Although there are only two commands in OSLO that specifically display paraxial ray data, paraxial_trace and paraxial_constants, the data that are displayed often provide the most important performance evaluation of an optical system. It is noteworthy that in a system with aberrations, the paraxial behavior provides a first order description of its performance, and in fact the goal of the design process is usually to make aberrated rays go where the paraxial rays go, not the other way around.
Paraxial optics is only applicable to centered systems (i.e. systems that have an optical axis). If the system contains tilted and decentered elements, the paraxial routines will simply ignore the existence of the tilt and decenter data. Similarly, if a system contains other special data, such as holographic or gradient index data, such data will be ignored by the paraxial ray trace.
True paraxial rays travel infinitesimally close to the optical axis. Formal paraxial rays are displaced from the optical axis by large amounts, but are refracted at (imaginary) tangent planes to surfaces, as shown below.
Paraxial optics is often considered to be linear, in the sense that if two base rays (e.g. the axial and chief rays) are traced through a system, the trajectory of any other paraxial ray can be predicted by writing its ray height and slope as combinations of corresponding quantities for the two base rays - and the expansion coefficients are the same at any point in the system.
A prime feature of paraxial ray tracing is that the rays never fail, i.e. there is always a solution of the ray height and slope, regardless of the curvature of a surface. In OSLO, the paraxial ray height, ray slope, and angle of incidence are described by PY, PU, and PI for the axial ray, and PYC, PUC, and PIC for the chief ray. The axial ray, also called a-ray or the marginal ray, is a ray from the center of the object surface through the edge of the paraxial entrance pupil. The chief ray, also called b-ray or the principal ray, is a ray from the edge of the object surface through the center of the entrance pupil. The paraxial ray data is computed and displayed using the paraxial_trace (pxt) command.
OSLO computes seven numbers, labeled paraxial constants, when the paraxial_constants (pxc) command is given. It important to understand the meaning of these numbers. For this purpose, a modified version of the demotrip lens can be used.
magnification = -0.5, Gaussian image height = 25, object NA = 0.5
Next, open the Surface data spreadsheet and change thickness 6 to 60.0.
*PARAXIAL TRACE
SRF PY PU PI PYC PUC PIC
0 -- 0.577350 0.577350 -50.000000 0.340048 0.340048
1 78.849791 -1.064379 4.287929 -3.559045 0.273978 0.172563
2 76.721034 -2.024752 -1.547965 -3.011089 0.455732 0.292958
3 64.572522 -0.036239 -5.213518 -0.276697 0.276697 0.469396
4 64.536282 2.003206 3.307610 -- 0.447307 0.276697
5 76.555519 1.028723 2.545192 2.683840 0.268771 0.466307
6 78.612965 -1.154701 -3.519323 3.221382 0.319894 0.082402
7 9.330932 -1.154701 -1.154701 22.414996 0.319894 0.319894
*PARAXIAL CONSTANTS
Effective focal length: 50.000541 Lateral magnification: -0.500000
Numerical aperture: 1.000000 Gaussian image height: 25.000000
Working F-number: 0.500000 Petzval radius: -149.381547
Lagrange invariant: -28.867513
The data show several interesting features. First, note that the height of the axial ray on the first surface is 78.8 mm, far greater than the radius of curvature of the surface. It would not be possible to trace any real rays through this system. Since all the data is paraxial, however, this is not important.
The effective focal length is the distance from the second principal point to the focal point, according to the customary definition.
The numerical aperture of the system is 1.0, according to the input specification of the object NA as 0.5, and the magnification as -.5. However, there is a subtlety here, caused by the fact that the numerical aperture is defined in terms of the sine of the axial ray angle, whereas the paraxial axial ray slope is defined in terms of the tangent of the angle. Thus when the object NA is .5, the axial ray angle in object space is 30 degrees, so the paraxial ray slope is tan(30) = 0.57735.
In order to calculate the numerical aperture in image space, it is necessary to make some assumptions about the optical system. OSLO assumes that the system is aplanatic. Then the numerical aperture in image space must be the object NA divided by the magnification, or 1.0 in the present case. This implies that the real ray angle emerging from the system would be 90 degrees.
Because of the way OSLO defines the paraxial constants, it is possible to set up a system having a numerical aperture (in air) greater than unity. This implies that the system cannot be aplanatic.
The Working f-number is defined to be 1/(2*NA). For the same reasons noted above, it is possible to set up a (non-aplanatic) system having a Working f-number less than 0.5.
It is apparent from the above discussion that there are basic differences between paraxial optics and real-ray optics. The paraxial model does not correspond to a perfect lens, but is only a first-order approximation to a perfect lens.
The Gaussian image height is the height of the chief ray on the paraxial image plane. If the system is out of focus, as is the current system, then this is not equal to the chief ray height on the actual image surface.
The Petzval radius is computed from the surface curvatures and refractive indices. It is not strictly a paraxial constant. The Petzval curvature (1.0/PTZRAD) can be used as an indicator of the complexity of the lens. The ratio of the Petzval radius to the focal length is sometimes called the Petzval ratio. It is easy to show that the Petzval ratio of a single thin lens is equal to the refractive index of the lens.
The above paraxial constants are computed for focal systems. For an afocal system, the angular magnification, the eye relief, the paraxial invariant, and the Petzval radius are computed.
Paraxial data can be computed in any wavelength by setting the wavelength number to the desired value before giving the paraxial commands. In this connection, it should be noted that a lens is always set up using wavelength 1, so if a solve, for example, is set on a curvature or thickness, the solve condition will only be satisfied for wavelength 1.
The paraxial data for a system that contains special data can be obtained by tracing an exact ray at a very small fractional aperture (e.g. FY = 0.0001) and scaling the resulting ray data. OSLO contains two star commands (*pxt and *pxc) that do this. These commands are included on the User >> General Analysis menu as First-order raytrace and First-order constants.
Next Page | Previous Page | Chapter Summary | Table of Contents
Copyright © 1997 Sinclair Optics Inc. All rights reserved.
Page last updated 19970601