Photograph of a retinoscope and schematic layout of the elements that compose it. Light coming from the source S reaches the retina of the examined eye. 

Contexts in source publication

Context 1

... images having the characteristics of spherical aberration. Three years ͑ and a lot of money ͒ were necessary to repair the telescope, which is still giving the best images of the universe ever seen. An- other example where spherical aberration is crucial arises in ophthalmology, 6 where advanced surgical procedures for re- ducing refractive errors are growing in popularity. With the development of sophisticated computer-controlled laser scanning and delivering systems, surgical techniques are be- coming more and more precise. However, if aberrations are not taken into account and refractive surgery is performed based on paraxial assumptions, the cornea takes an aspheric shape, which causes a dramatic increase ͑ up to ten times its natural level ͒ in the spherical aberration of the eye. The con- sequence is that visual performance, especially in a darkened environment, is highly affected. Cases like the ones mentioned, followed by the theory, can be complemented by some simple experiments on aberration. 7 These experiments, which are the natural way to verify both the failure of paraxial optics and the importance of aberrations in optical systems, result in a stimulating experience for the students. Although spherical and chromatic aberrations have been the subject of several educational papers during the past few years, 8 –11 other kinds of aberrations, such as astigmatism, have not been considered. Furthermore, to our knowledge there is no single optical arrangement that can be adapted readily to study several kinds of aberrations in the context of teaching. In this note we describe an experiment that is more versatile than previous setups and is an easily implementable adaptation of the Foucault ͑ or knife edge ͒ test. The Foucault test 12 was originally developed to assess concave mirrors for astronomical use and is still today one of the most widely used methods for the evaluation of the aberrations of optical systems. The technique is usually performed with a device that is especially designed for each application. Based on the same physical principles as the Foucault test ͑ although independently developed ͒ , the retino- scope is a simple and rather inexpensive optometric instrument mainly used to objectively measure the refractive state of the eye. 13 In the experiments reported here, a retinoscope is used to measure both longitudinal spherical aberration ͑ LSA ͒ and astigmatism of a lens. In Secs. II and III we revisit the fundamentals of the Foucault test and the retinoscopic technique, and show that, apart from minor differ- ences, both methods are based on the same optical principles. In Sec. IV we present the optical setup and the experimental results. In the Foucault test a collimated light beam is used to illuminate the optical element to be examined ͑ see Fig. 1 ͒ . A knife edge that can be displaced both axially and transversely is placed in the focal region, perpendicular to the optical axis. Depending on the position of the knife edge, different parts of the illuminating beam are blocked out. An observa- tional system—the eye, for example—is focused onto the exit pupil of the optical element. The form of the observed shadow pattern depends on several factors, namely the distance between the focus and the knife edge, the transverse position of the knife edge, and the aberrations of the optical element. 12 For an aberration-free optical element, the shadow pattern consists of a dark and a bright region separated by a straight edge parallel to the knife edge. If the knife edge is located behind the focus ͓ see Fig. 1 ͑ a ͔͒ and is transversely displaced, the observed dark region moves in the opposite direction of the knife edge. On the contrary, when the knife edge is placed in front of the focus, the dark region moves in the same direction as the knife edge ͓ see Fig. 1 ͑ b ͔͒ . Finally, if the knife edge is located at the focal plane of the system and is transversely displaced, the observed pattern changes suddenly from bright to dark without any apparent motion ͓ see Fig. 1 ͑ c ͔͒ . Thus, the location of the focus can be determined accurately by observation of the shadow patterns. To make the observed pattern as bright as possible, the illumination beam is formed in practice using an illumination slit, parallel to the knife edge, instead of a point source. 12 For an aberrated lens, the shadow-pattern structures are no longer divided by a straight edge. Consequently, it is no longer possible to see a completely bright or dark pattern at a given axial position of the knife edge. In addition, each kind of aberration produces a different pattern ͑ see Fig. 2 ͒ . The retinoscope is a simple self-luminous hand-held instrument used by optometrists in standard clinical procedures to measure the refractive state of the eye. A standard, commercially available streak retinoscope has a light source ( S ), consisting of a straight-filament bulb that is located in the handle of the instrument ͑ see Fig. 3 ͒ . The light from the source is reflected in a beam splitter toward the patient’s eye. Then, a real, and, in general, defocused image of the glowing filament is formed at the retina. This stripe-like spot of light acts as a secondary source of light, playing the role of the illumination slit in the Foucault knife edge. The observer ͑ the retinoscopist ͒ views through the retinoscope pupil ( P ) light that is diffusely reflected by the retina and emerges from the pupil of the eye. In retinoscopy the observed pattern at the patient’s pupil is called ‘‘the reflex.’’ Tilting the retinoscope ͑ for example, about an axis perpendicular to the plane of Fig. 3 ͒ causes the spot to shift its eccentricity from the pupil of the retinoscope and, depending on the refractive state of the eye, to change the appearance of the reflex, or equivalently, the appearance of the shadow pattern that surrounds it. Simi- lar to the Focault test, the form of the observed pattern depends on the location of the focus of the eye, that is, on the location of its far point O R in the object space relative to the axial position of the retinoscope. 16 In spite of not being a straight edge, the edge of the retinoscope pupil acts as the knife edge, because it blocks out the light coming from the secondary source at the retina. Let us consider the particular case of a myopic eye having its far point O R in front of the retinoscope ͓ see Fig. 4 ͑ a ͔͒ . In this case, a reflex pattern is observed through the pupil of the retinoscope consisting of a central bright fringe surrounded by dark regions at the eye pupil. When the instrument is tilted up, the spot A also moves up, and its back image O R moves down. Hence, the reflex is seen as moving ‘‘against’’ the movement of the instrument ͓ see Fig. 4 ͑ b ͔͒ . This situation is equivalent to the one represented in Fig. 1 ͑ a ͒ . On the other hand, if the far point is behind the retinoscope ͓ see Fig. 4 ͑ c ͔͒ , when the retinoscope is tilted up, the observed reflex moves in the same direction. In this case, the movement of the reflex is called ‘‘direct.’’ The analogous situation for the Focault test is the one represented in Fig. 1 ͑ b ͒ . Finally, as happens with the knife edge in Fig. 1 c , when O R coincides with the retinoscope pupil and the retinoscope is tilted, the observed pattern suddenly switches from bright to dark without any apparent motion ͓ see Fig. 4 ͑ d ͔͒ . In optometry, this situation is known as the ‘‘neutralization’’ of the reflex movement, and is the basis of the retinoscopic technique. Thus neutralization is obtained when the object plane ͑ the retina of the eye ͒ is conjugate to the retinoscope pupil plane. This situation can be achieved in practice either by placing trial lenses in front of the eye, or by moving the retinoscope back and forth until the proper distance to the eye is attained. The values of the focal length of the trial lens and the eye– retinoscope distance are sufficient to determine the refractive state of the eye. 13 The retinoscope is easily operated. In fact, the technique simply consists of the observation of the reflex through the retinoscope pupil while swinging the instrument until neutralization is obtained, that is, until the point is reached where no apparent movement of the reflex is seen. In summary, the neutralization obtained with a retinoscope is equivalent to the location of the focus of an optical element with the Foucault knife test. Probably because retinoscopy and the Foucault test do not share a common origin, it is not widely recognized that both techniques are in fact based on the same physical principle. Thus, the retinoscope can be used for testing the aberrations of conventional optical elements, as we will show next. Consider the schematic layout in Fig. 5 in which rays pro- ceeding from the axial point of the screen O impinge on a spherically aberrated lens at different heights h . The inter- section of the refracted rays with the optical axis depends explicitly on the value of h . For a given pupil radius, the longitudinal spherical aberration ͑ LSA ͒ is defined as the axial distance between the paraxial image, O Ј , and the image given by the outmost rays O Ј h , that is, LSA ϭ s Ј Ϫ s h Ј . ͑ see Fig. 5 ͒ . The LSA is related to the geometrical parameters of the lens through the following equation ͑ see, for example, Refs. 1, 2, or 8 ͒ : L S ϭ s 1 Ϫ s 1 ϭ 8 h f 2 3 n ͑ n 1 Ϫ 1 ͒ ͫ n n ϩ Ϫ 2 1 q 2 ϩ 4 ͑ n ϩ 1 ͒ ...

Context 2

... takes an aspheric shape, which causes a dramatic increase ͑ up to ten times its natural level ͒ in the spherical aberration of the eye. The con- sequence is that visual performance, especially in a darkened environment, is highly affected. Cases like the ones mentioned, followed by the theory, can be complemented by some simple experiments on aberration. 7 These experiments, which are the natural way to verify both the failure of paraxial optics and the importance of aberrations in optical systems, result in a stimulating experience for the students. Although spherical and chromatic aberrations have been the subject of several educational papers during the past few years, 8 –11 other kinds of aberrations, such as astigmatism, have not been considered. Furthermore, to our knowledge there is no single optical arrangement that can be adapted readily to study several kinds of aberrations in the context of teaching. In this note we describe an experiment that is more versatile than previous setups and is an easily implementable adaptation of the Foucault ͑ or knife edge ͒ test. The Foucault test 12 was originally developed to assess concave mirrors for astronomical use and is still today one of the most widely used methods for the evaluation of the aberrations of optical systems. The technique is usually performed with a device that is especially designed for each application. Based on the same physical principles as the Foucault test ͑ although independently developed ͒ , the retino- scope is a simple and rather inexpensive optometric instrument mainly used to objectively measure the refractive state of the eye. 13 In the experiments reported here, a retinoscope is used to measure both longitudinal spherical aberration ͑ LSA ͒ and astigmatism of a lens. In Secs. II and III we revisit the fundamentals of the Foucault test and the retinoscopic technique, and show that, apart from minor differ- ences, both methods are based on the same optical principles. In Sec. IV we present the optical setup and the experimental results. In the Foucault test a collimated light beam is used to illuminate the optical element to be examined ͑ see Fig. 1 ͒ . A knife edge that can be displaced both axially and transversely is placed in the focal region, perpendicular to the optical axis. Depending on the position of the knife edge, different parts of the illuminating beam are blocked out. An observa- tional system—the eye, for example—is focused onto the exit pupil of the optical element. The form of the observed shadow pattern depends on several factors, namely the distance between the focus and the knife edge, the transverse position of the knife edge, and the aberrations of the optical element. 12 For an aberration-free optical element, the shadow pattern consists of a dark and a bright region separated by a straight edge parallel to the knife edge. If the knife edge is located behind the focus ͓ see Fig. 1 ͑ a ͔͒ and is transversely displaced, the observed dark region moves in the opposite direction of the knife edge. On the contrary, when the knife edge is placed in front of the focus, the dark region moves in the same direction as the knife edge ͓ see Fig. 1 ͑ b ͔͒ . Finally, if the knife edge is located at the focal plane of the system and is transversely displaced, the observed pattern changes suddenly from bright to dark without any apparent motion ͓ see Fig. 1 ͑ c ͔͒ . Thus, the location of the focus can be determined accurately by observation of the shadow patterns. To make the observed pattern as bright as possible, the illumination beam is formed in practice using an illumination slit, parallel to the knife edge, instead of a point source. 12 For an aberrated lens, the shadow-pattern structures are no longer divided by a straight edge. Consequently, it is no longer possible to see a completely bright or dark pattern at a given axial position of the knife edge. In addition, each kind of aberration produces a different pattern ͑ see Fig. 2 ͒ . The retinoscope is a simple self-luminous hand-held instrument used by optometrists in standard clinical procedures to measure the refractive state of the eye. A standard, commercially available streak retinoscope has a light source ( S ), consisting of a straight-filament bulb that is located in the handle of the instrument ͑ see Fig. 3 ͒ . The light from the source is reflected in a beam splitter toward the patient’s eye. Then, a real, and, in general, defocused image of the glowing filament is formed at the retina. This stripe-like spot of light acts as a secondary source of light, playing the role of the illumination slit in the Foucault knife edge. The observer ͑ the retinoscopist ͒ views through the retinoscope pupil ( P ) light that is diffusely reflected by the retina and emerges from the pupil of the eye. In retinoscopy the observed pattern at the patient’s pupil is called ‘‘the reflex.’’ Tilting the retinoscope ͑ for example, about an axis perpendicular to the plane of Fig. 3 ͒ causes the spot to shift its eccentricity from the pupil of the retinoscope and, depending on the refractive state of the eye, to change the appearance of the reflex, or equivalently, the appearance of the shadow pattern that surrounds it. Simi- lar to the Focault test, the form of the observed pattern depends on the location of the focus of the eye, that is, on the location of its far point O R in the object space relative to the axial position of the retinoscope. 16 In spite of not being a straight edge, the edge of the retinoscope pupil acts as the knife edge, because it blocks out the light coming from the secondary source at the retina. Let us consider the particular case of a myopic eye having its far point O R in front of the retinoscope ͓ see Fig. 4 ͑ a ͔͒ . In this case, a reflex pattern is observed through the pupil of the retinoscope consisting of a central bright fringe surrounded by dark regions at the eye pupil. When the instrument is tilted up, the spot A also moves up, and its back image O R moves down. Hence, the reflex is seen as moving ‘‘against’’ the movement of the instrument ͓ see Fig. 4 ͑ b ͔͒ . This situation is equivalent to the one represented in Fig. 1 ͑ a ͒ . On the other hand, if the far point is behind the retinoscope ͓ see Fig. 4 ͑ c ͔͒ , when the retinoscope is tilted up, the observed reflex moves in the same direction. In this case, the movement of the reflex is called ‘‘direct.’’ The analogous situation for the Focault test is the one represented in Fig. 1 ͑ b ͒ . Finally, as happens with the knife edge in Fig. 1 c , when O R coincides with the retinoscope pupil and the retinoscope is tilted, the observed pattern suddenly switches from bright to dark without any apparent motion ͓ see Fig. 4 ͑ d ͔͒ . In optometry, this situation is known as the ‘‘neutralization’’ of the reflex movement, and is the basis of the retinoscopic technique. Thus neutralization is obtained when the object plane ͑ the retina of the eye ͒ is conjugate to the retinoscope pupil plane. This situation can be achieved in practice either by placing trial lenses in front of the eye, or by moving the retinoscope back and forth until the proper distance to the eye is attained. The values of the focal length of the trial lens and the eye– retinoscope distance are sufficient to determine the refractive state of the eye. 13 The retinoscope is easily operated. In fact, the technique simply consists of the observation of the reflex through the retinoscope pupil while swinging the instrument until neutralization is obtained, that is, until the point is reached where no apparent movement of the reflex is seen. In summary, the neutralization obtained with a retinoscope is equivalent to the location of the focus of an optical element with the Foucault knife test. Probably because retinoscopy and the Foucault test do not share a common origin, it is not widely recognized that both techniques are in fact based on the same physical principle. Thus, the retinoscope can be used for testing the aberrations of conventional optical elements, as we will show next. Consider the schematic layout in Fig. 5 in which rays pro- ceeding from the axial point of the screen O impinge on a spherically aberrated lens at different heights h . The inter- section of the refracted rays with the optical axis depends explicitly on the value of h . For a given pupil radius, the longitudinal spherical aberration ͑ LSA ͒ is defined as the axial distance between the paraxial image, O Ј , and the image given by the outmost rays O Ј h , that is, LSA ϭ s Ј Ϫ s h Ј . ͑ see Fig. 5 ͒ . The LSA is related to the geometrical parameters of the lens through the following equation ͑ see, for example, Refs. 1, 2, or 8 ͒ : L S ϭ s 1 Ϫ s 1 ϭ 8 h f 2 3 n ͑ n 1 Ϫ 1 ͒ ͫ n n ϩ Ϫ 2 1 q 2 ϩ 4 ͑ n ϩ 1 ͒ ...