Laura Kranendonk

Basic Optics

Laura Kranendonk

ERC, UW – Madison

2004

Table of Contents

Basic Optics

1. Introduction…………………………………………………1
2. Properties of Light…………………………………………..1
3. Color………………………………………………1
4. Spectral Power………………………………………2
5. Polarization………………………………………….4
6. Optical Fundamentals……………………………………….4
7. Fermat’s Principle……………………………………5
8. Snell’s Law………………………………………….6
9. Fresnel Equations…………………………………….7
10. Etendue………………………………………………9
11. Coherence……………………………………………11
12. Scattering…………………………………………….13

Optical Laboratory Toolbox

1. Introduction……………………………………………….15
2. Creating Light…………….……………………………….15
3. Lasers……………………………………………..15
4. LEDs……………………………………………..19
5. Directing Light……………………………………………..19
6. Lenses/Mirrors…………………………………….19
7. Fibers……………………………………………..22
8. Controlling Wavelength…………………………………….24
9. Diffraction Grating………………………………….24
10. Etalons……………………………………………..26
11. Harmonic Generation………………………………27
12. Measuring Light……………………………………………..28
13. Detectors……………………………………………..28
14. Cameras/Dyes………………………………………28

Appendix

1. SI Prefixes……………………………………………..29
2. Laser Classifications………………………………………..29
3. Corning SMF-28 Optical Fiber Product Data sheet………NA
4. Helpful websites……………………………………………..30

Basic Optics

Introduction

In 1960, the first Ruby laser was built [1], opening the door for many new optical inventions and research topics. Mechanical engineers in particular are currently working on optical methods for measuring fluid and gas properties, measuring mechanical stresses, and manufacturing techniques. In order to understand these new developments, a basic understanding of light and light properties is required. These notes are meant only as a review of light topics in order to understand principles commonly used in the mechanical engineer’s optical laboratory. The physics behind the materials being presented are generally covered in introductory physics courses.

Properties of Light

Visible light is a small part of the electromagnetic spectrum. Electromagnetic waves have electric and magnetic fields perpendicular to the direction of motion. They are able to travel through a vacuum or a medium. There are several properties commonly used to describe light such as color (wavelength, wavenumber, frequency, energy), and power (intensity).

Figure 1.1: Electromagnetic spectrum,

Wavelength (): measured in nanometers (nm), micrometers (m), or angstroms (, 1= 0.1 nm), the distance from one peak to the next.

Frequency (f): measured in Hertz (Hz, 1Hz = 1s-1). Frequency is the inverse of the time it would take a wave to travel 1 wavelength. To calculate frequency from wavelength:

(1.1)

Speed of light in a vacuum (c)  3 x 108 m/s

Wavenumber (): measured in inverse centimeters (cm-1). The wavenumber is how many waves fit in the distance of 1 cm.

Energy (E): measured in J/mole, J/photon, or electron volt (eV, 1eV=1.6 x 10-19 J per photon or per mole of photons). Energy of the wave can be calculated directly from the wavelength or frequency using the following equation:

(1.2)

Planck’s constant, h=6.626 x 10-34 Js.

______

Example 1.1:

a. Consider a 1550 nm wave. What are the frequency, wavenumber and energy (in a vacuum)?

Frequency:

= 194 THz

Wavenumber:

= 6451.61cm-1

Energy:

= 1.285x10-19J/photon

b. Consider the diagram below. What is , , and f?

When calculating differences, it is important convert into common units before taking the difference.

 = 700nm-600nm = 100nm

 = = 2380.95cm-1

(Note: realize that directly converting 100nm to cm-1 2380.95cm-1)

= 7.14x1013Hz

(Note: again, converting 100nm to Hz does not equal 7.14x1013Hz)

______

Spectral Power: measured in W/nm (power per wavelength). Light bulbs have much less spectral power than lasers since light bulbs emit many wavelengths – this is known as broadband light, as opposed to laser light, which is essentially a single wavelength.

Figure 1.2: Spectral power comparisons of lasers versus light bulbs.

______

Example 1.2:

The diagram in figure 1.2 shows the difference in spectral power of a 1 mW laser pointer versus a typical light bulb. Considering that an average laser pointer has a bandwidth of 100MHz (this is f) centered at 650 nm, compute the spectral power.

First, we need to convert the bandwidth (the wavelength range of the laser) into nm:

Then,  = 0.0001nm

Then, we can compute the spectral power:

=10 W/nm

______

Polarization: When the electromagnetic waves composing a light beam vibrate in the same direction, the light is said to be polarized. Polarized light is generally classified into 2 groups depending on how the electric waves are aligned with the plane of incidence. The plane of incidence is the plane composing the incident, reflected, and transmitted rays. Transverse magnetic (TM), parallel (||), P, and O are all ways to refer to waves that are polarized such that the electric field is in the plane of incidence. Transverse electric (TE), perpendicular (), S, and E are all ways to refer to polarization perpendicular to the plane of incidence. The polarization of light can affect many aspects of optics. Materials can have different properties for different polarizations, such as indexes of refraction (birefringents) or reflectivity. In addition to these polarized cases, light can be randomly polarized (also called natural light), or circularly or elliptically polarized.

Figure 1.3: Parallel polarization diagram.

Figure 1.4: Perpendicular polarization diagram.

Optical Fundamentals

In this section, fundamental optical principles will be reviewed in order to understand light behavior. Once these basic principles are understood, complex optical applications and tools can be studied. Therefore, it is very important to have a good understanding of these fundamental principles.

Fermat’s Principle

Fermat’s principle states that light will take path with the shortest travel time to go from one point to another, as shown in figure 4.

Figure 1.5: Illustration of Fermat’s principle, light will travel in a straight line from one point to another since a straight line will be the quickest path.

Fermat’s principle can be used to show the behavior of reflection. Depending on the incident angle of the light onto a surface (say a mirror), light will only reflect at that same incident angle. Therefore:

(1.3)

Figure 1.6: Fermat’s principle on a mirror. The beam from the dashed path is impossible because it is a longer travel time than the solid line path.

Fermat’s principle can also be used to show how a lens focuses light. In order to understand this phenomenon, it is important to know that light travels at different speeds depending on the index of refraction (n) of the material. Glass typically has an index of refraction of about 1.5, whereas air has an index of refraction of essentially 1 (c = speed of light in a vacuum). The speed that light travels through a medium is:

(1.4)

In figure 6, it can then be deduced that light will travel from point A to point B in the same amount of time (and therefore the shortest amount of time) for all of the lines. (Hint: the thickest part of the lens has the overall shortest path length, but since the light will travel slower in the glass, the overall time to travel from A to B will be the same for the centerline).

Figure 1.7: Light pathways through a lens, point A and B are at twice the focus length of the lens.

Snell’s Law

Snell’s law predicts the direction of light as it travels through different mediums. Snell’s law can be stated by the following equation where n is the index of refraction, and i, t, and r stand for incident, transmitted, and reflected.

(1.5)

Figure 1.8: Diagram of Snell’s law, showing light traveling from air to glass.

Total internal reflection can occur when light in a high index medium reaches a low index medium (ni > nt). In these conditions, when t=90o, the corresponding incident angle is called the critical angle. Any incident ray at the critical angle or higher will cause total internal reflection, meaning all of the light is reflected and none is transmitted. This will become important when dealing with fiber optic cables.

______

Example 1.3:

1. Consider a beam traveling from air (n=1) to glass (n=1.5) at a 30o incident angle. Sketch the system, calculating the reflection and transmission angles.

= 30o(Fermat’s Principle)

(Snell’s Law)

=19.47o

1. What is the critical angle (onset of total internal reflection) for a glass/air interface? Zinc-Selenide/air (n=2.5)?

To find the onset of total internal reflection (TIR), set the transmission angle equal to 90o. Therefore:

Glass:

= 41.8o

Zinc-Selenide:

= 23.58o

______

Fresnel’s Equations

The amount of light that is transmitted or reflected can be determined by the Fresnel equations. The ratio of reflected to incident power (or the fraction of power reflected) depends on the polarization of the light, and can be written as:

(Parallel polarized light) (1.6)

(Perpendicular polarized light) (1.7)

which can be simplified to when i=90. R is also called the reflected power coefficient. If no light is absorbed in the material, transmitted power can then be calculated by initial power multiplied by:

(1.8)

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Example 1.4:

1. If a 1mW laser beam is directed through a glass microscope slide, what is the transmitted power? (Assume there is no power absorption in the glass)

Using the Fresnel equations, it is possible to calculate the reflected power coefficient for both the air/glass interface, and the glass air interface. It turns out that these are the same:

Since there is no absorbance, the transmitted power coefficient is then just 1-R. Therefore, 96% (1-0.04) of the power is transmitted, or 0.96mW.

1. What would be the transmitted power if there were 50 microscope slides between the laser and detector? (Assume an air gap between each slide)

Realizing that at each interface, 96% of the power is transmitted, and there are a total of 100 interfaces, it can be deduced that:

Therefore, 1.69% of the original power or 0.0169mW is transmitted.

______

Figure 1.9: Reflection versus incident angle for an air/glass system.

Equations (1.4) and (1.5) can be plotted versus incident angle, as shown in figure 1.9. At 56.4o, there is no reflection for one of the polarizations. This is a special incident angle known as Brewster’s angle. By manipulating the incident angle, it is possible to polarize unpolarized light.

Example 1.5:

If there were the same 50 slides as in example 1.3, but now the incident angle was at Brewster’s angle, what would the transmitted power be? (Assume the light is unpolarized, and equal amounts of perpendicular and parallel-polarized light)

Answer:

At Brewster’s angle, none of the parallel-polarized light will be reflected (100% transmission, or 50% net transmission). From equation 7 (or from the graph in figure 8), the reflection for 1 air/glass and glass/air = 0.149. Therefore, the perpendicular polarized light transmits 85.1% (1-0.149) at each interface, so over all 100 interfaces, 0.851100=9.84x10-8 is transmitted (essentially 0).

Overall, the total transmission is then 50%.

Etendue

Etendue, also known as the parameter product, can be thought of as the quality of light, and is quantified by size multiplied by divergence. The smaller the etendue, the better collimated the light can be. Etendue is the optical equivalent of entropy; at best it will remain constant, however it will never improve. Figure 1.10 shows how etendue is calculated and graphically represented.

Figure 1.10: Graphical representation of etendue.

The smallest etendue possible is called the diffraction limit. The diffraction limit is wavelength dependent, and can be calculated from the following equation, where d is the “spot size”, the diameter of the light source:

(1.8)

The diffraction limit therefore leads to the minimum spot size that a beam could produce (rather than an infinitely small point). Lasers are able to produce light that is near diffraction limited. A genuine increase in etendue is called beam steering.

Example 1.6:

1. Light at 610 nm, leaves a laser at the diffraction limit. The laser has a 1m diameter emitting area. What is the divergence angle of the laser?

= 11.125o

1. If the laser beam were collected in a 1 cm diameter lens at the focal length, how large would the projected spot be 10 km from the lens?

First, calculated the divergence angle leaving the lens:

= 0.002o

Next, using trigonometry, figure out the size of the spot:

= 0.379 m

= 0.45 m2

Coherence

When 2 or more light beams are superimposed on each other, they have the potential to be either coherent or incoherent. When coherent light has the same peaks and valleys, it constructively interferes, whereas if a peak lines up with a valley, there is destructive interference, which weakens the signal.

When waves of different wavelengths are traveling along the same path, it is useful to know at what length the waves become incoherent. This ‘coherence length’ can be calculated with the following equation:

(1.9)

Example 1.7:

Two 10 mW lasers emitting very nearly the same color are arranged as shown below. The top laser is a free-running laser, and the bottom is a stabilized laser; the stabilized laser has a significantly smaller linewidth. The ‘full width half max’ (FWHM) of the top laser is 0.1 cm-1, and the FWHM of the lower laser is 300 kHz. FWHM is the bandwidth at half the maximum, since most lasers and other optical sources produce waves on a Gaussian curve FWHM is a good approximation of bandwidth.

Free running laser: fr= 700.01 nm, FWHM = 0.1cm-1

Stabilized laser: s = 700 nm, FWHM = 300 kHz

1. What is the coherence length of the 2 lasers?

Free running laser:

fr = 0.0049 nm

= 0.1 m

Stabilized laser:

s=4.9x10-7 nm

= 1,000 m

1. The detector is not reading a signal. Why could this be?

Answer:

Since the coherence length in the free-running laser is shorter than the path it is traveling, the light becomes incoherent, and therefore cannot align with the stabilized laser.

Scattering

When light hits molecules, it can become scattered in several ways. When light is scattered by very small molecules (d < ) elastic (Rayleigh scattering), inelastic with increased energy (Anti-Stokes Raman scattering), or inelastic with decreased energy (Stokes Raman scattering) scattering can occur. When light is scattered by larger molecules (d>), Mie scattering occurs.

The incident power, density of molecules, volume of interest, and collection angle of the optics can determine the scattered power for each type of scattering. The equation for scattered power is as follows:

(1.10)

= ratio of scattered power to incident power

= differential scattering cross-section

 = solid angle of collection [Sr], 1 Steradian = 4*radians

n = molecular density

(at standard temperature and pressure, air has a molecular density of:

2.7*1019 molecules/cm3)

Rayleigh:

(1.11)

for eqn. (1.11), n = refractive index of molecule

No= number density at STP

This inverse relationship with wavelength can be used to explain why during the day, the sky is blue and not red. As the sun sets, the sun’s rays travel through more of the atmosphere, allowing more scattering, causing the sunset to be a redder color.

Raman:

Mie:

d = diameter of scattering molecule

Wavelength changes due to Raman scattering can be calculated by the following equations:

Anti-Stokes Raman:(1.12)

Stokes Raman:(1.13)

Example 1.8:

A 1 W laser at 500nm is directed into air at room temperature and pressure. If a detector could pick up 10% of the area of a sphere, and 1 cm long, how much power would be detected due to Rayleigh scattering?

Answer:

n=2.7*1019 molecules/cm3

 = 0.1*4 = 1.26

L=1 cm

Ps = 2.86x10-8 W

Conclusions

In this section, we have developed the tools necessary to understand phenomena dealing with optics. With these fundamental principles, we can now move ahead to learning about the instruments used in optical laboratories, and other optical applications.

Optical Laboratory Toolbox

In the optical laboratory, it is important to be able to control the direction, wavelength, and spectral power of light. Using the principles developed in the previous section, it is possible to do this in many ways. Lasers and LEDs can be used to create light at specific wavelengths and powers. Lenses, mirrors, and fiber optics can be used to direct light. Diffraction gratings and etalons are examples of ways to control the wavelength. Finally, various detectors can be used to measure light properties.

Creating Light

When we think of artificial light sources, the first thing that comes to mind is probably a light bulb. Light bulbs give off incoherent light over a wide range of wavelengths. Heating metal filaments creates light in standard light bulbs. When specific wavelengths or coherent light is required, lasers or LED’s are the usual method.

Lasers – Light Amplification by Stimulated Emission of Radiation

Lasers produce coherent electromagnetic waves at narrow wavelength bands, either continuously or pulsed. There are three parts to a laser: gain medium, pump, and feedback. The basic idea behind a laser is that first atoms in the gain medium are excited to higher than usual energy states. Once enough atoms are excited, a photon causes them to drop to a lower energy state. In going from a higher energy state to a lower energy state, the atoms give off another photon at a specific wavelength. These photons cascade throughout the medium between mirrors, causing more the process to continually repeat. Finally, one of the mirrors will have only partial reflection, allowing a beam to exit the cavity (see figure 2.1). Lasers have the potential to have very high power, and could be dangerous if proper safety precautions are not used. Appendix 3 breaks down the types of lasers into classifications based on power and potential harmfulness.

Figure 2.1: Simplified diagram of a Ruby laser. Ruby (Al2O3 + Cr) is the gain medium; lamps are used to pump the ruby to an excited state with mirrors on either end of the cavity.

Lasers are classified according to the medium that is being excited. The three basic categories are solid (i.e. Ruby), gas (i.e. He-Ne), and liquid (dye) lasers. The medium determines the wavelength that the laser produces. Gain mediums can be very different, but essentially they all need to do the same thing. First, a “population inversion” needs to occur. Individual atoms normally are found in a ground energy state. When 1 electron is excited and moves to a different orbiting shell, the atom has an overall higher energy state. Population inversion occurs when there are more atoms in the excited state than the ground state. An example of a noble gas (Ne) excitation is shown in figure 2.2, and population inversion is shown in figure 2.3.

Figure 2.2: In naturally occurring neon, there are 2 electrons in the inner shell (K), and 8 in the outer shell (L). In one excited state, one electron is moved to the third shell (M).

Figure 2.3: Population inversion occurs when more atoms in a sample are excited rather than in the ground (low energy) state.

Thermodynamically, it is very difficult to raise atoms to higher energy states, however optically it is much easier. The 3 main transitions of photons important to laser understanding are shown in figure 2.4. Stimulated absorbance pumps the atoms to the higher energy states. Stimulated emission produces the desired electromagnetic wave. Spontaneous emission occurs naturally. In a laser, relative to simulated emission, spontaneous emission is rare.