The Story of Optics - The Beginning of Modern Optics

Gautam Gangopadhyay

Translation: Sunando Patra

Photo: Sunando

It’s time we transitioned from the medieval to the modern era. In 1543, Nicholas Copernicus (1473-1543 CE) published his famous book “De Revolutionibus Orbium Coelestium”, where he proposed the heliocentric model of the universe. This can be marked as the commencement of the modern era in science. The two famous scientists who follow him are Johannes Kepler (1571-1630 CE) and Galileo Galilei (1568-1642 AD).

We all know about Kepler’s laws regarding the planetary motion. Kepler observed a solar eclipse using a pinhole camera on 10 July 1600 CE. Although the idea of such a camera was mentioned in Aristotle’s writings, it was lost and again came to Europe following Alhazen’s work. A few years before, Tycho Brahe (1546-1601 CE) had noted that the moon, when watched with the naked eye, appeared larger in the night sky than during a solar eclipse observed through a pinhole camera. Kepler realized that the pinhole in the camera is not exactly a point; it behaves more like an extended aperture. That’s why the moon appears larger through it. He experimentally determined how light travels from a light source through a pinhole. Kepler also said that our eyes work like the pinhole camera. He published his research on optics in 1604.

Galileo is famous for using the telescope for sky observation. Kepler was the next person who inspired the scientists of his age to use telescopes, although he never did so due to his poor eyesight. There was no one as knowledgeable in the field of optics as him in his time. Galileo made various attempts to improve the telescope’s performance using trial and error. Kepler was more interested in the theoretical working principle of the telescope. Lenses were used in telescopes of this era. The main problem with lenses was spherical aberration. Lenses are usually parts of spheres because they are easier to manufacture that way. However, the light rays don’t converge exactly at one point in a spherical lens, resulting in slightly blurred images. The spherical aberration of the eye is very low, and Kepler had already researched vision before. He observed the shape of the eye’s lens and concluded that it should be slightly different from a sphere. We now know that the ideal shape of a lens should be a paraboloid. However, it was not possible to make such shapes at that time. Galileo's telescope had one convex and one concave lens. Kepler suggested making a telescope with two convex lenses and advised to increase its length.

A simulation of spherical aberration in an optical system with a circular, unobstructed aperture admitting a monochromatic point source. The top row is over-corrected (half a wavelength), the middle row is perfectly corrected, and the bottom row is under-corrected (half a wavelength). Going left to right, one moves from being inside focus to outside focus. The middle column is perfectly focused. Also note the equivalence of inside-focus over-correction to outside-focus under-correction. (Source: Wiki)

A video about spherical aberration. (Source: YouTube)

Galileo’s telescope had an upright image like modern binoculars. In Kepler’s telescope, the image is upside down. This is no problem for celestial objects. The magnification of Kepler’s telescope, on the other hand, is much higher. Crosshairs in cameras or rifle sights help to aim correctly. Using them in Kepler’s telescope is straightforward. It helps to determine the position of the image very accurately. That’s why astronomers gradually started using Keplerian telescopes. As I mentioned earlier, ancient Greek scientists generally believed that the speed of light was infinite. Alhazen or Ibn Sina contradicted their view. In his book from 1604, Kepler wrote that the speed of light is infinite.

Johannes Kepler (Source: Wiki)

Galileo's "cannocchiali" telescopes at the Museo Galileo, Florence. (Source: Wiki)

Galileo is considered the father of modern experimental science; he was the first to attempt to measure the speed of light. A person suddenly removed a cover from a lantern on top of a mountain, and from a distance, his assistant—seeing the light—would immediately remove the cover of another lantern. The first person would measure the time between uncovering the lantern for the first time and seeing the assistant’s light, then measure the distance between the two mountains and use the time difference to deduce the speed of light. Galileo wrote that he could not determine the speed of light by this experiment because the time difference was so small that it could not be measured. It was understood however that the speed of light was very large.

The value of the speed of light was first roughly determined by the Danish astronomer Ole Rømer (1644–1710 CE). He observed the eclipse of Jupiter’s moons. Galileo was the first person to discover Jupiter’s four moons. When observed from different places on Earth, the time of an eclipse of any of these moons will be different, just as a solar eclipse can be seen from multiple places, but with different start times. Galileo proposed that we can determine the longitude of a place on Earth if we know the time of the eclipse of Jupiter’s moons from that location. At that time, it was almost impossible to do this, especially from a ship, in the middle of an ocean. Galileo’s method was not suitable for marine use—estimating the time of the eclipse proved difficult from the decks of ships rocking in ocean waves. On land, however, his advice was rock-solid (Ha Ha!). Rømer observed the eclipse of Jupiter’s moons from the island of Hven in Denmark and from Paris in France. He found that when the Earth moves away from Jupiter, the time difference between two eclipses increases, and when the Earth moves towards Jupiter, it decreases. He speculated that when Jupiter is near Earth, light takes less time to reach Earth, and when it is far away, it takes more time. In 1676-77 CE, he calculated the time it takes for light to travel a distance equal to the Earth’s orbit and found it to be twenty minutes. There is further discussion on this in the appendix of this chapter.

Rømer observing the sky. (Source: Wiki)

Rømer never gave a specific value for the speed of light. Giovanni Domenico Cassini (1625–1712 CE) had determined the distance from the Sun to Mars a few years earlier. According to Kepler’s third law of planetary motion, the square of a planet’s orbital period is proportional to the cube of its distance from the Sun. By using this law, if the distance of one planet is known, then the distance of all other planets can be known. In this way, the diameter of Earth’s orbit was roughly known. Christian Huygens (1629–1685 CE) calculated that the speed of light is approximately 220,000 km/s. We now know that the speed of light is approximately 299,792 km/s—Rømer and Huygens’ measurements, although slightly lower, were close.

Fifty years later, in 1728, James Bradley (1692–1762 CE) determined the speed of light by observing stellar aberration. He calculated the speed to be 301,000 km/s, very close to modern measurements. There is further discussion on how Bradley obtained this measurement in the appendix. The speed of light is one of the most important topics in physics, and we will come back to it later.

Another name comes up in the context of optics: Leonardo da Vinci (1452–1519 CE). Leonardo, born before the age of the scientific revolution and known more as an artist, was also exceptional as an engineer. His discussions about light, in his notebooks, came naturally in the context of painting. There he mentioned that the eye cannot be the source of light and the reason is that the speed of light is finite. We’ll see later that light is a kind of wave, which was first suggested by Christian Huygens. A century earlier, Leonardo compared light with waves or ripples in water. Some historians believe that Huygens was influenced from Leonardo; there is a letter from Huygens stating that he bought a manuscript on optics by Leonardo. We’ve seen this before—Hero assumed that light always takes the shortest path, but scientists had forgotten about it. In Leonardo’s writings, we see that idea again. He also wrote about pinhole cameras.

Pierre de Fermat (1607–1665 CE) proved the principle of refraction following Hero’s work. In the history of science of the seventeenth century, Fermat is a notable name. He is associated with one of the most famous problems in mathematics; he mentioned a theorem in 1637 that took over three hundred years to prove. He was well acquainted with research on the principle of refraction in optics. While Hero said light always takes the shortest path, Fermat proposed that light takes the path of least time . From there, he proved Snell’s law related to reflection. There will be further discussion on this in the appendix of this chapter.

We need to mention two instruments before finishing this chapter. One has already come in our story: the telescope. Many names come up as the inventor of the telescope, but the first to apply for a patent was a spectacle maker named Hans Lipperhey (1570–1619). His patent was not granted, however, and many others also made the device almost at the same time. Galileo’s use of the telescope for sky observation is considered by many as the birth of modern science. The other instrument is the microscope. The name of its first inventor is unknown to us, but Galileo’s writings show that he inverted the telescope, to use as a microscope, in 1610. Antony van Leeuwenhoek (1632–1723) made many advancements in this instrument. Just as the telescope revealed the vastness of the universe to us, microscopy also revealed a huge unseen world right before us. The role of these two instruments in the advancement of science and technology is immense.

Appendix

Determination of the speed of light by Rømer

Schematic diagram taken from Rømer's article. (Source: Wiki)

The picture here is taken from an article by Rømer. Here, \(B\) represents Jupiter, and \(C\) and \(D\) represent the positions of Jupiter’s satellites just before and after an eclipse. \(A\) represents the Sun, and \(F, G, H, L, K\) represent various positions of the Earth. When the Earth moves from point \(F\) to point \(G\), i.e., when it moves towards Jupiter, the time between two eclipses decreases because light has to travel a shorter distance. Conversely, when the Earth moves away from Jupiter, from point \(L\) to point \(K\), the time difference between two eclipses increases. For one orbit, the time difference is small, but if we consider several orbits together, that difference accumulates, becoming considerable enough for measurement.

Let’s simplify Rømer’s logic a bit. Let’s assume that the radius of Earth’s orbit is \(D\), the period of a satellite’s orbit is \(t\), and \(c\) is the speed of light. Then, when Earth is moving from point \(H\) to point \(E\) (meaning it’s moving from conjunction to opposition in astronomical terms), if we observe that it takes time \(T_1\) for a satellite to complete n orbits, we can write \(T_1 = n t + D/c\).

Again, if the Earth takes time \(T_2\) for \(n\) orbits while moving from point \(E\) to point \(H\), then we can write \(T_2 = n t - D/c\). Therefore, \(T_1 - T_2 = 2 D/c\), or \(c = 2D(T_1 - T_2)\). According to Rømer’s calculations, this time difference is the maximum for Jupiter’s moon Io: twenty minutes. So, it takes twenty minutes for light to travel across Earth’s orbit.

Rømer’s work, published by the Royal Society. (Source: Internet Archive)

Stellar Aberration and the velocity of light

What is Stellar Aberration? In the image above, light comes to the telescope from a star (\(S\)) directly above the head. The telescope has moved a bit forward from position \(T_1\) to \(T_3\) by the time the light travels from the first (\(p_1\)) to the last point (\(p_3\)), because Earth is in motion. Therefore, the telescope needs to be angled a bit away from directly toward the star, and the star appears to be at the point \(S'\). Similarly, six months later—when Earth’s speed is in the opposite direction—the telescope needs to be angled in the opposite direction. This is called aberration. Light takes \(h/c\) time to cross a distance of \(h\), and the telescope moves a distance of \(v h/c\) forward during that time. Here, \(c\) and \(v\) represent the speeds of light and Earth, respectively. From the image, it can be seen that \(\tan(\theta) = (v h/c)/h = v/c\). The angle is very small, so expressed in radians, \(\tan(\theta) ≈ \theta\). In that case, \(c = v/\theta\).

Bradley did not start with a star directly above the head; the calculation would be a little harder in that case. In modern calculation, the value of aberration is 20.479 seconds, which is approximately 9.9285 \(\times 10^(-5)\) radians. If we assume Earth’s average speed to be 29.78 kilometers per second, then the speed of light is calculated to be 2.99944 \(\times 10^8\) meters per second or 299944 km/s. The modern value for the speed of light is 299792 km/s; it differs from Bradley’s method by only 0.05% or one part in two thousand. Bradley’s time, however, did not allow for such precise measurements of aberration or Earth’s speed, so his measurement had an error of 0.4%.

Video of Stellar Aberration. (Source: YouTube)

Fermat’s Principle

Consider that light rays coming from the air are going into the water and getting refracted. The refractive index of water relative to air is denoted by \(\mu\), so if the speed of light in air is \(c\), it will be \(c/\mu\) in water. Fermat wrote that light travels slower in denser mediums, meaning the refractive index of water is higher than 1. Light travels from point \(A\), passes through point \(P\), and then goes to point \(B\). It is thus clear that the values of angles \(i\) and \(r\) depend on the position of point \(P\). The time it takes for light to go from point \(A\) to point \(B\) is \(l_1/c + l_2/(c/\mu)\). Fermat showed that the path for which the least time is taken, the position of \(P\) is such that the ratio of \(\sin(i)\) to \(\sin(r)\) becomes constant. In the case of reflection, light travels through the same medium, so the speed remains the same before and after reflection. Thus, when the length of the path between two points is at its extreme (i.e., maximum or minimum), the time taken for the path also becomes extreme. Hero assumed that the speed of light is infinite; time is not involved there because light takes no time to travel in that case. Hero’s principle is thus a special case of Fermat’s principle. Fermat’s method had the seed of the calculus of variations hidden in it, which Newton later applied effectively.

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