Rainbows and refraction of light

Staff Emeritus
Gold Member

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Hi,

I have always wondered this about the standard explanation for a rainbow. Okay, so individual water droplets refract light entering them, dispersing it, as well as changing its direction. Fine. Here's my question: why don't we see millions of little discrete spectra, one for each droplet. Why the circular arc? Why the continuity between bands of colour produced by one droplet, and those produced by all the others?

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Danger
Gold Member
I've never thought of that before; now that you brought it up, I'm not going to get any sleep until someone answers. Thanks a lot. :grumpy:

tiny-tim
Homework Helper
why don't we see millions of little discrete spectra, one for each droplet. Why the circular arc? Why the continuity between bands of colour produced by one droplet, and those produced by all the others?
Hi cepheid!

Same reason we don't see the dots on a television screen.

If the raindrops are "well-tuned", the rainbow colours are sharp. If not, they're fuzzy, because we do see the different spectra for each drop, and they overlap.

Circular because colour depends on the angle from us to the raindrop to the sun, so the same colour lies on a cone whose axis goes through the sun and the back of our head (so the shadow of our head would be in the centre of the arc of the rainbow).

Staff Emeritus
Gold Member
Okay, I think I understand somewhat. I found this interactive Java applet:

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=44.0

You can drag the incident ray up and down, and it seems that light of any given colour is scattered at all angles, but there is an intensity maximum for the light from the ray that strikes at a certain point along the circumference of the circle. For red light, this occurs for the reflected ray at 42 degrees below the horizontal. Of course, you could always drag the incident ray downwards until you get a symmetric reflected ray at 42 degrees above the horizontal. By the same argument (spherical symmetry, essentially), this circle could represent any given crosssection of the sphere (just rotate it around an axis going through its centre parallel to the incident ray). So the reflected red light with maximum intensity lies on a cone. That is enough to convince me that the refracted light from ONE droplet will appear in the form of concentric rings. Presumably the radius can be determined from the distance from the observer to the droplet and the angle of that cone.

It's still not clear to my why the rings from different droplets *coincide*.

Also, maybe this is a dumb question, but if (for example), all the red sunlight is reflected back to the observer along a cone consisting of all rays whose angle from the optical axis is 42 degrees, why does any of that light reach the observer's eyes, given the distance involved?

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Staff Emeritus
Gold Member
It's still not clear to my why the rings from different droplets *coincide*.
In other words, why aren't there many different sets of rings, each set centred on the droplet producing it?

tiny-tim
Homework Helper
That is enough to convince me that the refracted light from ONE droplet will appear in the form of concentric rings.
No … if the droplet is in the red part of (our) rainbow, then we see red light.

All the other colours from that droplet are on a cone, but they miss us by many metres.

If the droplet is 5 kilometres away, and the pupil of our eyes is 5mm, then the angle is 1/1,000,000 radian … that's still red!
It's still not clear to my why the rings from different droplets *coincide*.
At our eyes, there are no rings … our eyes see one colour for each droplet (very much like a television CRT!).
Also, maybe this is a dumb question, but if (for example), all the red sunlight is reflected back to the observer along a cone consisting of all rays whose angle from the optical axis is 42 degrees, why does any of that light reach the observer's eyes, given the distance involved?
I don't follow … why wouldn't it?

Staff Emeritus
Gold Member
No … if the droplet is in the red part of (our) rainbow, then we see red light.

All the other colours from that droplet are on a cone, but they miss us by many metres.
I have no idea what this means, particularly the part about the droplet being on the red part of our rainbow.

If the droplet is 5 kilometres away, and the pupil of our eyes is 5mm, then the angle is 1/1,000,000 radian … that's still red!
Sorry for being dense, but *what's* still red?

I don't follow … why wouldn't it?
If there's a droplet straight ahead of you, 5km away, and it refracts incident light back in the direction toward you, but at an angle of 42 degrees above the horizontal, it stands to reason that that particular ray doesn't reach you.

tiny-tim
Homework Helper
… 42º from the centre of the rainbow …

I have no idea what this means, particularly the part about the droplet being on the red part of our rainbow.
Hi cepheid!

I mean that, from our point of view, there is a red band in the sky, an orange band, and so on.

Any droplet in the red band will appear red to us (slightly different shades of red, of course).

To someone a few hundred metres away, it may appear green or blue … it is not intrinsically red … it is like a little prism, and its colour depends on where the observer is.

That why I put the word "our" in …
No … if the droplet is in the red part of (our) rainbow, then we see red light.
… in someone else's rainbow, the same droplet may be green or blue!
Sorry for being dense, but *what's* still red?
In the rainbow of someone only 5mm away, the droplet is still red.
If there's a droplet straight ahead of you, 5km away, and it refracts incident light back in the direction toward you, but at an angle of 42 degrees above the horizontal, it stands to reason that that particular ray doesn't reach you.
Yes, you're right … but you've been misled … the 42º angle is relative to the axis of the cone of the rainbow, not the horizontal … in other words the centre of the rainbow is directly opposite the sun (so the centre is below the horizon!), and the red band is 42º from that centre.

Yes, tiny-tim is right. Just draw a line from light source to receiver (the eye or camera or whatever)

The rainbow is always directly behind the observer, and centred around that line. Like a shadow is always directly behind an object. Each 'pixel' of the rainbow is a just reflection of the light source. Each droplet is behaving like a tiny reflector selecting out one specific wavelength(colour) for our observer's point in space.

The rainbow only exists for the observer, because the circle of light at the base of the cone is a unique reflection, at a unique position, unique to his own position in space. You have to be exactly where he is to see those specific reflections. Somebody standing next to him will see a slightly different rainbow (generated by a different group of reflectors). Somebody standing on the other side of the cloud won't see any rainbow at all... You have to see the reflections.

And your rainbow follows you around wherever you go. Just like a shadow does.

And each of our 2 eyes sees its own unique rainbow. lol

And if the Earth wasn't in the way, we'd obviously see a complete circle. And obviously it doesn't have to be water droplets either, oil mist can easily make rainbows too.

But how far away is the rainbow ?

I think it must be sitting on the front of the cloud, right on the tiny water droplets at the front.

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jtbell
Mentor
Maybe this diagram will help. White light is coming in from the right, from behind the observer's head. There are six example water droplets (out of about a bazillion), with red, green and blue rays refracted/reflected from each, each color at a different angle. You have to imagine the rays of a single color from a single drop forming a cone, of which the diagram shows a cross-section.

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jtbell's diagram is a good one. Every droplet separates all colours, but not all colours are going to be at the right angle to hit out eyes. The colour which hits our eyes depends on the position of the droplet relative to the eye (more accurately the agle the droplet makes with the centre of our vision). Thus the circular rainbows (or semicircle, if the ground gets in the way).

Staff Emeritus
Gold Member
Yes jtbell, the diagram is immensely helpful, thank you!

tiny-tim

Even though you said outright that each droplet only 'produces' one colour from each observer's perspective, I did not get it until I saw the diagram. Now I also see what you meant in post's 6 and 8

tiny-tim
Homework Helper
Yes … it's always easier with diagrams!

Re: Rainbow

To address that most important of reasons (proving to she who must not be named that I am not talking out my butt), can someone please post me a link and explanation:

I'm aware there are lots of varieties of rainbow (having just googled and failed to find what I was after). Don't see them often around here though. The last one i saw was very bright with strong solid colouring, ut the effect included a wider but vague/faint warm red band on one side and a faint but rich blue on the other.

Please chuck me a link so I can demonstrate my bottom has indeed not been flapping ;) ta

Re: Rainbow

Hi,

I have always wondered this about the standard explanation for a rainbow. Okay, so individual water droplets refract light entering them, dispersing it, as well as changing its direction. Fine. Here's my question: why don't we see millions of little discrete spectra, one for each droplet. Why the circular arc? Why the continuity between bands of colour produced by one droplet, and those produced by all the others?
If you stand with your back to the Sun the rainbow is formed straight in front of you.
When viewed from above say in an aeroplane it forms a circle which is still dirctly opposite to the Sun.
I imagine that the (and will probably get shot down in flames)circular arc is made by the shape of the Sun by the light emitted.
The continuity of the bands of colour is caused by the refraction of the light from spherical shape of the Sun.
The outer edge of the Sun being further away creates the red band of longer wavelength.
Whilst the other colours arrive at the raindrops a little quicker because of it's spherical shape so they appear a different colour.
So the outside of the rainbow is red because it is red shifted whilst the inside is violet because it has a shorter wavelength.

Re: Rainbow

Many thanks.

So where the wider outer red band shows very faintly as per the blue, it's likely to be related to the sun's shape ie corona? or is this distortion likely to normally be there, just often overlooked as it's not very right in relation to the other bands?

Re: Rainbow

Many thanks.

So where the wider outer red band shows very faintly as per the blue, it's likely to be related to the sun's shape ie corona? or is this distortion likely to normally be there, just often overlooked as it's not very right in relation to the other bands?
Yes it's wider band is probably related to the width of the corona.
There are a lot of variables like Sun activity weather on earth,size of raindrops.
Here is a link to a good picture showing the wider red band.
You can see the spherical shape of the sun if you look closely.
http://www.atoptics.co.uk/rainbows/supers.htm" [Broken]

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Lok
Re: Rainbow

If the raindrops are "well-tuned", the rainbow colours are sharp. If not, they're fuzzy, because we do see the different spectra for each drop, and they overlap..
About this well tuning, the size of the droplets does not matter a lot(unless rain where the shape is not as spherical) but quantity does. The depth of the droplet cloud is very important as a narrow cloud produces a much sharper rainbow while a deeper one is fuzzy.

The middle interior of any rainbow appears brighter than it's outside due to reflection.

Re: Rainbow

Maybe this diagram will help. White light is coming in from the right, from behind the observer's head. There are six example water droplets (out of about a bazillion), with red, green and blue rays refracted/reflected from each, each color at a different angle. You have to imagine the rays of a single color from a single drop forming a cone, of which the diagram shows a cross-section.
Your drawing is very good, but I don't believe it answers why the arch is being formed. Actually, it is explained at all in the thread. Can you guys give a hint?

Lok
Re: Rainbow

The arch is the geometric locus of all possible angles where the ~42'deg angle is satisfied.

If the diagram was in 2D, imagine rotating it around the axis made of the sun and observer. By that you get a circle ( cone in 3D cut by the cloud "plane" ).

Redbelly98
Staff Emeritus
Homework Helper
Re: Rainbow

By the way, even a single color gets refracted through different angles by a spherical drop. Here is a ray trace diagram; incident rays are traveling horizontally to the right, and the exiting rays are the diagonal rays traveling towards the lower left:

The key is the higher concentration of exit rays along the bottom. For this angle there is a brighter intensity. For other angles the intensity is too dim to be visible, at least when viewed against a reasonably bright background.

Re: Rainbow

Ahh, I see. Thanks.

Lok
Re: Rainbow

By the way, even a single color gets refracted through different angles by a spherical drop. Here is a ray trace diagram; incident rays are traveling horizontally to the right, and the exiting rays are the diagonal rays traveling towards the lower left:

The key is the higher concentration of exit rays along the bottom. For this angle there is a brighter intensity. For other angles the intensity is too dim to be visible, at least when viewed against a reasonably bright background.
The total internal reflection of water has ~ 48'deg critical angle and play's an important role into what gets reflected on the back side and what passes through (a beauty of statistics).

Redbelly98
Staff Emeritus
Homework Helper
Re: Rainbow

Yes, definitely the angle-dependence of the transmission and reflection would play a role in the intensity.

Re: Rainbow

Hi,

I have always wondered this about the standard explanation for a rainbow. Okay, so individual water droplets refract light entering them, dispersing it, as well as changing its direction. Fine. Here's my question: why don't we see millions of little discrete spectra, one for each droplet. Why the circular arc? Why the continuity between bands of colour produced by one droplet, and those produced by all the others?
Long time this question is still with me if you got an answer plz forward it to me or lets ask more people together.