Optically from a paraxial perspective, the camera is an exceptionally simple instrument. Its purpose is simply to image light from an object located at the infinite conjugate onto the focal plane, where the sensor is located. As such, from a system perspective one might regard the camera as a single lens with the sensor located at the second focal point. This is illustrated in Figure 2.10.
If this system is the essence of simplicity, then the Pinhole Camera, a very early form of camera, takes this further by dispensing with the lens altogether! A pinhole camera relies on a very small system aperture (a pinhole) defining the image quality. In this embodiment of the camera, all rays admitted by the entrance pupil follow closely the chief ray. However, light collection efficiency is low. Whilst in the paraxial approximation, the camera presents itself as a very simple instrument, as indeed early cameras were, the demands of light collection efficiency require the use of a large aperture which results in the breakdown of the paraxial approximation. As we shall see in later chapters, this leads to the creation of significant imperfections, or aberrations, in image formation which can only be combatted by complex multi-element lens designs. Thus, in practice, a modern camera, i.e. its lens, is a relatively complex optical instrument.
Figure 2.10 Basic camera.
In defining the function of the camera, we spoke of the imaging of an object located at infinity. In this context, ‘infinity’ means a substantially greater object distance than the lens focal length. For the traditional 35 mm format photographic camera, a typical standard lens focal length would be 50 mm. The ‘35 mm’ format refers to the film frame size which was 36 mm × 24 mm (horizontal × vertical). As mentioned in Chapter 1, the focal length of the camera lens determines the ‘plate scale’ of the detector, or the field angle subtended per unit displacement of the detector. Overall, for this example, plate scale is 1.15° mm−1. The total field covered by the frame size is ±20° (Horizontal) × ±13.5° (Vertical). ‘Wide angle’ lenses with a shorter focal length lens (e.g. 28 mm) have a larger plate scale and, naturally a wider field angle. By contrast, telephoto lenses with longer focal lengths (e.g. 200 mm), have a smaller plate scale, thus producing a greater magnification, but a smaller field of view.
Modern cameras with silicon detector technology are generally significantly more compact instruments than traditional cameras. For example, a typical digital camera lens might have a focal length of about 8 mm, whereas a mobile phone camera lens might have a focal length of about half of this. The plate scale of a digital camera is thus considerably larger than that of the traditional camera. Overall, as dictated by the imaging requirements, the field of view of a digital camera is similar to its traditional counterpart, although, in practice, equivalent to that of a wide field lens. Therefore, in view of the shorter focal length, the detector size in a digital camera is considerably smaller than that of a traditional film camera, typically a few mm. Ultimately, the miniaturisation of the digital camera is fundamentally driven by the resolution of the detector, with the pixel size of a mobile phone camera being around 1 μm. This is over an order of magnitude superior to the resolution, or ‘grain size’ of a high specification photographic film.
Further Reading
1 Haija, A.I., Numan, M.Z., and Freeman, W.L. (2018). Concise Optics: Concepts, Examples and Problems. Boca Raton: CRC Press. ISBN: 978-1-1381-0702-1.
2 Hecht, E. (2017). Optics, 5e. Harlow: Pearson Education. ISBN: 978-0-1339-7722-6.
3 Keating, M.P. (1988). Geometric, Physical, and Visual Optics. Boston: Butterworths. ISBN: 978-0-7506-7262-7.
4 Kidger, M.J. (2001). Fundamental Optical Design. Bellingham: SPIE. ISBN: 0-81943915-0.
5 Kloos, G. (2007). Matrix Methods for Optical Layout. Bellingham: SPIE. ISBN: 978-0-8194-6780-5.
6 Longhurst, R.S. (1973). Geometrical and Physical Optics, 3e. London: Longmans. ISBN: 0-582-44099-8.
7 Smith, F.G. and Thompson, J.H. (1989). Optics, 2e. New York: Wiley. ISBN: 0-471-91538-1.
3 Monochromatic Aberrations
3.1 Introduction
In the first two chapters, we have been primarily concerned with an idealised representation of geometrical optics involving perfect or Gaussian imaging. This treatment relies upon the paraxial approximation where all rays present a negligible angle with respect to the optical axis. In this situation, all primary optical ray behaviour, such as refraction, reflection, and beam propagation, can be represented in terms of a series of linear relationships involving ray heights and angles. The inevitable consequence of this paraxial approximation and the resultant linear algebra is apparently perfect image formation. However, for significant ray angles, this approximation breaks down and imperfect image formation, or aberration, results. That is to say, a bundle of rays emanating from a single point in object space does not uniquely converge on a single point in image space.
This chapter will focus on monochromatic aberrations only. These aberrations occur where there is departure from ideal paraxial behaviour at a single wavelength. In addition, chromatic aberration can also occur where first order paraxial properties of a system, such as focal length and cardinal point locations, vary with wavelength. This is generally caused by dispersion, or the variation in the refractive index of a material with wavelength. Chromatic aberration will be considered in the next chapter.
A simple scenario is illustrated in Figure 3.1 where a bundle of rays originating from an object located at the infinite conjugate is imaged by a lens. Figure 3.1a presents the situation for perfect imaging and Figure 3.1b illustrates the impact of aberration.
In Figure 3.1b, those rays that are close to the axis are brought to a focus at the paraxial focus. This is the ideal focus. However, those rays that are further from the axis are brought to a focus at a point closer to the lens than the paraxial focus. In fact, the behaviour illustrated in Figure 3.1b is representative of a simple lens; marginal rays are brought to a focus closer to the lens than the chief ray. However, in general terms, the sense of the aberration could be either positive or negative, with the marginal rays coming to a focus either before or after the paraxial focus.
3.2 Breakdown of the Paraxial Approximation and Third Order Aberrations
In formulating perfect or Gaussian imaging we assumed all relationships are linear. For example, Snell's law of refraction was reduced in the following way:
(3.1)
In making the paraxial approximation, we are considering just the first or linear term in the Taylor series. The next logical stage in the process is to consider higher order terms in the Taylor series.