Gaussian Beams and the Knife-Edge Measurement

Gaussian Beams:

Gaussian beams propagating in the z-direction may be represented mathematically as:

U = U0ei(kz−ωt) e

ikr2/2q

qwhere q = z − zwaist − izR (1)

Here U is the electric field amplitude of the wave, q is known as the ‘complex radius’, zwaist is a realconstant indicating the position of the beam waist and zR known as the ‘Rayleigh range’.Substituting the expression for q into the equation for U , and assuming that the waist is at the origin(zwaist = 0), it is straightforward to show that:

U =U0i

zR

-Amplitude on axis at waist (constant)

×w0

w(z)-Amplitude gets smaller as beam spreads

×ei(kz−ωt) -Wave travelling to positive z

×e−iα(z) -Phase change with z

×e−r2/w(z)2 -Gaussian profile

×eikr2/2R(z) -Curved wavefronts

where the 1/e2 radius in the intensity distribution or ‘spot size’ w(z) obeys:

w(z) = w0

√

(

1 +z2

z2R

)

with zR =πw2

0

λ(2)

and the radius of curvature of the wavefronts R(z) obeys:

R(z) = z +z2

R

z(3)

and the phase factor (often referred to as Guoy’s phase) takes the form:

α(z) = tan−1(z/zR) (4)

The Gaussian beam is therefore narrowest at z = 0 (or z = zwaist) where it is said to have a ‘waist’ofw0, as illustrated in Fig. 1. Away from the waist the beam spreads with a hyperbolic outline, theasymptotes of which define the far field beam divergence such that the half-angle at the 1/e2 radius inthe intensity distribution is given by:

θ = limz→∞

w(z)

z=

λ

πw0

(5)

At the position of the beam waist the wavefront is planar corresponding to an infinite radius of curvature.Away from the waist, where the beam diverges (or converges) the wavefronts are of course curved.The Rayleigh range zR relates to the distance over which a Gaussian beam can be collimated beforeit spreads significantly due to diffraction. In particular, the Rayleigh range is the distance which thebeam travels from the waist before the beam diameter increases by

√2 or before the beam area doubles

in size. The Rayleigh range marks the approximate dividing line between the ‘near-field’ or Fresnel

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-3 -2 -1 0 1 2 3

Position in units of the Rayleigh Range zR

w(z)

1/e2 intensity radius

w0

waist

21/2

w0

θ

R(z)

Figure 1: Notation used in describing a Gaussian Beam

and the ‘far-field’ or Fraunhofer regions for a beam propagating out from a Gaussian waist. Note: textbooks often refer to the ‘Confocal Parameter’ b = 2zR, so named because it is equal to the length of asymmetrical confocal mirror cavity into which the beam fits.It is striking that the entire range of properties of the Gaussian beam depend solely on the size ofthe waist w0 and the wavelength λ. It is therefore possible to fully characterize a Gaussian beam bydetermining the size and location of the beam waist. This is readily achieved by making a series of‘knife edge’ measurements to determine the spot size as a function of the distance z.

Beam width measurement using the Knife Edge:

Method: Record the total power in the beam as a knife edge is translated through the beam using acalibrated translation stage. The power meter records the integral of the Gaussian beam between −∞and the position of the knife.Analysis: Assume a beam propagating in the z-direction with a Gaussian intensity profile:

I(x, y) = I0e−2x2/w2

xe−2y2/w2y (6)

where wx and wy are the 1/e2 radii of the beam in the x and y directions respectively. I0 is the peakintensity.The total power in the beam is:

PTOT = I0

∫

∞

−∞

e−2x2/w2xdx

∫

∞

−∞

e−2y2/w2ydy =

π

2I0wxwy (7)

Consider the knife edge being translated in the x-direction as shown in Fig 2.The transmitted power is then:

2

y

x

Figure 2: Principle of the knife edge measurement.

P (X) = PTOT − I0

∫ X

−∞

e−2x2/w2xdx

∫

∞

−∞

e−2y2/w2ydy

= PTOT −√

π

2I0wy

∫ X

−∞

e−2x2/w2xdx

= PTOT −√

π

2I0wy

[∫ 0

−∞

e−2x2/w2xdx +

∫ X

0

e−2x2/w2xdx

]

= PTOT −√

π

2I0wy

[√

π

8wx +

∫ X

0

e−2x2/w2xdx

]

=PTOT

2−√

π

2I0wy

∫ X

0

e−2x2/w2xdx (8)

Consider now the integral in Eq.8 which we wish to cast in a standard form. Making the substitutionu2 = 2x2/w2

x so that dx = wxdu/√

2 and making the necessary change to the limits of the integral leadsto:

P (X) =PTOT

2−√

π

2I0wy

∫

√

2Xwx

0

e−u2 wx√2du

=PTOT

2−

π

4I0wywx

2√π

∫

√

2Xwx

0

e−u2

du (9)

Using the standard definition of the Error Function listed in the Appendix and the expression in Eq. 9for the total power in the beam we arrive at our final result:

P (X) =PTOT

2

[

1 − erf

(√2X

wx

)]

(10)

Fitting: To fit the data in Origin, use a fitting function of the form:

Pmeasured =P1

2

[

1 ± erf

(√2(X − P2)

P3

)]

(11)

where P1 corresponds to the power, P3 corresponds to the 1/e2 radius of the Gaussian beam and the-(+) is chosen when the knife is translated in the positive (negative) direction.

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Shorthand method for determining beam widths

Taking a full set of knife edge measurements and fitting as above can be tedious and time consuming.Therefore in cases where only the width of the beam, rather than its full profile, is required (for example,when making measurements of width with distance from a source/a lens in order to determine the focalpoint of the beam), it is quicker to use the 90% − 10% method outlined below.Method: Measure the total power in the beam when fully exposed. Translate the knife edge acrossthe beam and measure the distance between the points at which the power output is 10% and 90% ofthis value, X10 and X90 respectively. This distance is X10−90.Analysis: According to Eq. 10, at 10% power:

0.1PTOT =PTOT

2

(

1 − erf

(√2X10

wx

))

(12)

Rearranging gives

erf

(√2X10

wx

)

= 0.8 (13)

Using Eq. 31, this can be related to the Gaussian probability so that

P (t10) = 0.9 where

√2X10

wx

=t10√

2(14)

Using standard probability tables, this gives

t10 = 1.28 (15)

Therefore from Eq.14X10 = 0.64wx (16)

From the symmetry of the Gaussian function it is straightforward to see that:

X10−90 = 2 × 0.64wx = 1.28wx (17)

Thus there is a ‘calibration’ from which the 1/e2 radius may be determined by making two measure-ments only. A similar treatment could be applied to X80 and X20 etc. However the 90%-10% methodis favored as it uses points at the extremes of the region of maximum change in the Error Function (seeFig. 4).

Real Gaussian Beams and M2

Real laser beams will deviate from the ideal Gaussian. A measure of their quality is given by M2 whichis defined such that

M2 = 1 for an ideal Gaussian beam M2 > 1 for a real beam

The definition of M2: Consider a Gaussian beam propagating from a source. In the far-field regionbeyond the Rayleigh range, from Eq 5 and Fig. 2:

θ = limz→∞

w(z)

z=

λ

πw0

(18)

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Consider the product of this with the waist w0:

w0θ =λ

π(19)

This is a lower fundamental limit which can only be approached by ideal Gaussian beams. For a realbeam

w0θ =λ

πM2 (20)

Therefore the Rayleigh range of a real beam can be rewritten

zR,real =πw2

0

λM2(21)

Method for measuring M2: The set up is as in Fig. 3. Using the 90% − 10% method, the beamwidth w(z) at at least 6 points across the range shown should be measured.Fitting: According to Eq. 2 and applying an arbitrary offset to take account of the initial position ofthe knife edge, the fitting equation to be used is:

w(z)2 = P1

(

1 +(x − P2)2

P3

)

(22)

whereP1 = w2

0 and P3 = z2R,real (23)

Using these values, M2 can then be calculated using Eq. 21

Lens

Laser sourceOutput beam

Beam narrowest at

focus

z

x

Figure 3: The set up for determining M2 by measuring the beam width with distance from a lens

Appendix: The Gaussian Distribution and the Error Function

The Normal or Gaussian distribution function is defined as:

φ(t) =1√2π

e−t2/2 (24)

where the pre-factor ensures the correct normalization. Statistical tables give the value of

P (t) =

∫ t

−∞

φ(t)dt (25)

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A perhaps more useful definition includes the width σ and non-zero mean µ of the distribution explicitly:

Φ(t) =1

σ√

2πe−(x−µ)2/2σ2

(26)

The full width at half maximum is then related to σ by:

FWHM = 2√

2ln2σ ≃ 2.35σ (27)

Unfortunately Gaussian beams are defined in yet another way, namely in terms of the 1/e2 radius or‘spot size’ w:

I(r) =2P

πw2e−2r2/w2

(28)

where P is the total power in the beam. Comparison of Eqs. 27 and 29 shows that w = 2σ.The Error Function is defined as follows:

erf(x) =2√π

∫ x

0

e−u2

du (29)

and has the properties that:

erf(∞) = 1 and erf(−x) = −erf(x) (30)

The Gaussian probability distribution is related to the error function by:

P (x) =1

2+

1

2erf

(

x√2

)

(31)

-3 -2 -1 0 1 2 3

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Position

w

-3 -2 -1 0 1 2 30.0

0.1

0.2

0.3

0.4

0.5

0.6

Position

w

Figure 4: Plot of the Error Function and the corresponding normalized Gaussian distribution showingboth σ and the ‘spot size’ w.

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