ABSTRACT
This section is about underwater vision. In order to be able to see
underwater we need a diving mask. This section explains why we need a
diving mask and why fish and other underwater animals do not. Furthermore
it describes how vision under water (using a diving mask) is distorted
compared to vision above water. First the principle of refraction and
lenses is explained.
This section is meant as a very short introduction to the physics of refraction and lenses. A lot more can be said about this subject. Please refer to other literature (or the internet of coarse) for a more full-fledged description, e.g. [2].
Refraction occurs when light travels from one medium to a medium with a different optical density. For example, when light travels from water to air. Optical density is characterized by the refractive index. Vacuum has a refractive index 1.0, air has a refractive index slightly higher than 1.0. The refractive index of water is 1.33. Water is said to have a higher optical density than air. Light travels slower in a optical more dense medium (in fact the refractive index is the velocity of light in vacuum divided by the velocity in the medium).
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A ray of light coming from a medium with refractive index n enters one with refractive index n'. The angle of incidence φ is defined as the angle between the incident ray of light and the normal of the plane between the media. The angle of refraction φ' is defined as the angle between the refracted ray and the normal. The relation between these angles is defined by Snell's law in equation (1).
Going to a optical denser medium (n'/n>1) means the refracted ray bends towards the normal (compared to the incident ray), whereas going to a less dense medium (n'/n<1) means the refracted ray bends away from the normal. This is shown in Figure 1. |
Figure 1: Going from a less dense medium to a denser medium (left) and going from a denser medium to a less dense medium (right) |
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A special situation exists in the case where light goes from an optical denser medium to a less dense medium. This is shown in Figure 2. As we have seen the incident ray is refracted away from the normal (green ray). There exists a critical angle of incidence φcritical for which the angle of refraction becomes 90°. This is shown as the purple ray. This critical angle of incidence is given by equation (2). It can be derived from equation (1) by setting φ' equal to 90°. The sine of φ' becomes 1. In this way we find the critical angle for going from water to air to be 48.8°. If the angle of incidence is larger than this critical angle, total reflection occurs. This is shown as the light blue ray in Figure 2. We can notice this as diver when we try to look through the surface at a distant point: the surface acts as a mirror. |
Figure 2: Total reflection when going from a denser medium to a less dense medium. |
| (2) |
A lens is an optical device with such a shape that it converges every ray of light it receives from an object point to one point called the image of the object point. Two types of lenses exist: positive lenses and negative lenses. In the upper section of Figure 3 a positive lens is shown on the left, a negative lens is shown on the right.
Figure 3: Positive and negative lens and their simplification
A positive lens usually has convex surfaces. In the upper left part of Figure 3 a positive lens is shown. The object point is at the left side of the lens. All rays of the object point are refracted by the two surfaces (air-to-glass and glass-to-air) of the lens. The refracted rays converge into the image point on the right side of the lens. If we would place a screen parallel to the lens coinciding with the image point, we would see a perfect image or projection of the object. As we will see in next section about why we need a diving mask the lens system of our eye is positive. It projects objects in the world around us on the eye's retina. This projection enables us to see.
In the lower left part of Figure 3 a simplified model is shown of the lens (this model applies to thin lenses). The lens is represented by a vertical line. The yellow horizontal line represents the optical axis of the lens, i.e. the axis through the center of the lens. Two focal points are associated with the lens. The focal point has the property that any incident ray that is perpendicular to the lens is refracted towards the focal point. In the model the three construction rays are shown. Construction rays are used to construct the image of any arbitrary object point. Following construction rays are shown:
So the construction rays are defined by the lens and the object point. The construction rays cross in one point. This is the image point.
A negative lens usually has concave surfaces. Rays from the object are refracted. The refracted rays cross in one point. However, now the point lies at the same side of the lens as the object. It is not possible to place a screen to see the projection of the object. Hence the image is called a virtual image. In the same way as for a positive lens we can model thin negative lenses and use construction rays to determine the image of an arbitrary object point. This is shown in the lower right half of Figure 3.
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We cannot see under water without goggles or a diving mask. To understand this, we have to understand how the human eye works. Figure 4 shows the main components of the eye. Light enters the eye on the left side through the cornea. The space between the cornea and iris (the anterior chamber) and the space between the iris and lens (posterior chamber) are filled with fluid called aqueous humor. The iris works like a diaphragm on a camera: it regulates the amount of light passing into the eye by changing the size of its opening. This opening is called the pupil. The diameter of the pupil varies from 2 to 8 mm (compared to a camera: using a focal length of 17 mm of the eye, the eye minimum and maximum opening correspond to about f/8 resp. f/2 stops). The factor of 4 between maximum and minimum opening corresponds to a factor of 16 between maximum and minimum illumination level at the retina (the amount of light entering the eye depends on the pupil area, which is proportional to the square of the diameter). Through the crystalline lens the light enters the third chamber of the eye, the vitreous body. This chamber is filled with a jelly called vitreous humor. Finally the light falls on the photosensitive layer called the retina. Light is converted to electrical signals that are passed to the brain through the optical nerve. Through the processing by the brain we can see. Next table provides refractive indices for the optical components of the eye.
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Figure 4: Anatomy of the eye |
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During the passage through the eye, the light is refracted to form an perfectly focussed image on the retina. Figure 5 shows normal operation of the eye in air. The main refraction (about 2/3) takes place at the air-cornea surface. The cornea has a refractive index of about 1.38, whereas air has a refractive index of 1.0. The fine-tuning needed for focussing is done by the crystalline lens. For this the lens can change shape: for nearby objects we need a stronger lens (more convex) than for far away objects. A circular ring containing muscles is present around the lens. This ring is called the ciliary body. When ciliary muscles are relaxed the lens suspension fibres (between ciliary body and lens) are stretched and the lens is flattened. When the muscles contract tension is removed from the lens suspension. Due to the elasticity of the lens, the lens becomes more convex. In this way nearby objects and far objects can both be seen sharp. This process is called accommodation. In Figure 5 the eye is perfectly accommodated. It projects the object (red dot on the left) on the retina in focus: each ray of light of the red dot that enters the eye is refracted by the cornea and lens. All these rays cross in one focal point that coincides with the retina. Two of these rays are drawn in the picture. A perfect image is projected on the retina. The owner of the eye sees the dot clearly. |
Figure 5: Normal refraction by the eye in air |
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Figure 6 shows what happens when we swim under water: the main refraction at the cornea surface has vanished almost completely: water has a refractive index of 1.33, whereas the cornea has a refractive index of 1.38. These refractive indices are almost identical. According to Snell's law, angle of incidence and refraction are almost identical: almost no refraction occurs. Since the refraction at the cornea is the main part, it cannot be corrected through accommodation by the lens (the lens only fine-tunes). The rays of the object no longer cross in a point that coincide with the retina. Instead the rays cross in a (virtual) point that is far behind the retina. This means under water we are extreme far sighted. Instead of one point, the 'projection' of the object point is a larger area shown in red in Figure 6. The image is blurred. |
Figure 6: Refraction by the eye in water |
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When we put on our diving mask, we have an air filled space in front of our eyes. This is drawn in Figure 7. The main refraction of incident rays at the cornea is back. A bit of refraction occurs at the face plate of the mask: at the water-to-glass transition and at the glass-to-air transition. The water-to-glass refraction is small, since water has about the same refractive index as glass. The glass-to-air refraction is more substantial, since refractive indices of glass and air differ. When the face plate is thin, we can regard the face plate as a transition from water to air and neglect the effect of the glass. This distortion due to refraction at the face plate is small. Hence, it can be compensated by the accommodation of the eye. So by using a diving mask we can project the image of an under water object on the retina. Hence we can see clear. Because of the refraction at the face plate the object looks different as the object would look above water without a mask. The object as it appears to the diver is called the virtual object. |
Figure 7: Thanks to the diving mask we can see under water |
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Fish can see underwater without a diving mask. So nature apparently has provided a solution for the problem humans cope with under water. Since refraction at the water-cornea surface is not significant, in the fish eye the crystalline lens performs most of the refraction needed for focussing. For this, the crystalline lens of a fish eye is almost spherical (high curvature) and has a relative high refractive index, ranging up to 1.65. In Figure 8 a fish eye is drawn. The lens itself is placed close to the cornea, bulging through the iris out of the eye. The reason for this placement is twofold: first, more light enters the lens since due to lack of refraction at the cornea, the cornea does not funnel down (concentrate) light on the lens. Second, it produces the well known fish-eye-view: a wide field of view. This is advantageous, since a fish cannot turn its neck to look what is behind it. Since the lens bulges through the iris, the iris cannot close. The iris cannot be used to regulate light. Besides that, the fish cannot close its eye. So do not shine your ultra bright diving light right into the eyes of a fish: this cannot be healthy for the fish. Note: The fish uses synchronized movement of the light sensitive cells (rods, cones and pigment granules) in the retina to control the amount of light that is processed. |
Figure 8: Fish eye |
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One problem with a spherical lens is distortion of the image called spherical aberration. This results from the fact that not all refracted rays cross in the same focal point. This is shown in the left picture of Figure 9: the focal point of parallel rays entering the lens on the outside are bent to a focal point that is closer to the lens than rays that pass through the center of the lens. In a fish eye a perfect solution has been provided: from the center of the lens to the outer surface the refractive index decreases. This results in all rays being refracted towards the same focal point. This is shown in the right picture of Figure 9. Note: like a cat some fish have eyes that are provided with a tapetum lucidum. This is a reflective layer behind the retina. It reflects the light so that is passes another time through the retina, increasing the chance of capturing by the retina. This enhances the sensitivity in poor light conditions. The presence of a tapetum shows when you shine your dive light on the fish: the eyes seem to light up. |
Figure 9: Spherical aberration (left) and the solution for it in the fish eye crystalline lens: decreasing refractive index from the center to the surface |
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Figure 10 shows a 1.20 m dolphin at a distance of about 2 m, as seen by a diver. Move your mouse over the image in order to lower the water level and see what the scene looks like without water. The image has been calculated with a raytracing program. The dolphin definitely looks bigger under water. We can calculate how large the dolphin looks (virtually) under water. For this we calculate the size of the retinal image above and under water (using a diving mask) and compare these image sizes. |
Figure 10: under water vs. above water - move your mouse over |
Figure 11: imaging in air
In Figure 11 the eye is shown schematically. The lens with focal distance f in the figure represents the combined crystalline lens and cornea of the eye. Two construction rays are drawn. Given the distances Di, Do and the object size ho, the image size h i-air is given by comparison of congruent triangles:
| (3) |
The magnification mair (ratio between image size and object size) can be calculated by comparing congruent triangles and is given by:
| (4) |
In this equation Di is the effective distance between the eye lens and the retina (about 17 mm). In Figure 12 the same situation as in Figure 11 is given, but now under water. We have the same object at the same distance from the eye (hence Do1 + Do2 = Do)
Figure 12: imaging in water
By comparing triangles and applying trigonometric rules and Snell's law we find following equations:
| (5a) |
| (5b) |
| (5c) |
| (5d) |
| (5e) |
In these 5 equations following parameters are given:
Following parameters are unknown:
Five equations with five unknowns can be solved. When the object size ho is small, the angles φ and φ' are small. We can make the approximation:
| (6a) |
| (6b) |
Combining equations (5a), (5b) and (5d) leads to
| (7) |
Together with equations (5c) and (6) this results in:
| (8) |
The magnification (image height divided by object height) is given by:
| (9) |
Finally we can calculate the factor M between hi's found in the under water and air situation (ho is equal in both situations and Do1 + Do2 = Do):
| (10) |
Rewriting this a bit results in
| (11) |
We can numerically solve equations (5) for a small object height ho using e.g. MathcadTM. Alternatively, we can simply calculate equation (11). In Figure 13 the magnification of the image under water is plotted versus the distance of the object to the diving mask. This is done for three distances between face plate and the eye.
Figure 13: Magnification as function of object distance, for 3 eye-face plate values
We can conclude from this Figure (and equation (11) of course):
Other sources [1] use another abstraction of the eye called the reduced eye. The reduced eye is shown in Figure 14 and 15. The eye is represented by a spherical surface (of the aqueous humor) with radius R (R=0.8 cm). Any ray passing the surface perpendicular will pass through a point at a distance R behind the surface on the optical axis. Such a ray from the object is shown in the Figure. The derived result for the magnification is equal to equation (11) if we state O1=Do1 and O2 + R = Do2 (compare Figure 15 to Figure 12).
Figure 14: The Reduced eye, above water situation
Figure 15: The Reduced eye, under water situation
Summarizing: under water the image that is projected on our eye's retina is enlarged by a factor of at most 1.33 when compared with the above water situation. How our brain perceives this image enlargement is quite complicated. In next section we try to say something about the way under water vision messes around with the stereoscopic view of our eyes.
Figure 16: Stereoscopic imaging by two eyes |
The dolphin of Figure 10 is seen by both eyes. From the rays to both eyes we can construct the virtual object. This is the object (dolphin) as it appears to the diver. In the figure beside, the object (dolphin) is represented by the horizontal red arrow. For the ease of calculation we only regard the left half of the object. Rays occurring from the left endpoint (say, the dolphin's head) enter the airspace in the diving mask. Refraction takes place. The refracted rays are received by the left and right eye of the diver. The actual way the light travels are represented in the figure by thick green lines. The virtual object is constructed by extending the refracted rays. The extended rays of both eyes intersect at one point. At this point the diver 'sees' the head of the dolphin. |
For the right eye we can write down three relations:
| (12a) |
| (12b) |
| (12c) |
In these relations the following symbols are defined:
Three unknowns (φright, φ'right and xright) can be solved. Consequently, similar relations can be stated for the left eye:
| (13a) |
| (13b) |
| (13c) |
Similar φleft, φ'left and xleft can be found. Knowing all incident and refraction angles, we can find the size ho' and distance Do1' from diving mask to the virtual object.
| (14a) |
| (14b) |
hereby is
| (14c) |
It is quite a nuisance to solve the equations. A program like Mathcad TM can solve these equations for you numerically. For our dolphin we find in this way:
| (15a) |
| (15b) |
What we find is that the size of the virtual object doesn't differ much from the actual size. The size appears to be slightly smaller. The distance from the diving mask, however, appears to be much smaller than the actual distance: we expect the dolphin to look much closer to the diver than it actually is.
In practice perception by the brain of the situation under water is more complicated and appears to depend on a number of factors as object distance, visibility, etc. Divers appear to get used to the distorted view and adapt their eye-hand coordination for example.
