coma and aberrations with 2nd order field dependence.
Aberrations of imaging systems can be described using a polynomial expansion of the dependence on field position. Aberrations on axis and those with linear field dependence can be calculated and controlled using Fermat's principle and the Abbe Sine Condition. We now present a powerful new set of relationships that fully describe the aberrations that depend on the second power of the field. A simple set of equations, derived using Hamilton's characteristic functions, which we call the Pupil Astigmatism Criteria, use on-axis behavior to evaluate and control all aberrations with quadratic field dependence and arbitrary dependence on the pupil. These relations are explained, validated, and applied to design optical systems that are free of all quadratic field dependent aberrations.
We reply to the comments on our paper previous paper. While the results obtained are the same as ours, we hold that, by using homogeneous coordinate notation, our method enables first-order and second-order derivatives of non-axially symmetrical systems to be computed numerically (such as [J. Opt. Soc. Am. A 28, 747 (2011)]), which are necessary for automatic optical design [Appl. Opt. 2, 1209 (1963)].
