4. LASER MODES

Originally by Tim Saint and Asa Hopkins (students)

(Revised by Jerry Gollub, 2002; further revised by Suzanne Amador Kane 2005)

“Lasers are amazing devices which emit beams of light powerful enough to vaporize a

bulldozer, yet are so precise that they can be used in delicate optical surgery, provided the

surgeon remembers to change the setting on the laser to ‘delicate optical surgery’ from

‘vaporize bulldozer.”

- Dave Barry, Haverford alumnus, on the topic of lasers

Introduction and Background

Dave Barry is right about one thing: Lasers are useful. Their applications are myriad

and diverse- but in all cases lasers are useful for the simple reason that they emit light

with a narrowly defined wavelength with a well-defined direction, as opposed to the

traditional light bulb, which emits a broad spectrum of light wavelengths in diffuse

directions. (Scientifically, we call this temporal coherence—a well-defined frequency of

light is emitted with a well-defined phase—and spatial coherence—the light is also

emitted in a highly parallel beam.) However, the wavelength distribution of laser light is

more complicated than you might initially think. Rather than emitting at a single

wavelength, the laser instead emits light at several distinct wavelengths, in a relatively

tight, Gaussian distribution centered at the ideal wavelength.

This distribution of wavelengths has important ramifications – but primarily for

people who have to spend a lot of time with lasers. As long as your CD player is working,

you don’t need to worry too much about the results of this distribution. Rather, as a

physics student, you should be concerned with the causes of this distribution, many of

which turn out to be (surprise!) quantum mechanical.

In this laboratory you will use a Fabry-Perot interferometer to measure the intensity

of light emitted from the laser as a function of frequency. You should find that the

radiation is not monochromatic, but rather that the radiation is concentrated in certain

discrete modes that are characteristic of the laser cavity.

Additional References:

Donald O’Shea, W. Russell Callen, and William Rhodes, Introduction to Lasers

and Their Applications, pp. 126-131 (Reserve Reading)

4-1

4-2

Laser Modes

Yariv, Introduction to Optical Electronics, Holt, Reinhart and Winston, New

York, 1975. (Reserve Readings)

Lengyel, Introduction to Laser Physics, John Wiley and Sons, New York, 1966.

(Reserve Readings)

T. Kallard, Exploring Laser Light, pp. 1-6

Frank J. Blatt, Modern Physics, pp. 191-198

The Helium-Neon Laser

The word laser stands for light amplification by stimulated emission of radiation.

The Helium-Neon laser is commonly used in instructional laboratory experiments and

applications. This laser utilizes many of the same strategies as other lasers do to cause

stimulated emission of light. The basic idea is as follows: the laser consists of a container

of Helium and Neon gas atoms (normally in a 10:1 ratio), with mirrors at both ends and a

voltage applied along its length. When the voltage is applied, an electric discharge causes

electrons to strike the Helium atoms and raise them to the excited 1s12s1 state. (See Fig.

1.) The goal is what is called a population inversion, where more Helium atoms are in an

excited state than in the ground state. This happens because the excited state is

metastable — atoms in that state decay relatively slowly to the ground state, so that the

excitation process causes them to accumulate in the excited state.

Laser Modes 4-3

Figure 1 Energy levels for the Helium-Neon laser. This exact energy level diagram shows the more commonly used red lasing line. The energy level diagram for the green lasing line used here is similar. Source: http://kottan-labs.bgsu.edu/teaching/workshop2001/chapter4a_files/image024.gif

Once the helium atoms are excited, the neon atoms come into play. The energy

needed to excite helium to the 1s12s1 state is almost exactly the same as the energy

needed to excite the neon to its 2p55s1 state. Once the helium population is successfully

inverted, excited helium atoms will strike neon atoms and transfer their energy to the

neon (the helium atoms then return to their ground state). Since the 2p55s1 neon excited

state is more stable than those below it, there is a buildup of neon atoms in this state,

while the atoms in other states decay to the ground state. Once a photon is released from

the decay of the neon 2p55s1 state to the neon 2p53p1 state through ordinary random

emission of a photon (called spontaneous emission), this photon can interact with another

neon atom and force it to de-excite in the same way. When a photon stimulates emission

of a second photon in this fashion, the second photon is emitted with exactly the same

energy, phase and direction as the first. This purely quantum mechanical effect is called

stimulated emission.

Many of the photons simply escape from the sides of the cavity, never to be seen

again. However those that happen to travel along the axis of the cavity and have the

4-4

Laser Modes

correct frequency to interfere constructively after reflection by the mirrors will survive

for a long time, and can induce even more neon atoms to radiate photons into the same

state, thus forming a phase coherent and parallel beam. Thus, a standing wave develops

between the two mirrors at either end of the cavity. The allowed wavelengths are those

for which the cavity length is an integral number of half wavelengths.

Photons are not trapped in the cavity completely, or no useful laser would exist, but

most photons must be reflected at the mirrors in order to maintain constructive

interference. For this reason, the mirrors are usually 99% reflective at the front mirror,

and 99.9% at the back one (from Exploring Laser Light by T Kallard). This means that a

particular photon will reflect about 100 times, giving a high probability to induce

stimulated emission, before emerging from the front mirror and leaving the cavity. This

abbreviated discussion is not meant to substitute for reading in the library!

Line Broadening Effects and Laser Modes

Since a laser works by producing many identical photons, each with the same

frequency E/h, the light from a laser exhibits that frequency. For the laser used in this

experiment (UniPhase model #1674P), the listed wavelength of emission is 543.5 nm.

Thus, its plot of intensity versus wavelength should be zero everywhere except at a

wavelength of 543.5 nm. However, several effects combine to produce broadening of the

emitted spectral line.

Figure 2 shows a plot of intensity versus frequency that demonstrates the presence of

more than one wavelength. The peaks are a result of longitudinal laser modes, as we will

explain shortly. This plot also shows a high level of resolution, with ten laser modes

visible. Although eight to ten laser modes are typical, you most likely will not be able to

attain such high resolution. The broadening occurs from a variety of effects, two of the

most important of which are Doppler broadening and lifetime broadening.

Laser Modes 4-5

Figure 2. High resolution intensity versus frequency plot, showing broadening and modes. (Taken from Yariv, Introduction to Optical Electronics)

Doppler Broadening

Recall the Doppler effect from first-year physics. An object at rest emits a wave of

some fixed frequency. An observer also at rest senses the wave at the same frequency.

However, if the source of the wave is moving with respect to the observer with the

frequency held constant at the source, the space between peaks of the wave will grow or

shrink for the observer at rest, resulting in a higher or lower frequency wave sensed by

the observer. For example, stars that are moving away from the Earth are seen to be “red-

shifted” because the Doppler effect has lengthened the wavelength.

Within the laser, a similar effect occurs, although at a much smaller scale. The

random thermal motions of the neon atoms in the hot gas affect the wavelength of light

emitted by the laser. While the energy of the atomic transition remains constant, the

wavelength will be observed to be shorter if the emitting atom is moving towards the

photodiode, and longer if it is moving away. The random distribution in the motions of

the neon atoms results in a smearing of the emitted frequency, an effect called Doppler

broadening.

Pre-lab question: Derive a formula for the Doppler broadening for a gas of neon atoms

at absolute temperature, T, using your intro physics expression for frequency (or

wavelength) shift as a function of speed of the emitting object and the equipartition

4-6

Laser Modes

theorem. Typical values quoted in the laser literature for helium-neon lasers are 800 to

1600 MHz FWHM (Full Width at Half Maximum), so you can extract an approximate

laser temperature from your formula.

Selected Readings covering Doppler Broadening:

Taylor, John R. and Chris D. Zafiratos, Modern Physics for Scientists and Engineers, pp. 35-39

Bransden, B.H. and C. S. Joachan, Introductory Quantum Mechanics, pp. 73-75, 513-514

O’Shea, Donald C., W. Russell Callen, and William T. Rhodes, Introduction to Lasers and Their Applications, pp. 81-83 (Library Reserve)

Lifetime Broadening

The Heisenberg uncertainty principle tells us that the product of the uncertainty of the

energy of a system (E) and the uncertainty of its lifetime (t) is always greater than or

equal to h/4 (Et ≥ h/4).

As described above, the photons in the cavity of a typical HeNe laser are produced

when 2p55s1 neon atoms decay to the 2p53p1 energy state, and the energy of each of these

photons is equal to the energy difference between these two energy states. The average

duration, t, of the neon excited state determines the uncertainty in the energy of the

photons emitted. (Long-lived states produce sharper spectral lines than do states with

short lifetimes.) Since E=hc/, the uncertainty in E also leads to an uncertainty in , since

h and c are known constants having little (h) or no (c is a standard for defining other

units!) uncertainty attached to their values. Thus, this quantum mechanical effect is a

further cause of the broadening of the laser spectrum, although much less significant than

Doppler broadening.

Pre-lab question: Look up a typical value of a radiative lifetime and the associated

broadening in frequency for a typical atom. You can find a value in Griffiths,

Introduction to Quantum Mechanics (pg. 359), Eisberg and Resnick, Quantum Physics of

Atoms, Molecules, Solids, Nuclei and Particles, (pg. 75-76) or other quantum textbooks

under discussions about the energy-time uncertainty principle, and other places.

Recalling that the helium-neon atoms emit light from a more stable state than is typical,

would you expect this lifetime broadening effect to be significant compared to the

Doppler broadening computed above?

Laser Modes 4-7

Selected Readings covering Lifetime Broadening: Saxon, David, Elementary Quantum Mechanics, pp. 200-201 Bransden, B.H. and C. S. Joachan, Introductory Quantum Mechanics, pp. 73-75,

508-512 O’Shea, Donald C., W. Russell Callen, and William T. Rhodes, Introduction to

Lasers and Their Applications, pp. 85-88

Laser Modes

As you may have noted from Figure 2, there is more going on than simple

broadening. The other main feature observed is the phenomenon of laser modes. These

peaks result from the constructive interference of many wavelengths within the laser

cavity with length d, but they are distinct from each other. Since the photons form a

standing wave in the laser cavity, they must satisfy the condition, . The frequency

difference, LM , between these modes is c/2d. The broadening from Doppler and

lifetime effects covers more than one of these modes, so the peaks you see are guided by

the overall Gaussian shape of the broadening effects.

Figure 3 clearly shows the fundamental physics of this process. Again, the peaks

shown in the last graph are not quite what you will see. Here each mode is represented by

a sharp line, but your spectra will have 2 to 4 (more only if you are very careful and

lucky) overlapping peaks.

4-8

Laser Modes

Figure 3. Illustration of how the broadened laser transition combines with the longitudinal cavity modes to create the laser output. (Taken from O’Shea et al., An Introduction to Lasers and Their Applications.)

Pre-lab question: Assuming your laser cavity is approximately 0.5 m long, what is the

number n in the equation for the standing wave criterion above? (Do not be surprised if n

is an extremely large number!) What would the spacing between wavelengths be in the

plot in Fig. 3 as a result? If temperature variations change the laser cavity length, they

will shift these frequencies by an amount that depends upon the cavity length, d, and the

thermal expansion coefficient of the laser cavity’s materials. Assume that your laser is

made out of Super Invar, a material selected for very low coefficient of thermal

expansion of less than 0.63 parts per million (PPM), by how much would the frequency

change for a one degree Celsius shift in temperature for a 0.5 m long laser? For a 10 cm

long laser pointer? Taking into account the numbers given above for linewidth

broadening, what problems might your last calculations imply?

Read all about laser modes in: O’Shea, Donald C., W. Russel Callen, and William T.

Rhodes, Introduction to Lasers and Their Applications, pages 89 to 97.

Laser Modes 4-9

Fabry-Perot Interferometer

The Fabry-Perot interferometer transmits light of very specific wavelengths only. As

is the laser itself, the FP interferometer is an optical resonator. It consists of a pair of

partially silvered plane mirrors. When a half-integral number of wavelengths exactly fits

in the distance L between the plane mirrors, radiation can build up in the FP cavity, and

an enhanced amount gets through the other side as well. Piezoelectric transducers, which

are attached to one of the mirrors, can make very small accurate changes in the distance

between the mirrors, thus changing the wavelength of light that is transmitted. Changing

the voltage on the piezoelectric transducers varies the distance at which the mirrors are

held apart. (Piezoelectric materials are also used in Scanning Tunneling Microscopes, to

precisely position the tip.)

The interferometer control circuit applies a voltage that varies with time to the

transducers. The time dependence is not sinusoidal; rather, the voltage increases slowly

from the minimum to the maximum voltage, then drops quickly back down to the

minimum (a sawtooth wave). Both the speed at which the voltage increases and the range

of voltages over which it sweeps can be adjusted, and you should take advantage of these

options.

When the voltage range is large enough, you will see two nearly identical sets of laser

modes. These correspond to differences of one in the number of half wavelengths in the

interferometer cavity. The separation (in frequency or wavelength) of these identical

copies is determined by the distance between the mirrors and is called the free spectral

range (FSR):

.

(1)

Knowing this, you can calibrate the horizontal axis of your plots by choosing a voltage

range that allows you to see two sets of modes, and then reading mirror separation, L, off

of the interferometer. In other words, a FP interferometer cannot make an absolute

frequency measurements. Since the entire pattern repeats as the FP cavity is expanded by

the applied voltage, the instrument only allows you to measure frequency differences.

4-10

Laser Modes

Learn your interferometer inside and out. We recommend the Instruction and

Maintenance Manual for Tropel Model 350 Fabry-Perot Interferometer and 351 Linear

Lamp Generator that is provided by your lab instructor.

Interpreting Fabry-Perot Spectra

The free spectral range (FSR) of a Fabry-Perot is one of two parameters that

characterize its performance. The following plot shows schematically what one would

see if you had a source of light that emitted one precisely defined wavelength, , with

absolutely zero width. (While this is a very unphysical situation that does not represent

your experimental situation, it does allow us to understand some features of the Fabry-

Perot.)

Figure 4 from Burleigh Fabry-Perot manual

Figure 4 represents the output of your Fabry-Perot if it is operated in the scanning

mode, analogous to your experimental situation. The y-axis represents the intensity of

light transmitted by the Fabry-Perot. The x-axis corresponds to wavelength of the

transmitted light. First, note that there is a repeat distance, FSR, between the peaks.

Second, note that there is a finite width, min, of each peak, even though the source of

light emits light with one precisely defined ; in other words, the width of each peak

must be a characteristic of the Fabry-Perot itself. Let us now examine the source of each

effect.

Laser Modes 4-11

When you scan the Fabry Perot, you change its length L. For a given length L1, there

is a resonant wavelength given by:

, where n is an integer. (2)After translation to a new length L2, you have a new resonant wavelength ,

and the difference in wavelengths is

(3)

Suppose that the translation distance L is exactly . (This corresponds to the

repeat distance in the spectrum.) Then:

.

(4)

In the last step, we dropped the distinction between the two lengths (which differ by only

one part in a million or so) and wrote it simply as L. This sets the scale of your spectrum.

This repeat distance is called the free spectral range (FSR).

I find the above derivation a bit confusing, since it’s a lot easier if you just do it using calculus. What is

really changing is n, the number of modes, as you change the cavity length, L. So, let’s compute:

We’ll use this to compute the absolute value of for discrete changes in n: n = 1:

The last step comes from using equation (2) to obtain n in terms of and L. We get equation (4) again

since the derivations are equivalent.

In the next step, you’ll also want to bear in mind the relationship: = c/so:

4-12

Laser Modes

The FSR can be expressed equally well as a frequency difference (rather than a

wavelength difference) by computing the change in frequency, to obtain:

.

(5)

These relationships allow you to calibrate the repeat interval in your spectrum in either

wavelength (or frequency) if you have measured the actual Fabry-Perot mirror separation,

L (and its associated uncertainty.)

This explains why the Fabry-Perot spectrum repeats itself every FSR FSR, but it

does not explain the nonzero width min of each peak. To understand this effect, we

turn to a discussion in a textbook (such as Introduction to Optics by Frank L. Pedrotti and

Leno S. Pedrotti, Prentice-Hall, Englewood Cliffs, NJ—a copy of the relevant sections is

to be found by the Fabry-Perot itself in lab.) There, the authors show how the

transmission between two parallel mirrors with a incident ray of light incident at some

finite angle depends upon the angle of incidence and the nonzero transmission

coefficients of the mirrors (and their consequent imperfect reflectivities). There you can

find worked out the minimum separation between two different wavelengths, 1 and 2,

that can be distinguished by an imperfect Fabry-Perot. This limitation does not depend

upon the FSR, but upon imperfections such as the nonideal angle of incidence and

reflectivities of the mirrors, misalignment, nonideal mirror surfaces, etc. A quantity, F,

called the coefficient of finesse, can be computed from these values to explain the

nonzero width min which even perfectly sharp sources with a single wavelength would

have in a Fabry-Perot spectrum. This relationship can be found in the source cited on

page 296 and following pages. (Fabry-Perot manuals may cite specifications such as the

finesse for calculations such as these.)

Experimental Procedure

WARNING: You should know that looking directly into a laser is a bad idea for

those of you who value your retinas. Serious and unrecoverable retinal burns can

result if you look directly into a laser or a strong laser reflection. If William Gibson

is right, your retinas should fetch you a fair price on the black market in about

Laser Modes 4-13

twenty years. User laser alignment goggles whenever possible and save those

retinas!

Alignment

Getting the FP interferometer to work requires careful alignment. and several

conditions must be satisfied simultaneously: (a) A sufficient amount of the incident light

must be parallel to the axis of the FP cavity; (b) The mirrors in the FP cavity must be

precisely parallel to each other. The method is described in the Burleigh interferometer

manual, pp. 7-10 (available in the lab).

1. Coarse alignment (may have been done already so it might be omitted; check with

instructor.) One first sends a laser beam through a 3x5 card with a small hole and then

through the BACK side of the FPI (the side without micrometers). You first adjust the

incident beam to be precisely perpendicular to the first mirror to be encountered (by

making its REFLECTION go back through a hole in a 3x5 card. Then you make the

TRANSMITTED beam look like Fig. 3 from the FPI manual (see below), by adjusting

the micrometer screws by SMALL AMOUNTS. Change only TWO of the micrometers

that rotate the mirror about orthogonal axes. Figure out which micrometers to turn

BEFORE doing so; don’t change the third one, and please don’t rotate them by more than

1 turn without checking with an instructor.

Figure 5

2. Fine alignment: (The use of the beam expander for this part may or may not be

necessary. Consult your instructor if you are not sure. If you are not using a

beam expander, proceed to the next paragraph and follow the instructions there

for fine alignment.) After the course adjustment is complete, make a FINE

adjustment by directing the laser beam through a beam expander. The beam

should be approximately collimated, but should diverge slowly (i.e. its diameter

4-14

Laser Modes

increases gradually with distance). Turn the FPI around, and let the incident

beam go through its FRONT side, the one with the micrometers. Adjust the beam

so that it is again perpendicular to the incident mirror. (Since the beam diverges

slightly, of course, only some of the light will be precisely perpendicular to the

mirror and parallel to the FPI’s axis.)

Figure 6: The experimental setup

Whether or not you are using the beam expander, put a white card at the exit of the

FPI, and look for light (in the dark). If you see linear fringes, the two FPI mirrors are

tilted with respect to each other. Eliminate the fringes by turning TWO of the

micrometers that rotate the mirror about orthogonal axes (as before). Figure out which

micrometers to turn BEFORE doing so; don’t change the third one. When correctly

aligned, the fringes disappear and the intensity will be fairly uniform across the image.

(See Figure 7 below.)

Laser Modes 4-15

Figure 7: (left) fringes appearing for imperfect alignment, while (right) a uniform beam

results for perfect alignment. (Reproduced from Burleigh Fabry-Perot manual)

3. Alignment of the photodiode: Place a focusing lens at the opposite side of the

laser. This should focus the beam onto the photodiode. The photodiode itself should be

shielded by a pinhole with a very small aperture, which will allow only properly aligned

laser light to strike the photodiode. The pinhole will also prevent other light sources from

affecting the reading. The light emitted from the Fabry-Perot Interferometer is focused

into concentric circles. You want to measure only the point at the center of these rings.

However, the rings may not be centered on your photodiode initially. Your photodiode

also is covered by a laser line filter, with a transmission curve that looks like Figure 8.

This enables you to take measurements with the room lights on, since only the laser light

is appreciably transmitted.

Figure 8: Transmission spectrum for the green laser line filter. (Thor Labs)

QUESTION: Why does the lens focus the transmitted light from the Fabry-Perot

Interferometer into concentric circles? Think about light traveling at different angles with

respect to the interferometer axis.

4-16

Laser Modes

4. Electronic detection: Set your oscilloscope to x-y mode, with the output from the

Fabry-Perot interferometer being x, and the photodiode output being y. You now have a

graph of intensity vs. voltage.

Easy Question: Why is this really a graph of intensity vs. wavelength?

Adjust slightly the position of the photodiode to maximize the amount of incident laser

light the photodiode is measuring, but be sure to capture only the center of the bulls-eye,

or you will measure the intensity at all wavelengths at once, losing the spectral

information. At this point you should see only a dot on the oscilloscope. This is because

you are not varying the voltage of the interferometer. Turn the interferometer on. You

should now see the dot sweeping back and forth across the oscilloscope. If you are lucky

you will see the dot describing peaks as it moves from left to right.

5. Superfine adjustment: The FPI control box has 3 potentiometers that add small

voltages to the three micrometers, and hence allow a superfine adjustment. The center

pot labeled “common” adds a voltage to all three, and hence translates the spectrum with

respect to the free spectral range of the FPI. Adjust these pots (and possibly the height

of the pinhole) until the peaks are as SHARP as possible.

6. Record data: Once you have observed peaks you should record data. To reduce

the noise use a low pass filter on the photodiode output, and scan slowly so as not to

distort the signal. Use the ‘Store’ function on your oscilloscope to save the trajectory of

the moving dot on the CRT of the oscilloscope. (You should notice that the dot traces a

different path as it returns quickly from right to left than it does when moving ‘forward,’

from left to right. This phenomenon is due to hysteresis in the piezoelectric stacks, and

you want to avoid looking at the return path.

Plot the spectrum on a plotter. Optimize the spectrum by going over the

adjustments again.

You should take at least two spectrum a defined length of time (at least fifteen minutes)

apart.

Final check before you leave the lab:

Note that you should measure the Fabry-Perot mirror spacing, L, know the wavelength of

your helium-neon laser, estimate your laser mirror spacing, d, from the length of the laser

Laser Modes 4-17

itself, and understand everything you need to know about the experimental part of the lab

before you leave lab. You should have at least two best quality Fabry-Perot spectra for

your system, and you should record how far apart in time and under what conditions

your two spectra were recorded for later analysis.

Experimental Results & Analysis

To interpret your Fabry-Perot spectrum, you should account for all sources of

possible broadening of your observed spectra, and you should either measure, estimate,

or do a literature search to identify, all quantities which might usefully enter into your

discussion.

Sources of useful information might include Eisberg and Resnick, Quantum Physics

of Atoms, Molecules, Solids, Nuclei and Particles (for lifetime broadening; are your

values for this effect likely to be greater, equal to or smaller than any estimates you might

find?), your intro physics textbook (for how to compute the Doppler broadening—you

should have an argument regarding what causes the Doppler broadening and how to

estimate how large this effect should be in the width of your wavelength; you should

look on the web for some typical numbers and compute your own estimate of your value

from your spectrum! These websites are quite useful for many issues:

http://www.repairfaq.org/sam/laserhen.htm

http://hyperphysics.phy-astr.gsu.edu/hbase/optmod/lasmodes.html

Be sure to include these elements in your final discussion and data analysis:

1. Make sure that you understand precisely how the Fabry-Perot Interferometer (FPI)

works in detail, including the concept of the free spectral range (FSR—see discussion

below).

2. Analyze the shape of the spectrum on your plots. (This is basically the point of the

lab.) What controls the shape of the spectrum? Some possibilities are: Doppler and

lifetime broadening; FPI misalignment; and the finite reflectivity of the FPI mirrors. You

may need to do additional research for this. Estimate the overall width of the laser line

(by comparing the FSR with the full-width-at-half-maximum (FWHM) of your laser

emission curve), and compare this value to your estimated values for each of the possible

sources of laser line broadening you have identified. Measure the widths of individual

peaks and explain what factors determine your value.

4-18

Laser Modes

3. Determine the wavelength differences between the laser modes. Explain how this can

be done by comparing the peak spacing to the FSR and compute your actual value in

terms of wavelength. Compare your computed laser cavity length, d, with your estimate.

4. What happens to your spectrum as the temperature of the laser drifts? Estimate the

change in cavity length that occurs in 15 minutes by comparing the drift in longitudinal

mode spacing over the same period.

5. Make a list of issues that you need to clarify by research or discussion with the

instructor. Don’t be satisfied with vague understanding!

Glossary

Absorption of radiationReceiving electromagnetic radiation by interaction with the material, and transforming it to different form, which is usually heat. The absorption process is dependent on the wavelength of the electromagnetic radiation and on the absorbing material.

Active Medium Collection of atoms or molecules which can be stimulated to a population inversion, and emit electromagnetic radiation in a stimulated emission. Amplification The process in which the electromagnetic radiation inside the active medium within the laser optical cavity increase by the process of stimulated emission.

Amplitude The maximum value of a wave, measured from its equilibrium.

Aperture A small opening through which the electromagnetic radiation pass.

Attenuation The decrease in radiation energy (power) as a beam passes through an absorbing or scattering medium.

Beam Diameter Defined as the diameter of a circular beam at a certain point where the intensity drop to a fraction of its maximum value. The common definitions are 1/e (0.368) and 1/e2 (0.135) of the maximum value.

Beam Divergence Angle of beam spread, measured in (milli)radians. Can be approximated for small angle by the ratio of the beam diameter to the distance from the laser aperture.

Black Body RadiationAny object surface can radiate heat to and receive heat from outside, if an object can absorb all the incident radiation, regardless of the frequencies and directions, this object is called Black Body. A ball cavity with a small hole can be regarded

Laser Modes 4-19

as a black body, since any radiation entering the ball cavity can only reflect inside it, thus totally absorbed.

BrightnessThe brightness of a light source is defined as the power emitted per unit surface area per unit solid angle.

CoherenceCoherence can be devided into spatial and temporal coherence. For any em wave, if at time t=0 and t0 the phase diference between two points in space remains the same, we say the em wave has spatial coherence; If at a point P, the em wave at t and t+dt has same phase difference if dt is the same, temporal coherence exists.

CompositeA "matrix" and an additional phase or additional phases consisting of particles, whiskers, fibres or any combination thereof, present for a specific purpose or purposes.

DOFThe depth of focus is the distance over which the focussed beam has about the same intensity, it is defined as the distance over which the focal spot size changes -5%~5%.

Electronic assemblyA number of electronic components (i.e., "circuit elements", "discrete components", integrated circuits, etc.) connected together to perform (a) specific function(s), replaceable as an entity and normally capable of being disassembled.

Evaporative Laser CuttingEvaporative laser cutting is the laser cutting process that target material is ablated through direct vaporization, typical applications are laser cutting of low vaporization temperature and low thermal conduction materials.

Excimer LasersLasers which use the noble gas compounds for lasing. Excimer lasers generate laser light in ultraviolet to near-ultraviolet spectra, from 0.193 to 0.351 microns. Gas Laser A laser in which the active medium is gas. The gas can be composed of molecules (like CO2), Atoms (like He-Ne), or ions (like Ar+).

Laser Fusion cuttingLaser fusion cutting is laser cutting through melting and gas jet blowing.

Ground StateLowest energy level of an atom or molecule.

Heat Affected ZoneHeat affected zone is the region close to the laser irradiated area that obvious temperature change from original area happens, or obvious strain state change happens.

HologramAn interference phenomena captured on a plate (or film). It can contain enormous amount of information and a 3 dimensional image can be constructed from it.

Knudesen layerIn laser processeing, strong evaporation occurs. The gas near the phase interface is not in translational equilibrium and the translational equilibrium is achieved

4-20

Laser Modes

within a few mean free paths by collisions between particles in a thin region. This region is called Knudsen layer

LaserLaser is the acronym of Light Amplification by Stimulated Emission of Radiation. Laser is light of special properties, light is electromagnetic (EM) wave in visible range. Lasers, broadly speaking, are devices that generate or amplify light, just as transistors generate and amplify electronic signals at audio, radio or microwave frequencies. Here light must be understand broadly, since lasers have covered radiation at wavelengths ranging from infrared range to ultraviolet and even soft x-ray range.

Laser machiningLaser machining is material removal accomplished by laser material interaction, generally speaking, these processes include laser drilling, laser cutting and laser grooving, marking or scribing.

Laser ModeLaser mode is the possible standing em waves in laser cavity.

Longitudinal (Axial) ModesAxial standing em waves within the laser cavity.

Laser Resonator or Laser CavityThe optical mirrors, active medium and pumping system form the laser resonator, which is also called Laser Cavity. Laser cavities can be divided into Stable Cavities and Unstable Cavities according to whether they make the oscillating beam converge into the cavity or spread out from the cavity.

LinewidthThe linewidth of laser is the width of laser beam frequency. Laser linewidth is much narrower than normal light.

Liquid LaserLasers which use large organic dye molecules as the active lasing medium.

M2 of the beamM2 is a beam quality index that measures the difference between an actual beam and the Gaussian beam.

MatrixA substantially continuous phase that fills the space between particles, whiskers or fibres.

Marangoni MechanismLiquid surface force due to temperature gradient (thermal) or composition gradient (chemical)

MicrocircuitA "monolithic integrated circuit" or "multichip integrated circuit" containing an arithmetic logic unit (ALU) capable of executing a series of general purpose instructions from an external storage. N.B.1: The "microprocessor microcircuit" normally does not contain integral user-accessible storage, although storage present on-the-chip may be used in performing its logic function. N.B.2: This definition includes chip sets which are designed to operate together to provide the function of a "microprocessor microcircuit".

Laser Modes 4-21

MultichipA "integrated circuit" where two or more "monolithic integrated circuits" bonded to a common "substrate".

Mode LockingA method to create very short laser pulses. It makes the phase difference of many modes (frequencies) in the laser cavity fixed, or locked, thus very narrow pulses (in time) are created.

Mushy regionPhase changes happen over a temperature region in general, thus solid and liquid state coexist during phase changes. The region of this mixture of solid and liquid is called Mushy region.

PhotonThe minimum quantity of light energy that can be exchanged is called a light quantum or photon.

Polarized LightIf the light has a dominant direction of the E vector, we say the light is polarized. Natural light is not polarized, while laser beam is polarized. Polarization can be created and adjusted by polarizer.

Population InversionNormally the number of atoms at high energy level(E1) is less than those in low energy level(E1), N2(E2) < N1(E1). If N2>N1, we say population inversion exists, which is a necessary condition for lasing.

PumpingThe process to raise atoms from lower level to upper level is called pumping.

Q-SwitchingA method to create laser pulses. It modualates the Q (Quality) of laser cavity to build population inversion first, then release the accumulated energy suddenly, in this way high energy pulses can be created.

Recombination RadiationIn semiconductors, when the electrons combine with the holes, photons are emitted, this is called Recombination Radiation.Semiconductor Lasers are based on this mechanism.

ResolutionThe least increment of a measuring device; on digital instruments, the least significant bit. (Reference: ANSI B-89.1.12)

Solid State LaserA laser in which the active medium is in solid state (usually not including semiconductor lasers).

Semiconductor LasersLasers which use semiconductor as active medium. The majority of semiconductor materials are based on a combination of elements in the third group of the Periodic Table (such as Al, Ga, In) and the fifth group (such as N, P, As, Sb) hence referred to as the III-V compounds.

Spontaneous RadiationAccording to quantum mechanics, the electrons of atoms can take different energy states, say E1, E2, E3, etc., E1<E2<E3<…. Lower energy level is more stable

4-22

Laser Modes

than higher energy levels, so electrons at high energy levels tend to decay to low energy levels, the energy difference between the two levels can be given out as electromagnetic radiation. This process is called Spontaneous Radiation.

Stable Cavity and Unstable CavityCavities can be identified as stable or unstable according to whether they make the oscillating beam converge into the cavity or spread out of the cavity, if converge it is stable, if spread out, it is unstable.

Stimulated AbsorptionWhen the atoms at lower energy levels absorb the incident energy with corresponding frequency, they jump to upper level states, this is called Stimulated Absorption.

Stimulated EmissionUnder the action of the incident electromagnetic field with the corresponding frequency, the atoms at upper level have a certain possibility to jump to the corresponding lower levels, emitting electromagnetic waves or photons with the same frequency, direction and phase with the incident waves. This process is called Stimulated Emission.

SubstrateA sheet of base material with or without an interconnection pattern and on which or within which "discrete components" or integrated circuits or both can be located.

SuperalloyNickel-, cobalt- or iron-base alloys having strengths superior to any alloys in the AISI 300 series at temperatures over 922 K (649º C) under severe environmental and operating conditions.

TEM ModeTransverse Electromagnetic Mode (TEM) of laser beam is called TEM mode. Three index are used to indicate the TEM modes. TEMplq, p is the number of radial zero fields, l is the number of angular zero fields, q is the number of longitudinal fields.

YAG yttrium/aluminum garnet

Ultrashort Pulsed LaserLaser whose pulse duration time is very short, below 1 ns, usually in the fs scale.

 

The above methods are not good enough for generating ultra-short high power pulses. The pulse duration is longer than ns level. If we want the pulse duration to be even shorter, mode locking should be used. With mode locking and other advanced techniques, we can generate pulses with durations from ps to fs level (10-12~10-15 second).

Step four: The idea of Mode Locking

Laser Modes 4-23

Although single mode oscillation for a laser system is desired for some cases, multi-modes oscillation has the advantage of possible high output power. If we can transform the high energy into the form of ultra-short pulses, we realize both objectives—ultra short pulse and high peak energy. Mode locking does this for us.

Consider a laser oscillating in a large number of longitudinal modes, i.e., standing light waves of different frequencies exist in the laser cavity. We suppose all the modes oscillating with same amplitude. Let D n be the frequency difference between consecutive longitudinal modes. If these modes have no phase correlation, the summation of these waves will appear as the figure below, it is some what random with time while has certain features—the waveform is periodic with a period t p=1/ D n , etc. The output power is N*E02, where N is the number of modes, E0 is the amplitude of the modes.

Figure2.24: Time behavior of the squared amplitude of the total electric field with random phases

Now let’s see what will happen if we let these modes have phase correlation, i.e., if the phases of the modes are locked. Suppose 2n+1 modes are oscillating with the same amplitude E0, and the phase of one mode differs by a constant phase from its next consecutive modes, i.e., j k-j k-1 =j , where we use k to indicate the kth mode. Then the total electric field E(t) is the summation of the modes. We have:

Where E0 is the amplitude, w 0+k*D w is the frequency of the kth mode, D w is the frequency difference between consecutive modes.

4-24

Laser Modes

Define A(t) as:

Then

E(t)=A(t) exp(j w 0 t)

Also define D w *t+j =t’, we can compute A(t’) to be:

A(t’)=E0 sin[(2n+1) D w *t’/2]/sin(D w *t/2)

We know the output laser intensity is proportional to E(t)2, E(t)2µ A(t’)2/E02, we plot the A(t’)2/E02 versus t’ in figure??

Figure2.25: Time behavior of the squred amplitude of the total electric field with locked phases

From above relations, some important conclusions can be drawn:

1. The peak values of pulse is: [(2n+1)E0]2, 2n+1 times of the normal value when the mode are not phase related or phase locked. The bigger the n, the higher the peak pulse energy;

2. Two successive pulses are separated by a time interval of tp=2p /D w =1/ D n . Between the big peaks, there are many small amplitude fluctuations;

Laser Modes 4-25

3. Define the pulse duration time to be the time interval between which the intensity of the beam decreases to half value, we find:

D t p @ 2p /(2n+1)D w = 1/D n L, D n L=(2n+1)D w /2p

So the bigger the number of modes locked, the sharper the pulse.

Note: we carry out our analysis for very simplified conditions, we have assumed all modes having same amplitude. For a Gaussian shaped amplitude distribution, the time duration is:

D t p @ 0.441/D n L

Now the idea of mode locking is clear. If we can lock the mode to make the modes have fixed phase correlation, we can generate sharp pulses with high peak energy.

Example: Suppose inside a laser cavity 100 modes are oscillating with same amplitude E02. Compare their peak intensity value and pulse duartion time for a) when these modes have random phases, b) when these modes have same phase.

Solve: Let the average amplitude of these modes be E02, the maximum intensity possible for the random phase is proportional to 100*E02. The laser intensity has a repetition rate of T0, pulse is not obvious for random phase of so many modes.

The peak intensity for 100modes with same phase is 10000*E02, which is about 100 times of the random case. The laser pulse duration time is T0/100.

 

Mode locking methods are also divided into active and passive methods. Active mode locking include Amplitude Modulator (AM) mode locking, Frequency Modulator (FM) mode locking and Synchronous Pumping mode locking, etc. We only discuss AM mode locking here, since they most widely used.

For AM mode locking, we insert an AM modulator into the cavity. This modulator adjusts the cavity loss. The cavity loss is very high for some time, light waves with certain phases during this period are attenuated, there is no laser output. Then the cavity loss is very low for some time, light waves with certain phases can oscillate in the cavity. The AM modulator period equals the cavity round trip time, T=2L/c=tp=2p /D w =1/ Dn . In this way the AM modulator locked the modes. AM modulator can be realized by a Pockels cell modulator, or an acoustooptic modulator, etc.

4-26

Laser Modes

Passive Mode Locking can be realized by fast saturable absorber modulator, slow saturable absorber modulator, Kerr lens mode locking modulator, etc. We introduce Kerr lens mode locking here.

Optical materials like quartz or sapphire have non-linear refractive indexes when sufficiently high intensity light is passing through,

n=n0+n2I

Where n2 is a positive number, I is the incident light intensity.

If the beam is of Gaussian transverse profile, the material has bigger refractive index in the center where the light is stronger, while in the wings of the beam the index is smaller, so the material acts just like a lens, the beam will be focused. We put a suitable sized aperture behind the Kerr optical material, when the light intensity is not big enough, the focus effect is not obvious, dispersion loss is huge, when the light intensity is strong, the beam is well focused into the aperture and passes through. When we place this device correctly in the cavity, mode locking happens. Kerr lens effect is very fast, it can be taken as instantaneous. The fastest mode locked pulses are achieved by this technique using ultra-broadband gain media.

There are more details in mode locking. We stop here hoping the readers have gained some basic knowledge on how pulse can be generated and on how ultra-short high energy pulses can be generated.

G2.10: Schematic of a typical mode-locked diode-pumped Cr:LiSAF laser (Courtesy of Laser Optics and Spectroscopy Group of the Physics Department at Imperial College,

London)

eneralized governing equations in LMP

Laser Modes 4-27

Laser machining processes have found wider and wider applications in the past few decades. Laser machining is assumed to be flexible, rapid and precise. However, process parameters such as laser intensity, laser beam size, laser pulse duration, depth of focus, beam quality, and scanning speed of the laser source must be carefully selected for each application to achieve optimized results. While empirical knowledge and experience have historically been employed to determine the process parameters, there has been continuous efforts in quantitatively relating the physics of laser machining process with the process parameters. The modeling of the laser machining process is a critical element for its successful operation, and these models can contribute a lot to the control and optimization of the process.

It is realized that laser machining processes are complicated due to their wide range of laser intensity, laser-material interaction time, scanning speed, beam size and wavelength. At an energy intensity of the order of 103 to 104 W/cm2, heat conduction and solute diffusion are dominant in determining the dwell time required for phase transformation in laser surface hardening. When laser intensity goes up to 107 W/cm2 level, as in laser deep penetration welding, conduction, convection, evaporation and plasma interaction all com to play an important role in the process. In laser machining, the energy level covers 106 to 109 W/cm2. In this energy range, conduction, convection, evaporation, plasma generation, gas dynamics and gas jet effects are important. With the development of ultrashort ( <10 psec) pulse lasers, the energy levels are further expand to 1012, the interaction mechanism is quite different because of the short pulse duration time.

As mentioned before, many models has been suggested, both analytical and numerical. Early models were mostly analytical in nature and primarily describe the conductive thermal fields induced in the solid by a moving laser source. The temperature dependent nature of the physical properties involved in the models and the complex geometry in reality make it necessary for numerical solutions.

The laser machining process is complex, so many models have been suggested, can we extract some general information from the existing works?

It is the purpose of this section that a general model be presented to give the reader an overall understanding of the laser machining processes.

Reliability: Flashlamps burn out unpredictably and often. Diode arrays degrade

gracefully over time--a very long time. Many last in excess of 20,000 hours!

Monitoring of LMP

Industrial laser machining has been plagued by the problems of reliability. Both in CO2 and Nd:YAG laser machining, this result in unnecessary costs due to the stoppage of production or creation of waste. Monitoring is an important way to inspect the process, give in-time feed back of machining quality. Some aspects of LMP monitoring have been covered in previous section. In this section, we introduce two topics: A monitoring

4-28

Laser Modes

system incorporated in laser heads and the plasma formation observation using nanosecond imaging techniques.

1. Process Monitoring through Special Laser Heads

Laser machining has become a well accepted common application in industry and laser processes is more and more involved in automated production systems, so they are expected to function in a more robust and repeatable manner. This increase of reliability and repeatability requirements prompts the development of suitable monitoring systems. Tools must be developed for the online condition monitoring of the process and the machine tool. By obsere certain parameter changes on either the process or the machine tool, quality faults and machine aging could be observed.

Machine faults involving standard peripheral equipment such as positing tables, robots or similar systems is not directly related to the use of laser and is not covered in this discussion. Changes in process parameters such as focal distance, power and speed can directly affect the process quality. Furthermore, the state of the optics, the process gas and beam alignment plays a major role in laser cutting. When considering the source of errors from the laser machine tools, the beam alignment and the optics are critical to the proper functioning of the laser.

It is clear that if sensors wer used to monitor process parameters and machining parameters then the process quality and the state of the laser could be monitored simultaneously. By implementing multi-sensor heads one could collect data relating to the process as well as the state of the laser.

Figure 10: A multi-sensor CO2 laser head [Tonshoff, et al., 1999].

Laser Modes 4-29

A CO2 cutting head equipped with multi-sensors was shown in Figure 10. A diffractive mirror was used to extract the process radiation. This proved to be a simple way for on-axis process monitoring. On the nozzle end, the microphone as well as a pressure guage is being used. The pressure guage measures the pressure inside the nozzzle. The microphone can either be positioned inside the nozzle or externally to it. The microphone was chosen so that it could withstand high sound pressure levels so that a change from 0-20 bar does not cause any damage.

In CO2 lasers, beam alignment is often a problem. The beam path can be several meters long, in which optical components can deteriorate and shoft the beam before it reaches the workpiece. The 8-quadrant thermopile is critical for finding any misalignments, power losses or beam distortions. The octavo sensor is positioned at the entry of the incoming beam, the use of ZnSe mirror allows a transmission of 1% CO2 radiation to the sensor. With this sensor the overall quality of the oncoming beam can be judged, but any deterioration caused by focusing lens cannot be assessed.

In laser cutting, even though a gas jet may be used to protect the lens, the lens still can get contaminated by molten metal sprays. So beam alterations due to impurities on the focusing lens should be monitored. The diffractive mirror serves part as the optical transmission of the laser energy to the focus lens, in the same time it diffracts part of the laser radiation on the process monitor sensor. Signal from the process monitor sensor tells about the laser-workpiece interaction and can be used as feedback signals for process control.

With this design, and with the assistance of other control units, laser machining failure can be detected.

2. Plasma Formation Observation in Laser Ablation of Metals

When a short, high intensity laser pulse is focused on a material surface, strong material ablation occurs. The ablation process is one of the major processes for laser micro-fabrication. Thus the observation of the dynamics of laser induced plasma is meaningful for process understanding as well as process control.

Plasma monitoring of polymers has been studied by Srinivasan et al.[ 1989], Furutani et al. [1998], laser ablation of metals in vacuum has also been studied[Chrisey, 1994]. In this section we reports plasma formation of metals in air and various gas environment.

The ablation process in gases was studied by photoacoustic detection technique. Photoacoustic signal intensity is a function of laser fluence, the shape of the function is similar to materials and wavelengths examined so far.

4-30

Laser Modes

Figure 11 Experimental set up for plasma observation [Yoshiro ITO, et al., 1999]

Figure 11 shows the experimental set up. The image system is a kind of pump and probe technique in which time resolution is manily determined by the pulse of the illuminating light. Light pulse from Q-switched Nd:YAG laser with second harmonic generator contains both converted second harmonic radiation and remaining fundamental light. Both of them are delivered though proper light separation apparatus. After passing the apparatus, separated second harmonic light (532 nm) was used as illuminating light and the remaining fundamental light (1064nm) was used as ablating beam. The 1064nm light was focused onto the sample and hit the sample perpendicularly. When the distance from the lens to the sample was changed, laser spot size was changed and laser fluence can be changed while laser pulse energy is kept constant. The 532nm light passed illuminate the machining position at an angle of 60 degrees from the surface normal. Observation direction was at the opposite half plane with an angle 30 degrees from the normal. Images were captured by CCD camera equipped gated image intensifier. A band pass filter of 532 nm was placed in front of the camera. The gate width of the camera was 3ns and its timing was monitored by digital oscilloscope. The Nd:YAG laser delivered 6ns pulses measured as FWHM by a fast photodiode.

The system was used for a delay times of -5 to 25 ns. For longer delays, a second laser was used as illuminating source and the delay time between the two laser pulses was controlled by a digital delay generator (DG 535, Stanform Inst. CO.). In following pictures, white image is jet-like plasma growing from the surface and its shadow is clearer than bright plasma image itself, which allows us to calculate the its height precisely from geometry calculation shown in the figure. Shadow length is elnongated than actual plasma height. The image of a ruler put at the sample position was recorded

Laser Modes 4-31

prior to measurement and used as scale on a display to get actual dimensions. Actual plasma height is calculated by:

H= OX / cos(300),

where OX is the observed shadow length.

Figure 12 shows typical images of laser irradiated aluminum surface arranged with delay time. Laser fluence was 32 J/cm2 and irradiated in air. Bright plasma spot appear at 2ns and grows in height with time until 15ns. After 5ns, clear black shadow of this plasma is observed.After 30ns, both brightness of the plasma and the thickness of its shadow become dim. Growth speed of the plasma decreases and the plsama begin to expand to radial direction. In photograph at 100ns, bright image and its clear shadow dissapear and irradiated part on the surface becomes visible. These pictures are evidence of jet-like plasma

http://sinhviendaminh.net/modules.php?name=News&op=viewst&sid=1766

IMPORTANT - This email and any attachments is intended for the above named

addressee(s), and may contain information which is confidential or privileged. If you are

not the intended recipient, please inform the sender immediately and delete this email:

you should not copy or use this e-mail for any purpose nor disclose its contents to any

person.

Tính từ đứng trước danh từ

Tính từ trong tiếng Anh thường đứng trước danh từ và được đặt ở vị trí "modifier" hay "attributive" - bổ nghĩa. Do đó chúng ta thường nói:

* Getting all the way round Brazil in five working days proved an impossible mission.* He asked me a number of difficult questions.* I was sitting next to the open window which I couldn't close.

Mission impossible, nếu tôi nhớ không nhầm thì ban đầu đây tên của một series trên truyền hình Mỹ và sau đó được chuyển thành phim. Trên thực tế chẳng có lý do gì cho việc đảo tính từ ra đằng sau danh từ trong trường hợp này ngoại trừ để tạo ấn tượng. Nó thu hút sự chú ý của người nghe.

Các trường hợp ngoại lệ: tính từ đứng dau sanh từ

Các tính từ bổ nghĩa được đặt sau động từ to be (và một số động từ đặc biệt khác - Copular verbs). Như vậy chúng ta có :

4-32

Laser Modes

* The mission was impossible.* All the questions he asked were difficult.* The window remained open.

Động từ liêh kết - copular verbs, nối tính từ với chủ ngữ, thường miêu tả trạng thái của một vật hay một người nào đó. Chúng gồm các động từ : be, seem, appear, look, sound, smell, taste, feel, get, become, stay, remain, keep, grow, go, turn. Chúng ta có các ví dụ sau:

* The policemen became angry.* The suspects remained calm although I could see that they were anxious.* The soup looked, smelt and tasted good.

Các tính từ bổ nghĩa cũng có các từ bổ nghĩa thêm cho chính nó, ví dụ để diễn tả nghĩa "capable of achieving first-class degrees" - có khả năng đạt được bằng hạng nhất, thì nó thường được dùng với cả cụm từ này đứng đằng sau danh từ, thay vì đứng trước danh từ mà nó bổ nghĩa cho:

* We are recruiting students capable of achieving first-class degrees.Không nói: We are recruiting capable of achieving first class degree students.Nhưng: She was a capable student.

* I used to live in a house next to the Royal Opera House.Không nói: I used to live in a next to the Royal Opera House house.Nhưng: I live quite near you. In the next street, in fact.

Tương tự các tính từ ở dạng phân từ 2 (phân từ quá khứ) cũng được đặt sau danh từ mà nó bổ nghĩa:

* The people questioned about the incident gave very vivid accounts of what had happened.* The issues discussed at the meeting all had some bearing on world peace.

Trong cả bốn ví dụ trên có lẽ cách thông thường hơn là dùng một mệnh đề quan hệ (a relative clause):

* We are recruiting students who are capable of achieving first-class degrees.* I used to live in a house which was next to the Royal Opera House.* The people who were questioned about the incident gave vivid accounts of what had happened.* The issues that were discussed at the meeting all had some bearing on world peace.

Và cuối cùng, các tính từ đi sau hầu hết các danh từ đo lường và sau các từ có tiền tố some-, any- và no- :

Laser Modes 4-33

* The fence around the estate was three metres high, thirty-five kilometres long and one hundred and twenty years old.* This place doesn't look very promising, but let's try and find somewhere nice for dinner.* I couldn't find anything interesting on the television so I had an early night.* There's somebody outside who wants to speak to you. Shall I let him in?* Nobody present at the meeting was able to offer me any useful advice.