“The Blind Girl” is a famous picture painted by an English painter John Everett Millais. Beautiful rainbows in the picture make an impression on the gallery and remain in their remembrance.

 This picture is a view of the English countryside. Two girls, probably are itinerants, sit side of a road short of a beautiful but empty meadow. The elder sister is blind. The younger sister cuddles to the elder and they sit on the grass together. Both sisters are poorly dressed.

There is a forest over the meadow, and two rainbows start from the forest to the dark gray sky. The colors of the rainbows are quite beautiful, and the rainbows are shining in the painting. The younger sister faces round the rainbows and says something to the elder. But the blind girl cannot see the beauty of the rainbows and retains her posture keeping her accordion on her knee.  They probably are orphans. This painting causes the audience’s pity on the blind girl of sisters who cannot even see the beauty of the rainbows.

 There is a small story on these two rainbows. As you know, a big secondary rainbow appears on the bright and beautiful primary rainbow. The luminance of the secondary rainbow is weaker than that of the primary rainbow when two rainbows appear. The picture caught the brightness of two rainbows. However, we have to consider another point on the two rainbows. That is the color order of the rainbows.

Most of the people believe the colors of a rainbow are the spectral seven colors. And they remind the word “the seven colors of the rainbow”. The colors of rainbows are different from the spectral seven colors in fact, and I will discuss this issue later. Here, I would like to mention the color orders of the primary and secondary rainbows. They are reverse. Red is outside, and purple is inside in the primary rainbow and purple is outside in the secondary rainbow. All rainbows show this sequence and have to be so.

However, Millais first unthinkingly painted rainbows in the same color sequence. This mistake became a matter, and finally, the second rainbow was repainted after various discussions.　The picture we see today is the one after repainting.

 Junior high school students learn the color orders of the primary and secondary rainbows in science class. So, some people may think such a mistake is ridiculous, and some of them complain about such an easy mistake on a matter of common knowledge. However, I believe such people do not know the nature of rainbows exactly. The complete picture of rainbows is much more complicated than the conventional explanations found in science textbooks.

Please look at a rainbow carefully when you find it next time.  Many people believe that the rainbow has seven colors. But only a few rainbows have seven colors ordering red, orange, yellow, green, blue, indigo, and purple. Many rainbows show beautiful red and yellow colors and hardly show blue. On the other hand, some rainbows show pale red, and some show wide ranges of yellow and greens and narrow ranges of red and purple. The seven colors of the rainbow are different from seven colors of visible spectra. But most of the people fall into the trap of the word seven colors and idly think that all rainbows have spectral seven colors. A person who carefully has observed a rainbow should have questions on the explanation of rainbows. There is no doubt to have such questions since the conventional explanations are not enough to explain the complete picture of rainbows.

 Please recall the image when you showed a brilliant bright rainbow. You might saw weaker narrow rainbows just inside the primary rainbow. Red and green lines are eye-catching and doubly or triply appear in this excess (supernumerary) rainbow. Supernumerary rainbows are one of the true characters of rainbows since it always appears concomitantly with a brilliant rainbow.  There are supernumerary rainbows just outer the secondary rainbow. Though the luminance of these supernumerary rainbows is weak and difficult to observe, they always exist. The complete explanation of rainbows should contain the explanation of these supernumerary rainbows.

Some people also noticed the difference in the brightness of the sky. The sky inside the primary rainbow is bright, and the sky outside of the secondary rainbow is a little bit darker but still bright. However, the sky between the first and second rainbows is dark.  This difference should be one characteristic of the rainbow phenomena. If you feel mystery on the beautiful red and green ring in the sky, the difference of the brightness also should be a mystery. For a scientific explanation, we cannot say we understand the phenomena until nothing unexplained remains. Beauty and eye-catching are problems concerning human feeling and have no meaning on the importance of the phenomena.

The shape of rainbows is also a problem. From the conventional explanation of rainbows, the shape should be perfectly circular, and the width of rainbows should be the same. However, some rainbows are ellipsoidal, and some are a little bit squashed. The width of the rainbows is different among rainbows. Sometimes the width can be different between parts of a rainbow.  Some people explain these differences from the difference of conditions of raindrops in the part of the sky. But he should explain the relationship between the conditions and the shape of rainbows to make his explanation perfect. Here, I will explain the rainbow phenomena step by step.  You finally understand the explanation of rainbow is not so easy as you have believed from the explanation of science textbooks in a junior high school.

 Let us start the explanation written in science textbooks. Spectral decomposition caused by reflection and refraction of light in raindrops acting as prism creates rainbows. This explanation of the origin of rainbows is generally correct. Though raindrops exist all around the sky, not all the sky but only a circular band of the sky shines. We start to discuss this point first. Then we will discuss the colorization of a rainbow.  If the sunlight has only a color, then a rainbow becomes a circular band of the color. For example, you can observe a yellow rainbow in a laboratory using the monochromatic yellow light source.

A part of the sunlight reflected at a surface of a raindrop, and most of the sunlight enters the droplet refracting at the surface. The incident light reflects in the droplet and goes out again, refracting at the surface. Figure 2 shows the simplest case of the light path. As shown in Fig. 2, the sunlight is a parallel beam, but the outgoing beam from the droplet is not a parallel beam. We could not observe a rainbow if there was not a special characteristic of the light path. If so, reflected light from every raindrop come to an observer and the observer see the uniformly bright sky.

Optical theory shows us that the light reflected by a raindrop has a special characteristic. Indeed, the reflected light is not a parallel beam, but the actual light path is different from that of Fig. 3. I purposely wrote the incorrect path in Fig. 3. Figure 4 shows the accurate light path inside and outside of a raindrop. A ray of parallel beam SP irradiating point P, paths inside the droplet through the arrows and goes out the droplet to the direction RT.  The other rays incident the droplet at the upper and lower sides of SP such as 1 and 2, and 3 and 4 exist to the upper side of the line RT.  It seems that rays 3 and 4 exit the lower side of RT if the rays 1 and 2 exit the upper side of RT. However, optical theory shows us all incident light exit the upper side of RT. P is a singular point of the raindrop. This singularity comes from the incident angle of the ray. The incident angle of a sphere is the angle between the ray and the tangential plane at the incident point.  Output ray goes to the direction RT when the incident angle is the specific value. Other parallel rays, 1, 2, 3, and 4 incident at any point except for P have different incident angles and go out upper side of RT. As incident angle increases, the outgoing rays once become closer to the RT from the upper side and then back away. The ray density becomes high just above the line RT. In other words, the outgoing light becomes strong around this direction.

 RT is such singular direction, and the angle between the parallel sunlight, and this direction depends on the refractive index of water and is independent of the size of a raindrop. The refractive index does not only depend on materials but also the wavelength of light. Averaged refractive index of water for white light is 1.333. Using this value, we obtain 42 degrees as the angle between SP and RT. The measured angle of the rainbow is approximately 42 degrees. Now, we can, at least, explain the origin of the rainbow.

Here, I would like to notice the angle of 42 degrees of the rainbow. In Fig 4, we wrote a raindrop as a circle, but the real shape of the raindrop is a sphere. Hence, the direction RT spreads in a 3D space making a circular cone. 42 degrees are the angle between the axis and face of the circular cone.  All innumerable raindrops in the sky reflect the sunlight in the cone shape, as shown in Fig. 5. Then the rainbow looks circular for human eyes like 1, 2, 3, and 4 in Fig. 6. H is the horizontal plane, O is the observer, and S means the direction of the sun.  In this case, the light comes not only from the raindrops on the arc 1, 2, 3, and 4 but also from all raindrops on the cone.  The angle of the rainbow is the sum of sun height “a” and the height of the rainbow “b” and the angle is always approximately 42 degrees from measurements. Hence, we can say the explanation in Fig. 4 is correct, and we now understand the basic mechanisms of the rainbow.

The secondary rainbows can be explained by the similar way. The second rainbow appears outside the first rainbow, and its angle is approximately 52 degrees. The second rainbow is the rays reflected twice in raindrops. Figure 7 shows the light path and a particular direction. In this case, rays 1 and 2, go on the upper, and lower sides of SP go out the lower side of RT.  The light intensity is strongest when the angle between the parallel incident ray and the output direction is 52.5 degrees.  Also, no ray returns to the direction smaller than that value. Optical theory can prove these results. The angle 52.5 degrees coincident with the measured value of the second rainbow. Now, we can successfully explain the second rainbow.

 You may think that there should be a rainbow created by rays reflected three times in a raindrop.  Not only the third rainbow but also the fourth and fifth rainbows exist. However, the third and fourth rainbows appear in the direction of the sun. The strong sunlight scattered by raindrops hides these rainbows, and we cannot observe them. The fifth rainbow appears in approximately the same direction with the secondary rainbow. The luminance of the fifth rainbow is much weaker than the secondary rainbow. Hence, we cannot observe the fifth rainbow. The supernumerary rainbow of the secondary rainbow is different from the fifth rainbow.

Now I have explained the basic origin of the primary and secondary rainbows. But I did not mention the spectral decomposition of the sunlight, i.e., the origin of colors of rainbows. Now, I will explain this point.

   The explanation is rather simple. The sunlight is composed of seven colors such as red and orange etc. Of course, the phrase “seven colors” means only the typical colors of spectra. Though real spectral color continuously changes from red to violet, I will use the phrase seven colors to make explanation simple. The reflective index of water depends on the colors of light. The reflective index of the red light is 1.331 and becomes larger toward the 1.344 of the purple light. The incident white light path through point P in Fig.8 separate into each color light.  The strongest directions of red and purple light are RT and R’T’, respectively. The calculation results using indexes for red and purple light, the output angle of the red and purple light for the primary rainbow are 42.22 and 40.36 degrees, respectively.  The angles of the red and purple light of the secondary rainbow are 50.24 and 53.36 degrees, respectively. Notice that the magnitude relation between the red and purple light inverts between the primary and secondary rainbows.  Therefore, the red and purple parts are outside of the primary and secondary rainbows, respectively.

 According to this explanation, the color order of rainbows should be the same. And all rainbows have spectral seven colors, and their width should be the same among rainbows. The whole width of the primary rainbow is 1.46 degree, and that of the second one is 3.12 degrees, as shown in Fig. 9. However, the width of rainbows can be different, and only a few rainbows show spectral seven colors. These disagreements clearly show that the conventional explanation written in the common textbook is not enough.

The conventional explanation cannot explain these features of rainbows, but it can explain the darkness of the sky between the primary and secondary rainbows. In Fig. 10, long and short arrows show the strong and weak outgoing rays. Part of light coming from raindrops inside the primary rainbow can reach point O.  Huge numbers of raindrops exist inside the primary rainbow and the sky becomes uniformly bright due to these raindrops. The same is true for raindrops outside the secondary rainbow. Dotted lines show the light paths.

No reflected rays from raindrops between the primary and secondary rainbow can reach the point O.  Rays from raindrops “A” and “B” go to the direction different from the point O irrespective of the strength of light. Hence, the sky between the primary and the secondary rainbows looks darker than the other parts. Light scattering by the other mechanisms makes the sky of this area not perfectly dark. The difference in the brightness of the sky is explained in this way.

Now, we start the explanation of the remaining problems, i.e., the width, color order, and supernumerary rainbows. Here, we have to introduce a different concept of light. Light beams are represented by arrows in explanations above. As you know, light has wave characteristics.  Therefore, optical phenomena should be explained using wave theory in the strict sense.  For example, the collection of light with a lens is typically illustrated as Fig. 12(a) in the geometrical optics way. However, it should be illustrated as Fig. 12(b) in the wave optics way. It is simpler to represent light beam by arrows perpendicular to the wavefront than by the wavefront themselves. Therefore, the representation using arrows is commonly used when the geometrical optics sufficiently explain the optical phenomena, such as focusing by a lens. Since the wavelength of visible light is as small as less than 1/1000 millimeter, wave natures of light do not become a matter for macroscopic objectives. Conversely, we should take wave nature of light into account, when the size of raindrop becomes as small as a cloud drop with a diameter of 1/100 millimeter.  The explanations of rainbows written above do not contain the factor of droplet size.  We have to fill in the gaps of explanations by geometrical optics by wave optics explanations. The appearance of supernumerary rainbows is well explained by considering the interference of light when the sunlight is reflected and refracted in a small raindrop.

Let us reconsider the origin of the primary rainbow from the wave-optical view.   The wave surfaces are superimposed on the rays SP and RT in Fig. 4 (Fig. 13). The only wavefront of a plane wave is added in the figure, in which nothing new exists in this case. The wave surfaces become curved surface when the raindrops become so small that the wavelength of light cannot assume zero compared with the raindrop size. The shapes of the wave surfaces change from those in Fig. 13 to AOB in Fig. 14. This change is the consequence of the wave optics theory. The curvature of AOB depends on the size of a raindrop and the refractive index of water. The curved surface becomes a plane surface as the size becomes big enough.  The explanation based on the geometric optical theory is only valid when the size of raindrops is much bigger than the wavelength of light. The light beam between AO has convex wave surfaces and diverges.  The light beam between OB has concave wave surfaces and focuses on point F in Fig. 15. Beyond the point F, this beam diverges, and the wave surface becomes convex.  The light beam has double wave surfaces (O’A’ and O’B’) far from the raindrop, as shown in Fig. 15. The double wave surfaces cause interference.

Waves become strong when tops of two waves overlap and weak when the top and bottom overlap canceling each other.  It is interference of light waves. In Fig. 15, the tops and bottoms of waves are represented by solid and dashed lines, respectively.  The light becomes intense to the RT direction since tops of A- and B-waves overlap in this direction. This direction is the front direction of geometrical optics, and light strength is extremely strong. This direction corresponds to the corn surface of 42 degrees explained before. The adding interference happens in the directions of 2 and 4 in Fig. 15 and light intensity becomes strong in these directions. In the directions of 1 and 3, overlap happens between tops and bottoms and light waves of this direction become weak.  These interference phenomena create two or three circular light bands inside the primary rainbow.  These are the true nature of supernumerary rainbows. Usually, only two or three supernumerary rainbows are observed. Because the intensity of interference light becomes weak as the angle from the front direction becomes large. In rare cases, six bands of supernumerary rainbows are observed.

Above is the outline of the effect of interference. For the rigorous explanation, we have to calculate the angles of direction 2 and 4 between that of direction 0. We also have to check whether the direction 0 agrees with the angle of 42 degrees explained above. An astronomer of England, Airy had made this calculation and found that direction is a little smaller than the value calculated from the geometric optics.  The difference between the geometric and wave optical view result from the shape of the wave surface is flat or curved. The wave surfaces become curved planes as the size of raindrops becomes small. Airy showed the strongest direction correspond to 42 degrees in large raindrops and two to three degrees smaller in small raindrops. The drop size becomes a matter of the wave optics treatment.  Now, we can explain the deviation from the perfect circular shape. If the raindrop size of a part of the sky is big and small in the other parts, then the shape of the rainbow can be warped a little.

Though the above explanation is not simple, we only deal with monochromatic light at this time. To explain the spectral future of rainbows, we have to make figures similar to Fig. 15 for seven colors of spectra and superimpose these figures.  Up to here, most of the characteristics of rainbows can be explained. Still, we need one more important piece to explain the whole nature of rainbows.

As I wrote, supernumerary rainbows always exist and appear inside of the primary rainbow. The distance, actually the angle distance, between the primary and supernumerary rainbows causes a new issue. The width of the primary rainbow is 1.46 degrees. If the angle difference between the primary and supernumerary rainbows is bigger than this value, then we have the primary rainbow having bright seven colors and two or three bands of supernumerary rainbows having faint seven colors inside the primary rainbow.  The angle difference becomes smaller and bigger when the raindrop size is big and small, respectively. Therefore, we can observe the perfect primary rainbow involves supernumerary rainbows inside of it.

When the raindrop is rather big, then the angle difference between the primary and supernumerary rainbows becomes smaller than the width of the primary rainbow. In such case, the reddish part of supernumerary rainbow overlaps with the blueish part of the primary rainbow. In this type of region, different color light overlaps and the color reorganization of such mixed colors become a problem. This problem is different from the color mixing of paints. We have to know, for example, what kind of colors we recognize when spectral red and green light mix each other. The wavelengths of red and green light are different from each other, and they are unchanged in the mixed light. The green and red light keep their characteristics in the mixed light. However, we observed this mixed light quite different from the green and red light. We feel this mixed light as yellow light. The simultaneous excitation of red and green light on retina makes the feeling of yellow. Of course, the ratio between the green and red light should be appropriate; the mixed light is practically indistinguishable to the spectral yellow light.  

Many of rainbows are such mixed color ones. The red part of supernumerary rainbow overlaps with the green part of the primary rainbow makes this part looks yellow. The first rainbow has a yellow spectral band outside the green band, and the connection this original yellow band with the mixed yellow band makes full yellowish band wide in the primary rainbow. Such a situation happens when the raindrops are rather large.  Such kind of rainbows is frequently observed after the shower in the summer.  Rainbows tend to bright and beautiful when the raindrops are big.  Please remained brilliant rainbows after a shower, you will agree that the red and yellow parts are bright and beautiful and green and blue parts are faint.

 The raindrop sizes vary among rains and the color order of rainbows can be and should be different among rainbows.

 Now, I have explained the characteristics of rainbows I showed you at the beginning of this story. Most people will answer the reflective and refractive actions when somebody asks them the origin of rainbows. This answer is not precise as I wrote above, but it is understandable since they are not scientific scholars.  They may be persons who have better scientific knowledge. However, the reflective and refractive actions only make rainbows having seven colors; that is different from most of the real rainbows with brilliant red and yellow bands. They have to observe these real rainbows. Assuming that a rainbow has seven colors without observing real rainbows is a doing go against a scientific mind.  

Natural phenomena are very complicated and full of mysteries. Most of the phenomena require more than two pages to explain. Rainbows are a relatively simple case in the phenomena. But we still need all this explanation. We have remaining problems on rainbows I will not discuss here. I believe we can understand the true beauty of natural phenomena by understanding it one by one.

 Some persons do not try to observe real rainbows, saying that the rainbows are the spectral actions come from the reflection and refraction by raindrops. Such persons never understand the beauty of rainbows. They are who become vision-impaired by science, instead of getting better vision. Everyone feels pity for the blind girl. Persons who close their eyes to the natural phenomena are also wretch in other sense.

Feb 18, 1947

Many websites and popular science books explain the coloring of the rainbow. Though I have not checked all of them even in Japanese, I could not find a better explanation than that written by Ukichiro Nakaya in 1947. He wrote a science essay entitled “Niji” as a contribution to the first number of a science magazine “Niji” for young Japanese. “Niji” is a rainbow in Japanese.
Nakaya died more than 50 years before, and his author’s copyright has vanished in Japan. You can find his essays in Aozora Bunko. Here, I place my translation of “Niji”. I know my English writing is terrible, and I hope someone will revise this translation to better English.