Gaussian Beam Optics 

[Hecht  Ch. 13.1 pages 594­596 Notes from Melles Griot and Newport]  Readings:  For details on the theory of Gaussian beam optics, refer to the excerpts from the Melles Griot and Newport catalogs. Melles Griot, anlong with Newport Corporation, is a major manufacturer of optical components used for research. You will also find tutorials on other subjects of optics and photonics at their web site: http://www.mellesgriot.com/products/optics/toc.htm (CVI Melles Griot Optics Guides) http://www.newport.com/servicesupport/Tutorials/default.aspx?id=110 (Newport Tutorials). Gaussian Beam Calculators: There are many calculators or programs to determine Gaussian beam propagation in free space or through a combination of lenses. Below is a short list of program you may use to solve the homework problems. Gaussian Beam Calculator for simple lenses http://www.photonics.byu.edu/Gaussian_Beam_Propagation.phtml http://www.newport.com/OpticalAssistant/FocusCollimatedBeam.aspx http://www.originalcode.com/downloads/GBC8p2.zip (need LabView) http://www.novajo.ca/abcd/ (for Mac only, ABCD Gaussian Beam Propagation) Gaussian Beam Intensity/Irradiance http://www.calctool.org/CALC/phys/optics/gauss_power_dist Gaussian Beam and Spatial Filtering http://www.newport.com/OpticalAssistant/SpatialFilterPinhole.aspx Introduction One usually thinks of a laser beam as a perfectly collimated beam of light rays with be beam energy uniformly spread across the cross section of the beam. This is not an adequate picture for discussing the propagation of a laser beam over any appreciable distance because diffraction causes the light waves to spread transversely as they propagate, Fig. 1.

1

Figure 1. Divergence of a Laser Beam

Additionally, the energy (irradiance) profile of a laser beam is typically not uniform. For the most commonly used He-Ne lasers (operating in the TEM00 mode) the irradiance (the power carried by the beam across a unit area perpendicular to the beam = W/m2) is given by a Gaussian function:

( )2 2 2 22 / 2 /

0 22 ,r w r wPI r I e ewπ

− −= = (1)

where w is defined as the distance out from the center axis of the beam where the irradiance drops to 1/e2of its value on axis. P is the total power in the beam. r is the transverse distance from the central axis. w depends on the distance z the beam ahs propagated from the beam waist. w0 is the beam radius at the waist. [The beam waist is defined as the point where the beam wave front was last flat (as opposed to spherical at other locations).] For a hemispherical laser cavity such as the one used for the He-Ne laser used in the lab, the waist is located roughly at the output mirror. w0 is related to w by

Experimentally, one can use a CCD detector array to measure how the irradiance various across the beam for several values of z>>zR. Then fit the data for each z using Eq. (1), which will yield values for w(z). Then one can use Eqs. (3) and (4) to determine w0 and calculate zR.

2

plane (i.e. the two lenses are separated by f0+fe). The rays reaching the eye are again parallel, but appear to subtend a much larger angle than the original object. From Fig. 2 it is easy to see that the angular magnification is

Figure 2. The Astronomical Telescope

Beam expander: Because Gaussian beams do not follow the rules of ray optics, we cannot use the lens equation to design a beam expander. However, as discuss in the Melles Griot Optics Guide, if you consider the object to be the beam waist of the incoming beam and the image to be the beam waist after the beam passes through the lens, then you can use a modified lens equation:

3

Lets’ now apply this to an inverted astronomical telescope with the focal length of the first lens being 5 cm and the second 40 cm. In the astronomical telescope the two lenses are separated by the sum of the focal lengths of the two lenses – 45 cm in this case. We assume a red He-Ne laser (633nm) with beam waist radius of 0.4 mm. We first use Eq. (3) to get zR=0.80 m. For the first lens, f = 5 cm, and the beam waist for the laser is close to the exit of the laser. We put the lens as close to the laser as possible and assume s=0. The using Eq. (5) we get

4

5

BEAM WAIST AND DIVERGENCE

In order to gain an appreciation of the principles and limita-tions of Gaussian beam optics, it is necessary to understand thenature of the laser output beam. In TEM00 mode, the beam emit-ted from a laser begins as a perfect plane wave with a Gaussiantransverse irradiance profile as shown in figure 2.1. The Gaussianshape is truncated at some diameter either by the internal dimen-sions of the laser or by some limiting aperture in the optical train.To specify and discuss the propagation characteristics of a laserbeam, we must define its diameter in some way. There are two com-monly accepted definitions. One definition is the diameter at which

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2.2 1 A S K A B O U T O U R C U S T O M C A P A B I L I T I E SO E MGaussian Beam Optics

w w w . m e l l e s g r i o t . c o m

In most laser applications it is necessary to focus, modify, orshape the laser beam by using lenses and other optical elements. Ingeneral, laser-beam propagation can be approximated by assum-ing that the laser beam has an ideal Gaussian intensity profile,which corresponds to the theoretical TEM00 mode. CoherentGaussian beams have peculiar transformation properties whichrequire special consideration. In order to select the best optics fora particular laser application, it is important to understand thebasic properties of Gaussian beams.

Unfortunately, the output from real-life lasers is not truly Gauss-ian (although helium neon lasers and argon-ion lasers are a very closeapproximation). To accommodate this variance, a quality factor, M2

(called the “M-squared” factor), has been defined to describe thedeviation of the laser beam from a theoretical Gaussian. For a the-oretical Gaussian, M2 = 1; for a real laser beam, M2>1. The M2 fac-tor for helium neon lasers is typically less than 1.1; for ion lasers,the M2 factor typically is between 1.1 and 1.3. Collimated TEM00diode laser beams usually have an M2 ranging from 1.1 to 1.7. Forhigh-energy multimode lasers, the M2 factor can be as high as 25or 30. In all cases, the M2 factor affects the characteristics of a laserbeam and cannot be neglected in optical designs.

In the following section, Gaussian Beam Propagation, we willtreat the characteristics of a theoretical Gaussian beam (M2=1);then, in the section Real Beam Propagation we will show how thesecharacteristics change as the beam deviates from the theoretical. Inall cases, a circularly symmetric wavefront is assumed, as would bethe case for a helium neon laser or an argon-ion laser. Diode laserbeams are asymmetric and often astigmatic, which causes theirtransformation to be more complex.

Although in some respects component design and tolerancingfor lasers is more critical than for conventional optical components,the designs often tend to be simpler since many of the constraintsassociated with imaging systems are not present. For instance, laserbeams are nearly always used on axis, which eliminates the needto correct asymmetric aberration. Chromatic aberrations are of noconcern in single-wavelength lasers, although they are critical forsome tunable and multiline laser applications. In fact, the only sig-nificant aberration in most single-wavelength applications is primary(third-order) spherical aberration.

Scatter from surface defects, inclusions, dust, or damaged coat-ings is of greater concern in laser-based systems than in incoherentsystems. Speckle content arising from surface texture and beamcoherence can limit system performance.

Because laser light is generated coherently, it is not subjectto some of the limitations normally associated with incoherentsources. All parts of the wavefront act as if they originate from thesame point; consequently, the emergent wavefront can be preciselydefined. Starting out with a well-defined wavefront permits moreprecise focusing and control of the beam than otherwise would bepossible.

For virtually all laser cavities, the propagation of an electro-magnetic field, E(0), through one round trip in an optical resonatorcan be described mathematically by a propagation integral, whichhas the general form

where K is the propagation constant at the carrier frequency ofthe optical signal, p is the length of one period or round trip, andthe integral is over the transverse coordinates at the reference orinput plane. The function K is commonly called the propagationkernel since the field E(1)(x, y), after one propagation step, can beobtained from the initial field E (0)(x0, y0) through the operationof the linear kernel or “propagator” K(x, y, x0, y0).

By setting the condition that the field, after one period, willhave exactly the same transverse form, both in phase and profile(amplitude variation across the field), we get the equation

where Enm represents a set of mathematical eigenmodes, and gnma corresponding set of eigenvalues. The eigenmodes are referred toas transverse cavity modes, and, for stable resonators, are closelyapproximated by Hermite-Gaussian functions, denoted by TEMnm.(Anthony Siegman, Lasers)

The lowest order, or “fundamental” transverse mode, TEM00has a Gaussian intensity profile, shown in figure 2.1, which hasthe form

.

In this section we will identify the propagation characteristicsof this lowest-order solution to the propagation equation. In the nextsection, Real Beam Propagation, we will discuss the propagationcharacteristics of higher-order modes, as well as beams that havebeen distorted by diffraction or various anisotropic phenomena.

Gaussian Beam Propagation

E x y e K x y x y E x y dx dyjkp

InputPlane

( ),, , , ,1

0 0

0

0 0 0 0( ) = ( ) ( )− ( )∫∫

gnm nm nmE x y K x y x y E x y dx dyInputPlane

, , , , ,( ) ≡ ( ) ( )∫∫ 0 0 0 0 0 0

I x y ek x y

,( )∝ − +( )2 2

(2.1)

(2.2)

(2.3)

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1 2.3A S K A B O U T O U R C U S T O M C A P A B I L I T I E SO E M Gaussian Beam Optics

13.5

CONTOUR RADIUS41.5w

20

40

60

80

100

4

PER

CEN

T IR

RA

DIA

NC

E

0w 1.5w

1/e2 diameter 13.5% of peak

FWHM diameter 50% of peak

direction of propagation

Figure 2.1 Irradiance profile of a Gaussian TEM00 mode

Figure 2.2 Diameter of a Gaussian beam

the radius of the 1/e2 contour after the wave has propagated a dis-tance z, and R(z) is the wavefront radius of curvature after propa-gating a distance z. R(z) is infinite at z = 0, passes through a minimumat some finite z, and rises again toward infinity as z is furtherincreased, asymptotically approaching the value of z itself. Theplane z=0 marks the location of a Gaussian waist, or a place wherethe wavefront is flat, and w0 is called the beam waist radius.

The irradiance distribution of the Gaussian TEM00 beam,namely,

where w=w(z) and P is the total power in the beam, is the same atall cross sections of the beam.

The invariance of the form of the distribution is a special con-sequence of the presumed Gaussian distribution at z = 0. If a uni-form irradiance distribution had been presumed at z = 0, the patternat z = ∞ would have been the familiar Airy disc pattern given by aBessel function, whereas the pattern at intermediate z values wouldhave been enormously complicated.

Simultaneously, as R(z) asymptotically approaches z for largez, w(z) asymptotically approaches the value

where z is presumed to be much larger than pw0 /l so that the 1/e2

irradiance contours asymptotically approach a cone of angularradius

the beam irradiance (intensity) has fallen to 1/e2 (13.5 percent) ofits peak, or axial value and the other is the diameter at which thebeam irradiance (intensity) has fallen to 50 percent of its peak, oraxial value, as shown in figure 2.2. This second definition is alsoreferred to as FWHM, or full width at half maximum. For theremainder of this guide, we will be using the 1/e2 definition.

Diffraction causes light waves to spread transversely as theypropagate, and it is therefore impossible to have a perfectly collimatedbeam. The spreading of a laser beam is in precise accord with thepredictions of pure diffraction theory; aberration is totally insignif-icant in the present context. Under quite ordinary circumstances,the beam spreading can be so small it can go unnoticed. The fol-lowing formulas accurately describe beam spreading, making iteasy to see the capabilities and limitations of laser beams.

Even if a Gaussian TEM00 laser-beam wavefront were madeperfectly flat at some plane, it would quickly acquire curvature andbegin spreading in accordance with

where z is the distance propagated from the plane where the wave-front is flat, l is the wavelength of light, w0 is the radius of the 1/e2

irradiance contour at the plane where the wavefront is flat, w(z) is

R z zwz

w z wz

w

( ) = +⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥

( ) = +⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

1

1

0

22

0

0

2

2

p

l

l

p

and

⎤⎤

⎦⎥⎥

/1 2

(2.4)

I r I eP

wer w r w( ) / /= =− −

0

2

2

22 2 2 22

p , (2.6)

w zz

w( ) =

l

p0

(2.7)

vl

p= ( ) =

w z

z w0

. (2.8)

(2.5)

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2.4 1 A S K A B O U T O U R C U S T O M C A P A B I L I T I E SO E MGaussian Beam Optics

This value is the far-field angular radius (half-angle divergence)of the Gaussian TEM00 beam. The vertex of the cone lies at thecenter of the waist, as shown in figure 2.3.

It is important to note that, for a given value of l, variations ofbeam diameter and divergence with distance z are functions of a sin-gle parameter, w0, the beam waist radius.

Near-Field vs Far-Field Divergence

Unlike conventional light beams, Gaussian beams do not divergelinearly. Near the beam waist, which is typically close to the outputof the laser, the divergence angle is extremely small; far from thewaist, the divergence angle approaches the asymptotic limit describedabove. The Raleigh range (zR), defined as the distance over whichthe beam radius spreads by a factor of √

_2, is given by

.

At the beam waist (z = 0), the wavefront is planar [R(0) = ∞]. Like-wise, at z=∞, the wavefront is planar [R(∞)=∞]. As the beam prop-agates from the waist, the wavefront curvature, therefore, mustincrease to a maximum and then begin to decrease, as shown infigure 2.4. The Raleigh range, considered to be the dividing line

zw

R .=p

l

0

2

(2.9)

ww0

w0

zw0

1e2 irradiance surface

v

asymptotic cone

Figure 2.3 Growth in 1/e2 radius with distance propa-gated away from Gaussian waist

laser

2w0

vGaussianprofile

z = 0planar wavefront

2w0 2

z = zRmaximum curvature Gaussian

intensityprofile

z = qplanar wavefront

Figure 2.4 Changes in wavefront radius with propagation distance

between near-field divergence and mid-range divergence, is the dis-tance from the waist at which the wavefront curvature is a maximum.Far-field divergence (the number quoted in laser specifications)must be measured at a distance much greater than zR (usually>10#zR will suffice). This is a very important distinction becausecalculations for spot size and other parameters in an optical trainwill be inaccurate if near- or mid-field divergence values are used.For a tightly focused beam, the distance from the waist (the focalpoint) to the far field can be a few millimeters or less. For beams com-ing directly from the laser, the far-field distance can be measuredin meters.

Typically, one has a fixed value for w0 and uses the expression

to calculate w(z) for an input value of z. However, one can also uti-lize this equation to see how final beam radius varies with startingbeam radius at a fixed distance, z. Figure 2.5 shows the Gaussianbeam propagation equation plotted as a function of w0, with the par-ticular values of l = 632.8 nm and z = 100 m.

The beam radius at 100 m reaches a minimum value for a start-ing beam radius of about 4.5 mm. Therefore, if we wanted to achievethe best combination of minimum beam diameter and minimumbeam spread (or best collimation) over a distance of 100 m, ouroptimum starting beam radius would be 4.5 mm. Any other start-ing value would result in a larger beam at z = 100 m.

We can find the general expression for the optimum startingbeam radius for a given distance, z. Doing so yields

.

Using this optimum value of w0 will provide the best combina-tion of minimum starting beam diameter and minimum beam

w z wz

w( ) = +

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥0

0

2

21 2

1l

p

/

wz

0 optimum( ) = ⎛⎝⎜

⎞⎠⎟

l

p

1 2/

(2.10)

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spread [ratio of w(z) to w0] over the distance z. For z = 100 m andl = 632.8 nm, w0 (optimum) = 4.48 mm (see example above). If weput this value for w0 (optimum) back into the expression for w(z),

.

Thus, for this example,

By turning this previous equation around, we find that we onceagain have the Rayleigh range (zR), over which the beam radiusspreads by a factor of √

_2 as

If we use beam-expanding optics that allow us to adjust theposition of the beam waist, we can actually double the distanceover which beam divergence is minimized, as illustrated in figure 2.6.By focusing the beam-expanding optics to place the beam waist atthe midpoint, we can restrict beam spread to a factor of √

_2 over a

distance of 2zR, as opposed to just zR.

This result can now be used in the problem of finding the start-ing beam radius that yields the minimum beam diameter and beamspread over 100 m. Using 2(zR) = 100 m, or zR = 50 m, and l = 632.8 nm, we get a value of w(zR) = (2l/p)½ = 4.5 mm, and w0 = 3.2 mm. Thus, the optimum starting beam radius is the sameas previously calculated. However, by focusing the expander weachieve a final beam radius that is no larger than our starting beamradius, while still maintaining the √

_2 factor in overall variation.

Alternately, if we started off with a beam radius of 6.3 mm, wecould focus the expander to provide a beam waist of w0 = 4.5 mmat 100 m, and a final beam radius of 6.3 mm at 200 m.

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1 2.5A S K A B O U T O U R C U S T O M C A P A B I L I T I E SO E M Gaussian Beam Optics

STARTING BEAM RADIUS w0 (mm)

FIN

AL

BEA

M R

AD

IUS

(mm

)

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

Figure 2.5 Beam radius at 100 m as a function of startingbeam radius for a HeNe laser at 632.8 nm

beam expander

w(–zR) = 2w0

beam waist2w0

zR zR

w(zR) = 2w0

Figure 2.6 Focusing a beam expander to minimize beamradius and spread over a specified distance

w 100 2 4 48 6 3( ) = ( ) =. . mm

zw

w z w

R

R

with

=

( ) =

p

l

0

2

02 .

Location of the beam waist

The location of the beam waist is required for mostGaussian-beam calculations. Melles Griot lasers aretypically designed to place the beam waist very closeto the output surface of the laser. If a more accuratelocation than this is required, our applicationsengineers can furnish the precise location andtolerance for a particular laser model.

APPLICATION NOTE

Do you need . . .

BEAM EXPANDERS

Melles Griot offers a range of precision beamexpanders for better performance than can beachieved with the simple lens combinations shownhere. Available in expansion ratios of 3#, 10#,20#, and 30#, these beam expanders produceless than l/4 of wavefront distortion. They areoptimized for a 1-mm-diameter input beam, andmount using a standard 1-inch-32 TPI thread. Formore information,see page 16.4.

w z w( ) = ( )20

(2.11)

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BEAMSPLITTERS

WIN

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FILTERS & ATTEN

UATORSULTRAFAST LASER OPTICS

ACCESSORIESTECHN

ICAL REFERENCE

POLARIZATION OPTICS

P h o n e : 1 - 8 0 0 - 2 2 2 - 6 4 4 0 • F a x : 1 - 9 4 9 - 2 5 3 - 1 6 8 0 • E m a i l : s a l e s @ n e w p o r t . c o m • We b : n e w p o r t . c o m

O p t i c s — 555

Gaussian Beam Optics

The Gaussian is a radially symmetricdistribution whose electric fieldvariation is given by:

E Er

ωS = −

0

2

0

2exp

Its Fourier transform is also aGaussian distribution. If we were tosolve the Fresnel integral itselfrather than the Fraunhoferapproximation, we would find that aGaussian source distributionremains Gaussian at every pointalong its path of propagationthrough the optical system. Thismakes it particularly easy tovisualize the distribution of thefields at any point in the opticalsystem. The intensity is alsoGaussian:

I E E E Er

ωS S S= = −

η η* * exp0 0

2

0

2

2

This relationship is much more thana mathematical curiosity, since it isnow easy to find a light source witha Gaussian intensity distribution:the laser. Most lasers automaticallyoscillate with a Gaussiandistribution of electric field. Thebasic Gaussian may also take onsome particular polynomialmultipliers and still remain its owntransform. These field distributionsare known as higher-ordertransverse modes and are usuallyavoided by design in most practicallasers.

The Gaussian has no obviousboundaries to give it a characteristicdimension like the diameter of thecircular aperture, so the definition ofthe size of a Gaussian is somewhatarbitrary. Figure 1 shows theGaussian intensity distribution of atypical HeNe laser.

I r Ir

ω( ) = −

0

2

0

2

2exp

0

0.5

0.83

0.8651.0

1.52

0.710.59

0.23

00.01

NORMALIZED RADIUS (r/ω0)

RE

LAT

IVE

INT

EN

SIT

Y

EN

CIR

CLE

D P

OW

ER

21

1.0

0.5

0

e-1

e-2

Figure 1

The parameter ω0, usually called theGaussian beam radius, is the radiusat which the intensity has decreasedto 1/e2 or 0.135 of its axial, or peakvalue. Another point to note is theradius of half maximum, or 50%intensity, which is 0.59ω0. At 2ω0, ortwice the Gaussian radius, theintensity is 0.0003 of its peak value,usually completely negligible.

The power contained within a radius r,P(r), is easily obtained by integratingthe intensity distribution from 0 to r:

P r Pr

ω( ) = ∞( ) − −

1

2 2

0

2exp

When normalized to the total powerof the beam, P(∞) in watts, the curveis the same as that for intensity, butwith the ordinate inverted. Nearly100% of the power is contained in aradius r = 2ω0. One-half the power iscontained within 0.59ω0, and onlyabout 10% of the power is containedwith 0.23ω0, the radius at which theintensity has decreased by 10%. Thetotal power, P(∞) in watts, is relatedto the on-axis intensity, I(0)(watts/m2), by:

Pω

I

I Pω

∞( ) =

( )

( ) = ∞( )

π

π

0

2

0

2

20

02

The on-axis intensity can be veryhigh due to the small area of thebeam.

Care should be taken in cutting offthe beam with a very small aperture.The source distribution would nolonger be Gaussian, and the far-fieldintensity distribution would developzeros and other non-Gaussianfeatures. However, if the aperture isat least three or four ω0 in diameter,these effects are negligible.

Propagation of Gaussian beamsthrough an optical system can betreated almost as simply asgeometric optics. Because of theunique self-Fourier Transformcharacteristic of the Gaussian, we donot need an integral to describe theevolution of the intensity profilewith distance. The transversedistribution intensity remainsGaussian at every point in thesystem; only the radius of theGaussian and the radius ofcurvature of the wavefront change.Imagine that we somehow create acoherent light beam with a Gaussiandistribution and a plane wavefrontat a position x=0. The beam size andwavefront curvature will then varywith x as shown in Figure 2.

R(x)

ω(x)ωo

x=0

θ x

Figure 2

BEAM

SPLI

TTER

SW

INDO

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FILT

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& A

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UATO

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R OP

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ACCE

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TECH

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POLA

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OPT

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P h o n e : 1 - 8 0 0 - 2 2 2 - 6 4 4 0 • F a x : 1 - 9 4 9 - 2 5 3 - 1 6 8 0 • E m a i l : s a l e s @ n e w p o r t . c o m • We b : n e w p o r t . c o m

556 — O p t i c s

The beam size will increase, slowlyat first, then faster, eventuallyincreasing proportionally to x. Thewavefront radius of curvature, whichwas infinite at x = 0, will becomefinite and initially decrease with x.At some point it will reach aminimum value, then increase withlarger x, eventually becomingproportional to x. The equationsdescribing the Gaussian beamradius w(x) and wavefront radius ofcurvature R(x) are:

ω (x) ω 1x

ω

R(x) x 1ωx

202

02

2

02

2

= +

= +

λπ

πλ

where ω0 is the beam radius at x = 0and λ is the wavelength. The entirebeam behavior is specified by thesetwo parameters, and because theyoccur in the same combination inboth equations, they are oftenmerged into a single parameter, xR,the Rayleigh range:

xω

R0

2

= πλ

In fact, it is at x = xR that R has itsminimum value.

Note that these equations are alsovalid for negative values of x. Weonly imagined that the source of thebeam was at x = 0; we could havecreated the same beam by creating alarger Gaussian beam with anegative wavefront curvature atsome x < 0. This we can easily dowith a lens, as shown in Figure 3.

D

F

R(x)

ω(x)ωo

x=0

θ x

Figure 3

The input to the lens is a Gaussianwith diameter D and a wavefrontradius of curvature which, whenmodified by the lens, will be R(x)given by the equation above withthe lens located at -x from the beamwaist at x = 0. That input Gaussianwill also have a beam waist positionand size (or Rayleigh range)associated with it. Thus we cangeneralize the law of propagation ofa Gaussian through even acomplicated optical system.

In the free-space between lenses,mirrors and other optical elements,the position of the beam waist andthe waist diameter (or Rayleighrange) completely describe thebeam. When a beam passes througha lens, mirror, or dielectric interface,the diameter is unchanged but thewavefront curvature is changed,resulting in new values of waistposition and waist diameter (orRayleigh range) on the output sideof the interface.

These equations, with input valuesfor ω and R, allow the tracing of aGaussian beam through any opticalsystem with some restrictions:optical surfaces need to be sphericaland with not-too-short focal lengths,so that beams do not changediameter too fast. These are exactlythe analog of the paraxialrestrictions used to simplifygeometric optical propagation.

It turns out that we can put theselaws in a form as convenient as theABCD matrices used for geometricray tracing. But there is a difference:ω(x) and R(x) do not transform inmatrix fashion as r and u do for raytracing; rather, they transform via acomplex bi-linear transformation:

qq A B

q C Dout

in

in

=+[ ]+[ ]

where the quantity q is a complexcomposite of ω and R:

1

q x

1

R x

j

w x2( ) = ( ) −

( )λ

π

We can see from the expression forq that at a beam waist (R = ∞ and ω = ω0), q is pure imaginary andequals jxR. If we know where onebeam waist is and its size, we cancalculate q there and then use thebilinear ABCD relation to find qanywhere else. To determine the sizeand wavefront curvature of the beameverywhere in the system, you woulduse the ABCD values for eachelement of the system and trace qthrough them via successive bilineartransformations. But if you onlywanted the overall transformation ofq, you could multiply the elementalABCD values in matrix form, just asis done in geometric optics, to findthe overall ABCD values for thesystem, then apply the bilineartransform. For more informationabout Gaussian beams, see Chapter17 of Siegman’s book, Lasers.

Fortunately, simple approximationsfor spot size and depth of focus canstill be used in most optical systemsto select pinhole diameters, couplelight into fibers, or compute laserintensities. Only when f-numbers arelarge should the full Gaussianequations be needed.

At large distances from a beamwaist, the beam appears to divergeas a spherical wave from a pointsource located at the center of thewaist. Note that “large” distancesmean where x»xR and are typicallyvery manageable considering thesmall area of most laser beams. Thediverging beam has a full angularwidth θ (again, defined by 1/e2

points):

θ λπ

= 4

2 w0

BEAMSPLITTERS

WIN

DOWS

FILTERS & ATTEN

UATORSULTRAFAST LASER OPTICS

ACCESSORIESTECHN

ICAL REFERENCE

POLARIZATION OPTICS

P h o n e : 1 - 8 0 0 - 2 2 2 - 6 4 4 0 • F a x : 1 - 9 4 9 - 2 5 3 - 1 6 8 0 • E m a i l : s a l e s @ n e w p o r t . c o m • We b : n e w p o r t . c o m

O p t i c s — 557

We have invoked the approximationtanθ ≈ θ since the angles are small.Since the origin can beapproximated by a point source, θ isgiven by geometrical optics as thediameter illuminated on the lens, D,divided by the focal length of thelens.

θ = −D

Ff( /#) 1

where f/# is the photographic f-number of the lens.

Equating these two expressionsallows us to find the beam waistdiameter in terms of the input beamparameters (with some restrictionsthat will be discussed later):

24

0wF

D=

λπ

We can also find the depth of focusfrom the formulas above. If wedefine the depth of focus (somewhatarbitrarily) as the distance betweenthe values of x where the beam is √2times larger than it is at the beamwaist, then using the equation forω(x) we can determine the depth offocus:

DOFF

D=

8

2

λπ

Using these relations, we can makesimple calculations for opticalsystems employing Gaussian beams.For example, suppose that we use a10 mm focal length lens to focus thecollimated output of a helium-neonlaser (632.8 nm) that has a 1 mmdiameter beam. The diameter of thefocal spot will be:

4632 8

10

1π

( )

, ,nm

mm

mm

or about 8 µm. The depth of focusfor the beam is then:

8632 8

10

1

2

π

( )

, ,nm

mm

mm

or about 160 µm. If we were tochange the focal length of the lensin this example to 100 mm, the focalspot size would increase 10 times to80 µm, or 8% of the original beamdiameter. The depth of focus wouldincrease 100 times to 16 mm.However, suppose we increase thefocal length of the lens to 2,000 mm.The “focal spot size” given by oursimple equation would be 200 timeslarger, or 1.6 mm, 60% larger thanthe original beam! Obviously,something is wrong. The trouble isnot with the equations giving ω(x)and R(x), but with the assumptionthat the beam waist occurs at thefocal distance from the lens. Forweakly focused systems, the beamwaist does not occur at the focallength. In fact, the position of thebeam waist changes contrary towhat we would expect in geometricoptics: the waist moves toward thelens as the focal length of the lens isincreased. However, we could easilybelieve the limiting case of thisbehavior by noting that a lens ofinfinite focal length such as a flatpiece of glass placed at the beamwaist of a collimated beam willproduce a new beam waist not atinfinity, but at the position of theglass itself.