The tale of a rainbow
‘It is bedtime! Here comes a tale!’ the woman quickly placed herself on the edge of the bed. Two blankets were competing for the right to be called the “tangled” one. A small but exquisite lamp cast its soft light on the papers of the woman’s open notebook. Carefully written letters strived to tell the hidden story through their unique appearances.
‘I’m eighty-two, Lucy,’ a man looked offended, ‘I need no tales! The professor of physics is not going to be afraid of some darkness,’ he swept away invisible dust from the bed. ‘I don’t need it right here right now.’
‘You do not even know what the tale is all about. Do not you want to find out?’ a little smirk flashed on the woman’s face.
‘No desire at all!’
‘Ah, I was blown away by the twist of the story. However, if you do not want to hear it, I’m not going to insist.’
‘The twist, you say?’
‘It doesn’t really matter, does it?’ Lucy started standing up slowly.
‘Alright, alright!’ the man rolled eyes — the tale it is.
‘Right then!’ Lucy’s place on the bed was reclaimed, and the tale began. ‘Once upon a time…’
‘Really, “Once upon a time?”’
‘If you are going to interrupt me, I won’t be able to tell you the story.’
‘Okay, okay, continue…’
The woman cleared the throat and continued, ’once upon a time, there lived a bright fellow, whose name was Aiden. A life full of adventures was waiting for him.
He would hear the stories of his brothers and sisters. How they managed to heat a planet to the degrees that no life would be able to appear or, on the contrary, sustain an exquisite life on another one. The beauty of these stories startled our sunray every time, and he could not wait to experience them on his own.
The time has come for our ray to begin his immensely long and incredible journey. Breaking free from the Sun’s core and its surface, which took about 40000 years, Aiden was ready to head towards the unknown parts of the Universe. Little did he know that his future was a special one.
His destination was not immense planets nor hostile ones. It was the Earth. A rocky planet with diverse forms of life — animals, planets, humans. He was about to meet the alive part of the Universe. Something that was not as common as a giant planet.
On its way, he saw how his siblings were going on Mercury, other — on Venus, causing those little planets to heat up even more. “And I’m going to smash myself, giving away all my life and getting no glory of a burning planet at all!” — thought Aiden.
That is how he went — frustrated with destiny. After about eight minutes, he finally entered the Earth’s atmosphere, waiting to be absorbed.
However, his sadness did not last any longer after Aiden saw it. Something that made his non-existent heart bounce up.
He was little when he first found out the great story of seven ribbons. To witness them was a rare occasion but to participate in their weaving was everybody’s dream. All you need is a small drop of water, and Aiden was a lucky ray.’
‘Do not you even dare to say that he saw a rainbow,’ the man’s voice cracked. He leaned on his one arm in anticipation.
‘Aiden saw a small drop of water…’
‘And that is it?’ the man could not hold his disappointment.
‘A small drop of water near a rainbow.’
‘YES, AIDEN, MY BOY, GO!!!!!’ the happy scream hit the walls. ‘I’m sorry, I just wanted to say that it is good for Aiden, that is it,’ the man put a face full of fake composure.
‘Yes, you are right,’ a proud smile shined on Lucy’s face, ‘it was very good for Aiden. He would finally understand how one of the most terrific phenomena on the Earth occurred. He got near this tiny drop, burning with desire to unleash the mystery of the great ribbons.
Aiden did not have much time to ponder about the way he was going to produce seven colorful threads as he entered the drop…’
The man’s arm flew up in the air, demanding to answer the non-existent question.
‘Yes?’
‘Lucy, I think I know the rest. May I continue the tale?’ his zeal was so strong that the question was almost a demand. Knowing the character of the man, one would argue that it was the demand in the form of an almost polite question.
‘Well, I thought I was in charge of telling the story, but if you wish…’
‘I wish very much, thank you,’ the man cleared his throat. ‘So, Aiden did not have much time to understand the mechanism behind the birth of different threads. Little did he know that he was a source of all of them. You see, Lucy,’ the man looked at her, ‘all the way to the drop, the ray appeared to be white, no clue of what he was composed of. As he entered the drop, two processes happened at the same time: reflection and refraction. Some part of Aiden was reflected from the surface of the water drop, leaving for its adventure. The other part was refracted. Refraction, my dear, was the one that was responsible for Aiden’s division into seven colors.’ The man made a pause for a gulp of surprise from his one member of an audience. Lucy had no choice but to make it.
‘Why would the ray divide into seven colors, you ask? Well, sunlight appears bright white to our eyes. However, in reality, seven lights of different colors are so close to each other that we perceive them as white one. For instance, when sunlight enters a drop of water, each color gets bent to a different amount of angles from the line perpendicular to the surface. It happens because every color has its frequency. The bigger the frequency — the better light will be refracted. That is, the better it will be bent.’ The man was gesticulating everything throughout his explanation. However, he still found it necessary to grab paper and colored pencils, which he always carried around, to draw a little sketch, ‘you see?’
The man pointed at the violet color, ‘it has the biggest frequency of all of the visible colors presented in a ray — that is why it is bent in water so much! That is what has happened to our little Aiden. In the drop, he was divided into seven different colors. But that is not all, no, no!’ the man was excitedly smacking his lips, ‘no, Lucy, refraction was only the first step to the rainbow. Then, reflection and refraction followed again when the ray reached the drop’s inner boundary between water and air. However, in that case, reflection was a crucial step. Whereas the new refraction allowed some of the ray’s pieces to refract into the air and search for their ways of life.’ Lucy did not dare to interrupt the tale, which she heard so many times, ’the last step for Aiden was refraction once again. Once again, he went to the new side of the drop, and all of his colors were refracted, leaving the drop to create a rainbow. The angle of Aiden’s initial entrance to the drop was changed completely, as was his life. The end.’ The man took a deep breath, ‘I wish there were a sketch for this incredible process with all angles marked.’
‘You wish no more!’ Lucy turned the notebook, revealing a carefully drawn picture of the whole process.
‘Where did you find this masterpiece?’ gasped the man, taking the notebook so gently as if it was made of thin, magnificent glass.
‘You used to show me this drawing while you were telling me this tale,’ memories like a movie started playing in front of Lucy’s eyes, ‘you told me that this tale was a spell which protected me from darkness,’ a soft laugh, ‘I remember you telling mom that you were never afraid of it. However, later that night, you revealed to me that you had invented this story to keep both of us safe from the fear and darkness itself.’
‘I invented this story?’ it was a shock of a child who found out that his birthday present was a puppy.
‘Yes, dad, you did,’ whispered Lucy, as if dreading that the loud voice would scare away the precious moment. ‘You drew this picture with all the precision of a true physics professor. When I asked you about all of these angles, you promised me to tell about them when I would grow up,’ her eyes were sparkling under the lamp’s light. ‘I think I have grown up enough to hear the full tale.’
The man looked at his daughter. His lips were trembling, ’well, Lucy,’ he tried to calm down, ‘I think you have.’ The man put his hand on Lucy’s hair, ‘you have grown enough.’ Lucy once again was just a little girl, waiting for her beloved story, and the man was once again a father who was proud to tell it. The man finally broke the sweet silence, ’Lucy, prepare for the STORY OF YOUR LIFE!’
Lucy clapped her hands with excitement, leaning closer to her father.
‘Well, not exactly a story. More like a good small lecture of your life, if you know what I mean. Okay, well, do you see this angle α when the ray firstly enters the drop? This angle is between the line, which is perpendicular to the drop’s surface, and the ray itself. As you can see, some of the ray’s part is reflected by the drop on the same angle alpha between the line and the ray. The other part travels into the drop at the angle β between the line and the ray.’
‘I always wanted to ask you why there is only one line in the drop when refraction divides the ray into seven colors?’
‘There is only one line for an easy explanation. Sure enough, now there are seven colors inside the drop, but it would be cumbersome to draw all of them, as well as the angles for them. Okay, back to the story. The angle β is different from the angle α because the ray entered the new environment, which has a different index of refraction.’
‘Index of refraction?’
‘It is the measurement of the bending of the ray. More specifically, it is the ratio between the speed of light in a vacuum and the speed of light in an observable environment. Water has a greater index of refraction than air does since the speed of light in the water is lower than in the air, therefore, the ratio is going to be greater. Thus, the ray bends towards the line perpendicular to the surface when entering the water from the air since it meets the bigger index of refraction. Here comes the moment of truth!’ the man shook his finger, ‘there are slightly different indexes of refraction for each of the colors since every one of them has different frequencies. Therefore, they do not bend all at the same angle — but at different ones, which separates them and allows us to see all seven colors.’
‘That is why frequencies were important to the tale,’ at this time, Lucy made a sincere gulp of surprise.
‘That is right. So, some of the ray’s part was refracted by water at angle β. Now, the other part of the sunlight was reflected by the same initial angle α. After the reflection, this angle is called the angle of reflection. However, before it, the angle α, which is between the ray and the line perpendicular to the drop’s surface, is called the angle of incidence. As you guessed, the angle of reflection is the same as the angle of incidence.’ The man pointed at them, ‘do not confuse angles α and angle of refraction β. You have seen that they are two completely different angles.’
Lucy quickly nodded.
‘Then, let’s continue. Aiden, as you can easily remember, went to the inner surface of the water drop. You can see that the same two processes happen again — reflection and refraction at the point B. However, there are two differences from our first encounter with them. Firstly, the refraction is happening from the water to the air, which has a lower index of refraction. Therefore, the ray is not going to be bent so close to the line perpendicular to the water surface. In the air, it will have the same angle as it did right before entering the drop — the angle α.’
‘Why?’
‘Well, you see, this process looks like reverse refraction from the air to the water. If entering the water from the air, the ray bents from the angle α to the angle β, then, entering the air from the water at the angle β will lead the drop to have the angle α.’
‘Oh, I see,’ thoughtfully whispered Lucy, ‘and?’
‘And do not let me forget about the second difference,’ the slight hint of irritation appeared in the man’s voice. He prepared himself by coughing and continued. ‘Secondly, this time, reflection plays a crucial role in the gradual process of creating a rainbow. Now, the angle of reflection is β which must be the same as the angle of incidence.’ The man smiled, ‘now, Lucy, we have got to the last part of the “crafting the rainbow.” I think it will not shock you if I tell you, once again, about our old friends: reflection and refraction. Some of the light gets reflected by the inner surface of the drop and continues its journey inside the drop, constantly being refracted and reflected. The other part leaves the drop again at the angle α in the form of seven colors.’
‘Dad, you have not told me about those two angles,‘ Lucy pointed at angles γ and ẟ.
‘Oh,they are special,’ the man looked at them with love in his eyes. ‘See those two lines?’ he pointed at the black and the orange lines, ‘the black line is parallel to the initial direction of the ray. The orange line is the continuation of the ray which left the drop. The angle ẟ between those two lines is the angle by which the initial direction of the ray has been changed.
Moreover, since red light’s frequency is the smallest one among frequencies of other visible light, red light is going to have the smallest angle ẟ because it is not going to bent so much. That is, red light is not going to change direction as dramatically as it will the violet light, for example.’
‘Okay,’ Lucy was contemplating, ‘then what is the angle γ?’
‘It is just the difference between the angle of 180 and the angle ẟ. You can imagine that if red light has the smallest angle ẟ, then it has the greatest angle γ.’ The man gently scratched his head, ‘so you see, every light has its angle. Thus, for instance, red light has the biggest angle γ, and no other visible light will be able to get the same angle. That is why the red color is the top of the rainbow, and above it is just a dark region — because no light can “jump over” red light. Have you ever wondered why there is bright space under the rainbow, right below the violet color?’
‘Oh, right, I have never really wrapped my head around it…’
‘Well, now you will. If red light can make it to the highest angle γ, it can go on degrees lower than this angle when a ray hits another spot on the drop. The same goes, for instance, for the yellow color. It makes to its angle γ, but it also manages to go to degrees lower than its angle γ. At the end of it, all colors can go below their angles γ and mix up again, creating white light under the rainbow. ‘
‘Wow!’
‘Now, Lucy, you need to know where to look to find a rainbow. Firstly, a rainbow must be away from the Sun. It is, of course, for the light of the Sun to hit raindrops and produce this terrific phenomenon. Secondly, you should look at the angle γ from your head’s shadow to see the rainbow. Why, you would ask,’ Lucy closed her mouth. ‘The rays of the Sun that hits you are going to be parallel to the ones which hit drops. Thus, the ray that will hit you is going to be that black line, which we discussed, the orange line is going to be the ray, which left the drop, and the angle between those two lines is the angle γ. Therefore, by looking at the angle γ from your head’s shadow, you will see the whole rainbow.’
‘It is unbelievable,’ the daughter was sitting, shocked.
‘Do you remember that some of light will stay in the drop, being reflected and refracted?’
‘Yes,’ Lucy slowly nodded.
‘When the light will be reflected on this point,’ the man showed the point C. ‘When on the next point it will be refracted, the secondary rainbow will be created. However, it will have the violet color as a top one and the red color as a bottom.’ The man slowly closed his notebook, ‘I think it is the end of my tale.’
At last, Lucy’s longed desire was satisfied. For a moment, no words were uttered since the thoughts of gratitude found a silent way to show themselves — little droplets, so similar to the ones of the tale, were shining on Lucy’s eyes.
‘I had no idea that the lecture could stir up so many emotions in you,’ laughed the man. Even though he was also guilty of letting some of the diamonds escape his eyes.
‘Only one lecture is capable of doing it,’ Lucy took the man’s hand.
‘Oh, it is just a tale,’ his voice cracked: the words could not conceal the true value of the tale.
The warmth of the room was serving the role of a gentle blanket, whereas the warmth of the closeness played a role of a soft, hot tea that touched every corner of the soul.
Small little footsteps, which so eagerly run behind the door to another room, interrupted the silence.
‘Darlene is on her way to the bed,’ Lucy stood up, ‘your granddaughter is so scared of darkness, you know?’
The man’s face was beaming with a smile, ‘then you know what tale to tell her.’
