GES’s R. H. Sapat College of Engineering, Management Studies & Research, Nashik-422005, Maharashtra

Experiment No. 4: Laser beam divergence and spot size

Aim: To calculate the beam divergence and spot size of the given laser beam.

Objectives : a) To understand the concept of laser. b) To develop the experimental measurement skill. c) To explore application areas of laser. d) To calculate the beam divergence and spot size of the given laser beam e) Apparatus: He-Ne gas laser, light detector with current output, constant power supply.

Theory: Laser: The term LASER is the acronym for Light Amplification by Stimulated Emission of Radiation. It is a mechanism for emitting electromagnetic radiation via the process of stimulated emission. The laser was the first device capable of amplifying light waves themselves. The emitted laser light is a spatially coherent, narrow low-divergence beam. When the waves(or photons) of a beam of light have the same frequency, phase and direction, it is said to be coherent . There are lasers that emit a broad spectrum of light, or emit different wavelengths of light simultaneously. According to the encyclopaedia of laser physics and technology, beam divergence of a laser beam is a measure for how fast the beam expands far from the beam waist. A laser beam with a narrow beam divergence is greatly used to make laser pointer devices. Generally, the beam divergence of laser beam is measured using beam profiler. Divergence means “Departure from norm or Deviation”. The beam divergence of laser beam is the measure of increase of diameter or radius. A laser beam consists of very nearly parallel light rays, the beam diameter increases far more slowly with distance from the light source as compare to the light beam from a other light sources.

Beam GES’sSpot Size: Beam Diameter RHSCOEMSR, is defined as the distance across the centerNashik of the beam for which the irradiance (I) equals 1/e2 of the maximum irradiance. The spot size of the beam if the radial distance from the center of maximum irradiance to the 1/e2 points.

Beam Divergence:The beam divergence of an electromagnetic beam is an angular mea- sure of the increase in beam diameter with distance from the optical aperture from which the electromagnetic beam emerges. It is given by; w − w θ = 1 2 2 where and are the beam spot sizes of a laser bean mounted at two points separated by a w1 w2 distance d.

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Lasers usually emit beams with a Gaussian profile. A Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity (irradiance) distributions are described by Gaussian functions. Having, setup the definitions, let us try and look at the implications of both the quantities that we aim to measure. A Gaussian beam is a beam of electromagnetic radiation whose trans- verse electric field and intensity (irradiance) distributions are described by Gaussian functions. Many lasers emit beams with a Gaussian profile, in which case the laser is said to be operating on the fundamental transverse mode, or mode of the laser’s optical resonator. When refracted by a lens, a Gaussian beam is transformed into another Gaussian beam (characterized by a different set of parameters), which explains why it is a convenient, widespread model in laser optics. This is the reason why the dispersed beam which we get by using a disperser has a Gaussian Profile too.

The electric field amplitude of a Gaussian beam can be given by;

2 2 w∘ −r r E (r, z) = E∘ exp exp −ikz − ik + iζ(z) w(z) [ w2(z) ] [ 2R(z) ] where, r : radial distance from the centre axis of the beam z : axial distance from the beam's narrowest point i : imaginary unit (for which i2 = − 1) 2π k = : wave number (in radians per meter). λ w(z) : radius at which the field amplitude drops to 1/e and field intensity to 1/e2 of their axial values, respectively. : waist size. w∘

E∘ = |GES’sE (0,0)| RHSCOEMSR, Nashik R(z) : radius of curvature of the beam's wavefronts ζ(z) : Gouy phase shift. It is an extra contribution to the phase that is seen in beams which obey Gaussian profiles.

The corresponding time-averaged intensity (or irradiance) distribution is;

2 |E(r, z)|2 w −2r I (r, z) = = I ∘ exp 2η ∘ [ w(z) ] [ w2(z) ] where is the intensity at the center of the beam at its waist. The constant η is defined as I∘ = I (0,0) the characteristic impedance of the medium in which beam is propagating. For free space it is . η = η∘ ≈ 377Ω

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Let’s now see Beam Divergence.The divergence of a beam can be calculated if one knows the beam diameter at two separate points , and the distance between these points. The beam (w1, w2) (d ) divergence is given by; w − w θ = 2.arctan 2 1 d For the laser we observe that the divergence is very less. So, we can approximate tanθ to θ, hence giving; w − w θ = 2 1 2 The divergence of a laser beam is proportional to its wavelength and inversely proportional to the diameter of the beam at its narrowest point.

Beam parameters: Beam parameters govern the behaviour and geometry of a Gaussian beam. The important beam parameters are described below.

1. Beam divergence: The light emitted by a laser is confined to a rather narrow cone. But, when the beam propagates outward, it slowly diverges or fans out. For an electromagnetic beam, beam divergence is the angular measure of the increase in the radius or diameter with distance from the optical aperture as the beam emerges. The divergence of a laser beam can be calculated if the beam diameter and at two separate distances are known. Let and are the d1 d2 z1 z2 distances along the laser axis, from the end of the laser to points “1” and “2”.

2 1 Θ d2 − d1

LASER d Θ d d GES’s RHSCOEMSR,1 Nashik2

z1 z2 − z1 z2 Usually, divergence angle is taken as the full angle of opening of the beam. Then, d − d Θ = 2 1 z2 − z1 Half of the divergence angle can be calculated as; w − w θ = 2 1 z2 − z1 where and are the radii of the beam at and . Like all electromagnetic beams, lasers are w1 w2 z1 z2 subject to divergence, which is measured in milli-radians (m.rad) or degrees. For many applications, a lower-divergence beam is preferable.

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2. Spot Size:

The distance across the center of the beam for which the irradiance (intensity) equals 1/e2 of the maximum irradiance (1/e2 = 0.135) is defined as the beam diameter. The spot size (w) of the beam is defined as the radial distance (radius) from the center point of maximum irradiance to the 1/e2 point. Gaussian laser beams are said to be diffraction limited when their radial beam divergence is close to the minimum possible value, which is given by;

Θ λ θ = = 2 π w∘ where λ is the wavelength of the given laser and is the radius of the beam at the narrowest point, w∘ which is termed as the beam waist.

Quantitative Definitions of Beam Divergence: According to the most common definition, the beam divergence is the derivative of the beam radius with respect to the axial position in the far field, i.e., at a distance from the beam waist which is much larger than the Rayleigh length. This definition yields a divergence half-angle (in units of radians), and further depends on the definition of the beam radius. For Gaussian beams, the beam radius is usually defined via the point with 1/e2 times the maximum intensity. For non-Gaussian profiles, an integral formula can be used, as discussed in the article on beam radius. Sometimes, full angles are used instead, resulting in twice as high values. Instead of referring to directions with 1/e2 times the maximum intensity, as is done for the Gaussian beam radius, a full width at half- maximum (FWHM) divergence angle can be used. This is common e.g. in data sheets of laser diodes and light-emitting diodes. For Gaussian beams, this kind of full beam divergence angle is 1.18 times the half-angle divergence defined via the Gaussian beam radius (1/e2 radius).

Full Width at Half Maximum (FWHM): The full width at half-maximum, also called half-power beam width, is measured from the distribution curve of the beam’s intensity along a pre-defined axis, passing through the beam’s center, which is also usually its point of maximum intensity. The FWHM corresponds to the distance between the two points closest to the peak that have 50% of the maximum irradiance or intensity.GES’s One can prefer usingRHSCOEMSR, other percentages of a beam’s maximum intensityNashik to define its width. A common one is 13.5%, which leads us to our next beam diameter definition: 1/e2.

L A S E R S P O T

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Performing the real lab:

1. Arrange the laser and detector in an optical bench arrangement. 2. The laser is switched on and is made to incident on the photodiode.

3. Fix the distance, z between the detector and the laser source. 4. By adjusting the micrometer of the detector, move the spot in the horizontal direction, from left to right. 5. Note the output current for each distance, x from the measuring device. 6. Then the beam profile is plotted with the micrometer distance along the X-axis and intensity of current along Y-axis. We will get a gaussian curve as shown below. 7. The experiment is repeated for different detector distances.

8. Note the points in the graph where the intensity equals 2 of the maximum intensity, . 1/e Ie 9. Find the micrometer distance across the beam corresponding to these points (A-B) from the

figure for a pair of detector distances and . Half of this distance is noted as and . z1 z2 w1 w2 10. Then the divergence and spot size of the laser beam can be calculated from the equations.

Performing the simulator: 1. Visit: ->home->physical sciences->laser optics virtual lab->laser beam divergence and spot size 2. The experimental arrangement is shown in the simulator. 3. A side view and top view of the set up is given in the inset. 4. The start button enables the user to start the experiment. 5. From the combo box, select the desired laser source. 6. ThenGES’s fix a detector distance, RHSCOEMSR, say 100 cm, using the slider Detector distance, Nashik z. 7. The z distance can be varied from 50 cm to 200 cm. 8. For a particular z distance, change the detector distance x, from minimum to maximum, using the slider Detector distance, x. The micrometer distances and the corresponding output currents are noted. The x distances can be read from the zoomed view of the micrometer and the current can be note from the digital display of the output device. 9. Draw the graph and calculate the beam divergence and spot size using the steps given above. 10.Show graph button enables the user to view the beam profile. 11.Using the option Show result, one can verify the result obtained after doing the experiment.

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Observations: To find the Least Count of Screw gauge One pitch-scale division (n) = 1 mm; Number of divisions on head scale (m) = 100 Least Count n 1mm L . C . = = = 0.01mm m 100

For z = 50 cm For z = 100 cm Distance 1 2 Sr. No. mm Current Current m A m A 1 -0.10 0 0 2 -0.09 0 0 3 -0.08 0 0 4 -0.07 0 0.000252 5 -0.06 0 0.001199 6 -0.05 0 0.004495 7 -0.04 0.000099 0.013250 8 -0.03 0.002629 0.030716 9 -0.02 0.027418 0.055999 10 -0.01 0.111929 0.080292 11 0 0.178887 0.090539 12GES’s RHSCOEMSR,0.01 0.111929 Nashik0.080292 13 0.02 0.027418 0.055999 14 0.03 0.002629 0.030716 15 0.04 0.000099 0.013250 16 0.05 0 0.004495 17 0.06 0 0.001199 18 0.07 0 0.000252 19 0.08 0 0 20 0.09 0 0 21 0.1 0 0

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Graphs: A) For z1 = 50 cm

0.200 Z1=50cm 0.180 E = 0.1788m A 0.160 max

0.140

0.120

0.100 0.0894m A 50 % FWHM 0.080 Current I (mA) Current 0.060

0.040 d = 0.025mm 0.020 1

0.000 -0.10 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Detector Distance (mm)

B) For z2 = 100 cm 0.200 Z2=100cm 0.180 0.160 GES’s RHSCOEMSR, Nashik 0.140 d = 0.05mm 0.120 2

0.100 Emax = 0.090m A

0.080 Current I (mA) Current 0.060 0.045m A 50 % FWHM 0.040

0.020

0.000 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Detector Distance (mm)

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Calculation: From Graphs 1. Spot size: Diameters of the beam corresponds to spot size 2. d1, d2 1/e 1 Detector Distance Spot size = e2 For z1 = 50 cm d1 = 0.025mm For z2 = 100 cm d2 = 0.050mm

2. Radial beam divergence: Wavelength of He-Ne laser , λ = 6328Å(Std.)

d 0.025mm d 0.050mm w = 1 = = 0.0125mm and w = 2 = = 0.025mm 1 2 2 2 2 2

λ 0.0006328mm θ1 = = = 0.01612rad = 16.12mrad π w1 3.14 × 0.0125mm

λ 0.0006328mm θ2 = = = 0.08061rad = 80.61mrad π w2 3.14 × 0.025mm

3. Divergence angle (Θ)

d − d 0.050 − 0.025mm 0.025mm Θ = 2 1 = = = 5 × 10−5mrad z2 − z1 100 − 50cm 500mm

Results:GES’s RHSCOEMSR, Nashik 1 Radial beam Detector Distance Spot size = Divergence angle (Θ) e2 divergence For z1 = 50 cm d1 = 0.025mm θ1 = 16.12mrad 5 × 10−5 mrad For z2 = 100 cm d2 = 0.050mm θ2 = 80.61mrad

Conclusion: The Laser beam divergence and spot sizes have been calculated from graph and compared with the values from simulation and verified.

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Screen shot: For at 0mm detector distance. z1 = 50cm

For at 0mm detector distance. z2 = 100cm

GES’s RHSCOEMSR, Nashik

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