Geometrical Reasoning in Wave Situations: The case of light diffraction and coherent illumination optical imaging

Abstract

This particular study is part of a research programme on the difficulties encountered by students when learning about wave phenomena in a three‐dimensional medium in the absence or presence of obstacles. It focuses on how students reason in situations in which wave optics need to be used: diffraction of light by an aperture, imaging in the presence of diffraction, and coherent illumination imaging. Paper and pencil questionnaires were designed and two hundred French students (aged 19–23) were questioned after lessons on wave optics. Tendencies towards geometrical reasoning are shown to recur. Students reason at a macroscopic level, following the rays of the incident light, instead of reasoning at an elementary waves level in using the phase concept and the Huygens–Fresnel principle. Consequently, for them, the image of a point source located at infinity is behind the image focus plane of the lens when diffraction has to be considered. Moreover, it is not possible to have the image of the source and of an illuminated diaphragm behind one lens: these images cannot exist simultaneously or are merged. Some remarks are made on the way waves are taught in France and some pedagogical implications are discussed.

Keywords:

• Student
• University
• Reasoning
• Qualitative research
• Physics education
• Wave optics
• Diffraction
• Coherent images

Notes

1. Diffraction of visible light can be observed by the naked eye through apertures of a tenth of a millimetre or less. This is why this is not a phenomenon often observed in everyday life. As the wavelengths of the visible light lie between 400 and 800nm, many textbooks make an error when they write that the diffraction of light occurs through apertures of the same size as the wavelength of the light. This error, also made by students (see Ambrose et al., 1999; Maurines, 1997), indicates a tendency toward an object–notion‐based reasoning which consists of following the wave considered as a whole through the aperture.

2. When the point of observation is near the obstacle, another model has to be used, the electromagnetic model. It is based on two vectors: the electric field and the magnetic field, and on Maxwell equations (Born & Wolf, 1980).

3. Here, the amplitude is the complex amplitude of the function describing the field of the wave.

4. These diagrams are based on rays. There is also another type of diagram based on wave surfaces; see Maurines (1997, 1999a, 1999b).

5. We will consider here that the lens is corrected for the geometrical and chromatic aberrations. The position and the shape of an image with geometrical aberrations can be explained at first approximation by geometrical optics. For the irradiance distribution, it is necessary to use wave optics (Born & Wolf, 1980).

6. The size of the diffraction spot depends on the wavelength of the light, on the diameter of the lens and its focal length, and on the positions of the object, the lens and the image. As the lens always has a finite diameter, diffraction cannot be removed. It can be neglected in the usual observations but must be considered in the case of devices with high magnification (microscope and telescope). It also depends on the desired precision: an image could look like a point when observed with the eyes and to be surrounded with diffraction rings when observed with a magnifying lens.

7. When the lens is corrected for the geometrical and chromatic aberrations, the quality of the image is only limited by the diffraction caused by the finite size of the lens. In that case, the wave surface of the emergent wave is spherical (Goodman, 1972). This is why we draw Huygens sources on an arc of circle centred on the image focus of the lens on the second diagram of Figure 2.

8. More precisely, it depends on the width of coherence of the light wave in the plane where the illuminated object is (Born & Wolf, 1980).

9. The two types of images have the same general shape but not the same details (Goodman, 1972). For example, the coherent image of the edge of an aperture has sharper edges than the incoherent image but shows oscillations of the light intensity. The resolution of the details is complex because the distribution irradiance depends on the distribution of the phase of the object in case of coherent illumination. The incoherent image is not necessarily better than the coherent image. For example, the resolution of the images of two‐point objects obtained with coherent illumination can be the same as with incoherent illumination but can also be better or worse.

10. In the experiment discussed here, two types of diffraction occur: the diffraction by the illuminated object and the diffraction by the lens. In order to simplify the explanation, we only consider the diffraction by the illuminated object.

11. That does not mean that it is impossible to change an image in case of incoherent illumination (Goodman, 1972).

12. Coherent optical imaging is a quite recent field of physics and this is certainly why its teaching raises discussion. Thus, there are discussions of the way to present situations to students where diffraction and interference are observed in the presence of lenses. In his first publication, Colin (1999) and Colin and Viennot (2001) suggest the use of the two models, geometrical optics and wave optics, in the same prototypical situation involving an object illuminated by a plane wave, a lens and a screen. He proposes to use geometrical optics in order to explain the existence of the image of the illuminated object and wave optics in order to explain the existence of the diffracting pattern. Moreover, he suggests ascribing a different meaning to the “rays” drawn from a point source: in the first case, they are a path for energy; in the second case, a path for phase propagation. Our analysis of the physical content and the results of our enquiries lead us to think that this approach raises more difficulties than it resolves. In a later publication (Colin & Viennot, 2002), this point of view has been given up. The accent is only put on the fact that the rays to be considered depend on the point of observation.

13. It has to be noticed that the physics programme of the Grade 12 invited teachers to introduce the history of optics in their classes. So, they could have chosen to present students with the Huygens principle like some authors of textbooks do.

14. For more details, see the overview of our research programme (Maurines, 2001b).

15. Another scheme of reasoning is linked to this trend towards a geometrical reasoning: it consists in considering incoherent point sources and indistinguishable rays so that the image of a diaphragm is not changed in a filtering experiment: details can be found in Maurines (1999c, 2000, 2001a, 2002).

16. This tendency to reason on the basis of a travelling supply also leads students to answer that the centre of the image of a “diffracting” diaphragm is missing when a small stop is placed on the path of light, just as the centre of the image of an object is missing when a small mask is placed on the lens.

17. As students have difficulties to differentiate between interference and diffraction, we should rather use an interference device which is not based on diffraction for the analogy between interference in water and light interference. We prefer to use first a device such as the Meslin lenses and then the double‐slit.

18. There are also discussions on the method to use in teaching. For instance, Colin and Viennot (2002) do not agree with the use of drawings based on the Huygens principle. In fact, it seems to us that the two types of drawings, those based on wave surfaces and those based on “rays”, must be used. Each type of diagram has its advantages and disadvantages (see Maurines, 1997, 2000) but it seems to us that the first type of drawings emphasises the existence of two levels more easily than the second one. However, we must remember that it is impossible to know the phase and the amplitude of a wave in both diagrams.