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Beam Divergence

Definition : a measure for how fast a laser beam expands far from its focus

German : Strahldivergenz

Units: °, mrad

Formula symbol:

Author: Dr. Rüdiger Paschotta

Cite the article using its DOI : https://doi.org/10.61835/761

Get citation code: Endnote (RIS) BibTex plain text HTML

The beam divergence (or more precisely the beam divergence angle ) of a laser beam is a measure for how fast the beam expands far from the beam waist , i.e., in the so-called far field . Note that it is not a local property of a beam, for a certain position along its path, but a property of the beam as a whole. (In principle, one could define a local beam divergence e.g. based on the spatial derivative of the beam radius , but that is not common.)

A low beam divergence can be important for applications such as pointing or free-space optical communications . Beams with very small divergence, i.e., with approximately constant beam radius over significant propagation distances, are called collimated beams ; they can be generated from strongly divergence beams with beam collimators .

Some amount of divergence is unavoidable due to the general nature of waves (assuming that the light propagates in a homogeneous medium, not e.g. in a waveguide ). That amount is larger for tightly focused beams. If a beam has a substantially larger beam divergence than physically possibly, it is said to have a poor beam quality . More details are given below after defining what divergence means quantitatively.

Quantitative Definitions of Beam Divergence

Different quantitative definitions are used in the literature:

• 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 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 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 ( radius).

As an example, an FWHM beam divergence angle of 30° may be specified for the fast axis of a small edge-emitting laser diode . This corresponds to a 25.4° = 0.44 rad half-angle divergence, and it becomes apparent that for collimating such a beam without truncating it one would require a lens with a fairly high numerical aperture of e.g. 0.6. Highly divergent (or convergent) beams also require carefully designed optics to avoid beam quality degradation by spherical aberrations .

Divergence of Gaussian Beams and Beams with Poor Beam Quality

For a diffraction-limited Gaussian beam , the beam divergence half-angle is , where is the wavelength (in the medium) and the beam radius at the beam waist . This equation is based on the paraxial approximation , and is thus valid only for beams with moderately strong divergence. It also shows that the product of beam waist radius and the divergence angle (called the beam parameter product ) is not changed by any optical system without optical aberrations which transforms a Gaussian beam into another Gaussian beam with different parameters.

A higher beam divergence for a given beam radius , i.e., a higher beam parameter product , is related to an inferior beam quality , which essentially means a lower potential for focusing the beam to a very small spot. If the beam quality is characterized with a certain M 2 factor , the divergence half-angle is

As an example, a 1064-nm beam from a Nd:YAG laser with perfect beam quality ( ) and a beam radius of 1 mm in the focus has a half-angle divergence of only 0.34 mrad = 0.019°.

Beam Quality Calculations

Enter input values with units, where appropriate. After you have modified some values, click a “calc” button to recalculate the field left of it.

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Spatial Fourier Transforms

For obtaining the far field profile a beam, one may apply a two-dimensional transverse spatial Fourier transform to the complex electric field of a laser beam (→ Fourier optics ). Effectively this means that the beam is considered as a superposition of plane waves , and the Fourier transform indicates the amplitudes and phases of all plane-wave components. For propagation in free space, only the phase values change; it is thus easy to calculate propagation over large distances in free space, or alternatively in a homogeneous optical medium.

The width, measured e.g. as the root-mean-squared (r.m.s.) width, of the spatial Fourier transform can be directly related to the beam divergence. This means that the beam divergence (and in fact the full beam propagation) can be calculated from the transverse complex amplitude profile of the beam at any one position along the beam axis, assuming that the beam propagates in an optically homogeneous medium (e.g. in air).

Measurement of Beam Divergence

For the measurement of beam divergence, one usually measures the beam caustic , i.e., the beam radius at different positions, using e.g. a beam profiler .

It is also possible to derive the beam divergence from the complex amplitude profile of the beam in a single plane, as described above. Such data can be obtained e.g. with a Shack–Hartmann wavefront sensor .

One may also simply measure the beam intensity profile at a location far away from the beam waist , where the beam radius is much larger than its value at the beam waist. The beam divergence angle may then be approximated by the measured beam radius divided by the distance from the beam waist.

This encyclopedia is authored by Dr. Rüdiger Paschotta , the founder and executive of RP Photonics AG . How about a tailored training course from this distinguished expert at your location? Contact RP Photonics to find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, training) and software could become very valuable for your business!

More to Learn

Encyclopedia articles:

• beam radius
• laser beams
• collimated beams
• beam parameter product
• beam quality

Blog articles:

• The Photonics Spotlight 2007-07-11 : “What is a Beam Width, Beam Size, and a Beam Waist?”

Questions and Comments from Users

If I know the emitting area (2 μm × 1 μm) of a laser diode; how to calculate the divergence angle for that?

The author's answer:

From the emitting area alone you cannot do that; the transverse shape of the complex amplitude profile is also relevant. For a rough estimate, you may assume a Gaussian beam with = 1 μm for the wider direction, for example, assuming a single-mode profile.

What are some typical beam divergence values used for long distance satellite communications?

That differs between different usage scenarios. For example, if you want to reach a satellite with a sender on Earth, you can use a relatively large optical system, e.g. a with a 1-m diameter, and then achieve a correspondingly small beam divergence of the order of a microradian. For optical communications between different satellites, or for the backlink of a satellite, you typically need to use smaller optics and thus have a correspondingly larger beam divergence.

The size of the required optics is often the limiting factor. Another factor may be the highly precise orientation required for working with low-divergence beams.

Does the divergence depend on the medium, for example air vs. glass (no fiber optic)?

Yes. A beam with a certain initial focus will expand faster in air than in glass, for example. Its wavelength is shorter in glass.

What is the best way of accurately measuring the beam divergence of a large laser beam – for example a beam with w0 = 5 cm and around 120 µrad divergence (half angle) at 1500 nm?

A direct measurement is of course difficult, since you would need to use a very large propagation distance. (The Rayleigh length would be 5.24 km.) However, I am afraid that any other approach, introducing some additional optics (e.g. a telescope or an interferometer ) would introduce additional uncertainties. Therefore, it may be necessary to use that direct approach, somehow realizing that huge propagation distance of e.g. more than 15 km.

One might also think about where exactly and increased divergence could hurt you in your specific application. Maybe that would give you some useful hints.

If I use an aperture of 15 mm × 10 mm (v × h) at the output coupler of an excimer having a 3 mrad × 1 mrad beam divergence, how big is the beam at a distance of 4 meters?

A rough estimate: (3 mrad × 1 mrad) · 4 m = 12 mm × 4 mm for a tiny aperture; adding the initial aperture size, we arrive at 27 mm × 14 mm.

If I know the beam diameter at two points, can I compute the beam waist from those measurements alone? For example if the beam diameter is 1.06 mm at 62 cm from the laser head and 1.54 mm at 90 cm can I compute the w0?

You can estimate it that way, particularly if you also know the position of the beam waist (e.g. that it must be at a flat output coupler mirror). It would be better, however, to use more data points, reducing the impact of measurement errors.

Is there a simple relationship between r.m.s. phase error (compared to flat) and M 2 , divergence, BPP, or times diffraction-limited?

No, it is not just the r.m.s. phase excursion because it also matters a lot how quickly the optical phase varies across the beam.

How do I calculate the beam divergence of diode laser if I know the beam intensity profile at two different points separated by 50 cm, for example?

Provided that the beam focus is outside these two points, and that the beam diameter at the focus is much smaller than at those points, you can calculate the beam divergence angle as the difference of the beam radius divided by the distance of 50 cm.

How can I calculate the beam divergence from the beams size at the laser output and on the target at a known distance?

If the beam radius at the target is much larger than at the laser output, the half-angle divergence is just the beam radius at the target divided by the distance.

Would it be possible to estimate the M 2 factor of the beam emitted from a laser diode by using its FWHM divergence in combination with its emitting area dimensions?

Yes, for a rough estimate that can be used, although the exact beam radius will not be known and that kind of divergence value is not the variance-based value which is needed for .

Knowing the beam size at far field and the divergence angle, how to calculate the beam size at the laser exit?

That can be done if you know the beam quality factor (easily derived from the equation in the article): .

If I know the beam size at 2 different point such as at 1200 mm and 2 km, and the wavelength is also known, can I calculate the divergence and ? I am getting .

At least for the calculation, you need the beam waist radius at the focus, which you cannot get from those data alone. Getting the divergence as such should be possible, assuming that the beam monotonously expands between the two points.

Do you have experience with a beam which has different divergence before and after the waist in a homogeneous medium?

No, and I think this is not possible. At least not for the common definition of beam divergence based on the D4σ method.

As I am reading the section on spatial Fourier transforms, I had a thought: is the width of the spatial Fourier transform related to the linewidth of the laser? I.e., do higher Q resonators yield better beam quality? Or am I mixing things up here?

The linewidth is related to a temporal Fourier transform – that's really a different issue.

How can I calculate the beam divergence for a multimode fiber?

That depends on the launch conditions, but should in any case not be substantially larger than its numerical aperture .

I have a fiber coupled light-source with 105/125 μm fiber. I know that I wish to achieve a Beam Divergence of 1 × 1 mrad. What should be the fiber NA in order to get the Beam Divergence of 1 ×1 mrad?

In order to get such a low divergence of light directly from the fiber, you would need to have an extremely small numerical aperture – far outside the practical range. The solution must therefore be different: using a suitable lens behind the fiber end.

How to calculate the divergence angle of a Lorentzian beam?

What is the beam shape with the minimum divergence?

You can calculate the far field distribution essentially by applying a Fourier transform to the Lorentzian shape. Then you can calculate the divergence angle, based on some chosen criterion (e.g. FWHM).

The answer to the second question depends on the beam divergence criterion. For example, when using second-moment based width definitions, a Gaussian shape leads to minimum divergence for a given beam waist diameter.

I have the info of wavelength, beam quality in mm · mrad and minimum laser light cable diameter for laser cutting machine. May I suppose the minimum laser light cable diameter is the beam waist radius?

Yes, that way you can estimate the beam divergence obtained when focusing such that you get into the fiber. Compare that with the numerical aperture to check whether the launch efficiency can be high.

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Laser Beam Divergence Calculator

Table of contents

Lasers are good but not perfect : learn how — and why — a pointer can't work from the Earth to the Moon with our laser beam divergence calculator!

Keep reading to discover one of the most important features of a laser device. Here you will learn:

• An introduction to lasers and their operations;
• What is the laser beam divergence ;
• How to calculate the divergence of a laser beam; and
• How does the divergence of a laser affect the way it propagates .

You will also find some examples and, finally, why pointing at the Moon is rather tricky!

Explaining lasers at the speed of light

Lasers are devices that emit a highly coherent beam of monochromatic light thanks to a process of amplification of a single wavelength inside an energized medium.

Laser beams have three main properties. We already mentioned two; let's see all of them with an explanation.

• Laser radiation is nearly monochromatic . The characteristic of the medium where the light gets amplified allows selecting almost exclusively a single wavelength of light.
• Laser radiation shows a high directionality . The propagation of a beam happens along an axis around which the light expands rather slowly , even over long distances.
• Laser light is highly coherent . The photons emitted by a laser device are in phase with each other over both time and distance. Coherence allows laser beams to show interference .

The properties of laser radiation made the technology fundamental in the second half of the 20th century. However, when firstly devised, lasers were too "advanced": a scientist said of them that they were "a solution seeking a problem".

Eventually, the problems were found, which allowed the lasers to find applications in many high-technology sectors. The devices are now irreplaceable in medicine, optics, astronomy, manufacturing... you name it!

Inside the resonating chamber: at the origin of the divergence of a laser beam

Meet a laser . It looks like a uniform opaque cylinder from the outside, but you may notice a different material on one of the bases. That's where the radiation escapes the chamber : don't look at it when the device is on!

Inside the chamber, you can see two mirrors on opposite and parallel sides . One of them (the one at the opening of the resonator) is partially transparent, while the other is fully reflective.

Between the two mirrors, you can find a medium . This material constitutes the core of the device. If pumped with energy (either electricity or light), it can amplify the light of a specific wavelength. After bouncing back and forth many times, the photons escape the chamber, amplified and coherent.

The beam acquires its properties inside the optical resonator. We are talking about the three features listed above plus the way it propagates . Ideal lasers are Gaussian beams , which means that they move along a primary direction — or axis — with a cross-sectional intensity profile following a Gaussian curve .

The beam reaches its smallest size at a particular point in the resonating chamber. We call that point the waist of the beam . Once the beam passes the waist, it starts expanding in a cone . The angle of expansion is the divergence of the beam . Let's analyze it in detail.

What is the divergence of a laser beam?

The divergence of a laser beam measures how much the beam spreads with the distance, that is, the rate at which the laser diameter increases.

The diameter of the beam is measured at the 1 / e 2 1/e^2 1/ e 2 intensity point: the distance from the peak at which the intensity drops to 1 / e 2 1/e^2 1/ e 2 of the maximum value. The cone described by the divergence angle contains 86 % 86\% 86% of the total power of the laser.

🙋 Lasers always have a divergence, even if we can tweak some parameters to make it as small as possible. A laser with extremely small divergence is called a collimated beam .

All of the calculations for the divergence of a laser beam rely on the far-field approximation . Take a look at the diagram below. Close to the waist , the beam expands following a smooth curve. If you measured the divergence around that area, you would get the wrong result, underestimating the actual value. If you move far from the waist, the beam diameter increases almost linearly with the propagation direction. This situation is what we call far-field approximation .

How to calculate the laser beam divergence

The formula for the divergence angle of a laser beam is:

In the formula, we can identify:

• D f D_{\text{f}} D f ​ — The diameter of the beam measured in the final point;
• D i D_{\text{i}} D i ​ — The diameter of the beam measured at the initial point; and
• l l l — The distance between the initial and final point.

Notice how we measure twice the angle between the 1 / e 2 1/e^2 1/ e 2 asymptote and the propagation direction.

The previous formula for the divergence of a laser beam arises from geometrical considerations. As you can see in the diagram above, we ignore the structure of the Gaussian beam and only consider rays propagating in straight lines.

The divergence of a laser beam is limited by the physical characteristic of the laser itself. For ideal beams (propagating in a Gaussian beam with quality factor M 2 = 1 M^2=1 M 2 = 1 ), the divergence can't be lower than twice the value given by the formula:

• λ \lambda λ — Wavelength of the laser (or the frequency, if you apply the correct formula as in our wavelength calculator );
• w 0 w_0 w 0 ​ — Diameter at the waist ; and
• M 2 M^2 M 2 — Beam quality parameter .

🙋 The diffraction limit is an important concept in optics (or whenever you can find an oscillatory phenomenon). It defines the maximum resolution obtainable, fixing a lower limit to the detection capabilities of sensors and such. In a laser, the diffraction limit defines the smallest possible spot of a beam: the wavelength of the light would make it impossible to go lower than that!

💡 We talked about diffraction in our diffraction grating calculator .

How to use our laser beam divergence calculator

Our laser beam divergence calculator calculates the divergence of the beam in the far-field limit . You only have to input the values of the diameters at the initial and final points and the distance between the two. We will calculate the rest.

In the further properties section, you can find the fields for the optional variables wavelength , waist diameter , and quality factor of the beam. If you insert them, we will fire a warning if your calculations will return a value below the minimal theoretical limit of divergence . If you leave them empty, the only alert that will fire is if the divergence calculation returns a negative value.

Let's now calculate the divergence angle of a laser beam. Consider a beam with initial diameter D i = 4  mm D_{\text{i}}= 4\ \text{mm} D i ​ = 4   mm . Let's move away from the source by a distance l = 10  m l=10\ \text{m} l = 10   m , and measure the diameter again. We find D f = 7.5  mm D_{\text{f}}=7.5\ \text{mm} D f ​ = 7.5   mm . This information is all we need to calculate the divergence of that beam. Input the values in the LASER beam divergence calculator.

We will apply the formula for the divergence of a laser:

This value is small but not that small for lasers. You can use our calculator in reverse too: insert the distance, the initial diameter, and the divergence, and find out the final diameter of the beam. In this case, at the distance of 1  km 1\ \text{km} 1   km the beam would be more than 35  cm 35\ \text{cm} 35   cm in diameter.

We made other laser-related calculators! Discover more about those fascinating devices with our:

• Laser brightness calculator ;
• Laser linewidth and bandwidth calculator ; and
• Laser beam spot size calculator .

Why can't we use a laser pointer from the Earth to the Moon?

The lasers we see normally operate at relatively small distances, and we may be bound to think that they can propagate for an arbitrarily long distance without losing their characteristic "dot" shape.

Say this to an astronomer, and he will surely disagree. When the distances increase, lasers reach their limits while still maintaining an edge over traditional light sources.

Consider the process of lunar ranging , the measurement of the distance between Earth and Moon through the reflection of laser pulses on the reflectors left by space missions on the surface of our satellite.

In a typical lunar ranging experiment, a telescope collimates a laser (thus achieving a small divergence) and shoots the beam in the direction of the Moon, trying to hit the area of the reflector. This task is not like shooting at the broad side of a barn: the reflectors are incredibly tiny, and even though their locations are known, the sheer distance between them and us makes every hit a success.

The beam expands on its way to the Moon, reaching the surface of our satellite with a diameter of roughly 2  km 2\ \text{km} 2   km . The telescope sending the pulse has a diameter of 3.5  m 3.5\ \text{m} 3.5   m . Considering the mean value of the distance between Earth and Moon ( 384 , 400  km ) (384,400\ \text{km}) ( 384 , 400   km ) , we can calculate the divergence of the beam:

Which corresponds to barely more than a second of arc. This is an impressive feat! The pulse would be technically invisible by a human eye on the Moon, and its photons must still travel all the way back to Earth, undergoing further divergence. Catching a reflected photon is as hard as looking for a needle in a haystack. Luckily, scientists send a lot of photons at the same time, making detection possible.

The divergence of a laser beam is the measure of the increase in the beam diameter over distance . Even though lasers have a high directionality, the light propagates following a Gaussian beam, expanding over time.

The laser beam divergence is measured by an angle, often the full angle of the beam.

How to calculate the divergence of a laser beam?

To calculate the divergence of a laser beam in the far field approximation, be sure to know:

• The initial diameter of the beam Di ;
• The final diameter of the beam Df ;
• The distance between the two measurement points l .
• Compute the difference between final and initial diameter: Df - Di .
• Compute the ratio between the difference and twice the distance l : (Df - Di)/(2 × l) .
• Compute the arc tangent of the ratio and multiply by 2 .

Express your result in milliradians: the divergence is often pretty small.

How do you reduce a laser beam divergence?

You can reduce the divergence of a laser by acting on a single parameter, the initial diameter of the beam. Increasing it helps increase both the Rayleigh range (thus increasing the coherence length) and the divergence.

You can easily see this effect by observing the laser beam divergence formula and varying the value of the initial diameter Di :

ϴ = 2 × arctan((Df - Di)/(2 × l))

You can collimate the beam using a lens or a telescope to achieve such an effect.

What is the divergence of a laser with initial beam diameter of 1 mm and diameter at 10 meters of 5 mm?

0.4 mrad . To find this value, simply plug the values:

• Initial diameter Di = 1 mm ;
• Final diameter Df = 5 mm ; and
• Distance l = 10 m

...into the formula for the laser beam divergence:

ϴ = 2 × arctan((5 - 1)/(2 × 10,000)) = 0.4 mrad

Initial diameter (Di)

Final diameter (Df)

Distance (l)

Divergence (ϴ)

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Waist diameter (w 0 )

Beam quality parameter (M²)

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Quick guide on laser beam divergence measurement

Tuesday, July 11, 2023

• Laser experts

Lasers are in demand for their well-collimated, single-wavelength electromagnetic beams. The low-divergence, single-wavelength radiation and vast range of available laser power ranges make them one of the most valuable tools for material processing in industry and research.

Laser beam divergence is one important parameter to know for laser manufacturers and users alike who need to take control of their laser spot size . From laser welding applications to wafer dicing or eye surgery, knowing your beam divergence is a must for building a functional setup and directing the energy precisely where it is needed for a given process.

Here we present important concepts about laser beam divergence measurement. Understanding them will help you avoid wasting your energy where it is not needed!

First, what is laser beam divergence?

For a circular beam, divergence is defined as the angular measure of how the beam diameter increases with the distance from the laser aperture. It is measured in milliradians (mrad) or degrees ( ° ). Simply put, it tells you how the beam grows from the source to the target.

Laser specifications change over time for many reasons and it causes problems accross all industries. Learn about how laser output measurement solves numerous problems in YOUR industry. Download the guide below.

Gentec-EO's high-accuracy laser beam measurement instruments help engineers, scientists and technicians in all sorts of laser applications from the factory to the hospital, laboratory and research center. Learn about our solutions for these measurement types:

• Laser power meters
• Laser energy meters
• Laser beam profilers
• Terahertz power meters

Laser beams diverge because they would require an infinitely thin and long cavity of atoms emitting photons in resonance along one single direction to get a collimated beam over an infinite distance. That’s just not how things work in real life: for example, considering the growing number of industrial applications such as welding, cutting, or cladding, using fiber lasers that have a large divergence at the fiber output, dealing with divergence is inevitable.

More precisely, divergence is defined as the angle at which the beam expands in the far field (i.e. at a distance from the beam focus which is much farther than the Rayleigh length) by using the middle of the beam waist as reference.

The Rayleigh length Zr is the distance along the propagation axis from the beam waist to the place where the area of the beam cross-section is doubled.

All of this can seem a little confusing regarding Rayleigh length and far field. So, the key point to remember is that the beam behaves differently when checked close to the waist or in a far field. In the far field, the divergence of the beam tends to be linear, while it is not the case at the waist, like you can see in Figure 1.

Laser beam divergence measurement is all about beam size

By design, circular laser beams usually possess a Gaussian-shaped energy density distribution (or irradiance) from the center of the beam to the edges. You know, that bell-shaped curve telling you that most of the beam energy is located at the center of the beam, along the propagation axis.

Figure 2: theoretical Gaussian beam cross-section.

A proper laser beam profiling camera such as Beamage-4M will provide ISO-compliant beam size measurements such as the beam radius defined by the distance from the point of maximum energy density E max   in the energy distribution (at the center of the beam) to the point where the energy density equals E max / e 2 .

Figure 3: Example of a Gaussian fit made on x and y in Gentec-EO PC-Beamage software on a real beam. The X and Y beam dimensions are derived from this fit.

Have a look for yourself! You can download and try out the PC-Beamage software for free along with the available simulated beams.

The laser beam divergence formula

For a circular Gaussian beam, the minimum achievable value of divergence (half-angle) is given by this simple formula:

In the equation above, λ is your laser wavelength and ω 0 is the beam natural waist: its smallest dimension along the z-axis.

A Gaussian laser beam is said to be diffraction-limited when the measured divergence is close to θ 0 . To achieve the best aiming performance, that’s the goal!

You can do the calculations yourself, but you could also use our free calculators. You have two choices here. It depends if you’re in a far field or close to the waist. 

Try our laser beam divergence calculator

Gentec-EO's  beam divergence and diameter calculator is publicly available and free to use. Bookmark the page and use it whenever it can save you time.

If you are working around the beam waist, try our spot size calculator instead. It will not provide divergence information but rather give you the beam size in that close field (the calculator assumes a perfectly collimated beam prior to the lens, see the Formulas section for more details).

Stay still?

One would think that moving the camera along the propagation axis of the laser is required to measure divergence but it actually isn’t!

We all have in mind that to obtain the M 2  factor of a laser beam (also known as beam quality factor ), one has to measure both the divergence θ   and the beam waist  ω 0 , which does require to move along the z-axis to actually find the waist and measure its size.

The M 2  factor, which is dimensionless, is an indicator of the laser beam quality and quantifies how close your beam propagation is to the propagation of a theoretical Gaussian beam of the same wavelength.

Measuring the M 2  factor requires the use of a moving stage for your beam profiler, along with lenses and alignment mirrors, which is available as a complete package thanks to the Gentec-EO automated Beamage-M2 system. It adds an extra layer of information about beam quality but here, we don’t need to obtain several beam diameter values along the z-axis to calculate divergence.

Stay still!

First step is to place an aberration-free focusing lens between your camera and the laser. The key is to place the lens in the far field of the laser beam and the camera sensor exactly at the focal point of the lens (not at the beam waist).

At this point, it’s all about calculations! According to the ISO11146:2005 standard , the divergence in both main axes (x and y) is given by:

ω f   is the width of the focused spot at distance f  from the lens and  f  is the focal length of your lens at the wavelength of your laser. Beware that focal length is wavelength-dependent, so make sure you get the correct value from the lens supplier.

This method applies to non-Gaussian beams as well. No need to measure beam diameters before or after the focal point; that’s the beauty of this setup. If you change your lens for another focal length, the size of the beam at focal point will be different but if your camera is well positioned at focal point, the measured divergence will be the same.

Now that you have installed and launched PC-Beamage on your computer, open the Divergence tab and have fun!

Can we control the divergence of a laser beam?

For example, with high-power fiber lasers, your sample is usually processed up close to the fiber output, but the divergence is controlled and modified by using optical scanners that provide the required beam size on the sample to be processed.

Collimators can also be used in low-power fiber laser applications such as in telecommunications applications and with various diode lasers.

Go ahead and test various optics to evaluate how they affect your laser beam size, now that you know how to measure divergence easily.

Natural divergence of laser beams is often harnessed to enlarge the beam size and comply with power density damage thresholds for power measurement with a laser power meter .

Do you need to make sure how to use your beam profiling camera the fastest/easiest way and verify your power densities for safe profiling and power measurement? Contact us with your laser specs !

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How does diffraction cause laser beam divergence, and why will a laser beam always diverge, due to diffraction?

I have seen it said that diffraction causes laser beam divergence, or that a laser beam will always diverge, due to diffraction, or some variation of these statements. I understand diffraction in general, and I understand that the phenomenon applies to all waves, so I understand that it would also apply to laser beams; but it is not clear to me how it causes laser beam divergence, or why a laser beam will always diverge, due to diffraction. When trying to research to understand how diffraction causes laser beam divergence, I can't find anything that directly and clearly explains this – most results either just mention diffraction in the context of lasers without providing explanation, or mention 'diffraction-limited beams', which I think is something different to what I'm asking. So how does diffraction cause laser beam divergence, and why will a laser beam always diverge, due to diffraction?

• diffraction
• laser-cavity

• $\begingroup$ I would suggest that you look up Huygens principle and play with it to see how it would apply to a laser beam of various widths. $\endgroup$ –  S. McGrew Commented Apr 2, 2021 at 1:15
• $\begingroup$ It's not true that all laser beams diverge due to diffraction. What's true is that all laser beams with finite spatial extent diverge due to diffraction. $\endgroup$ –  The Photon Commented Apr 2, 2021 at 5:25
• 1 $\begingroup$ @ThePhoton Assuming "finite spatial extent" is the same as 'spatial confinement', don't all laser beams have "finite spatial extent"? Isn't this the assumption we use when solving Maxwell's equations, which then results in the 'Gaussian beam'? Or am I misunderstanding something? $\endgroup$ –  The Pointer Commented Apr 2, 2021 at 5:50
• $\begingroup$ Yes real laser beams all have finite extent. But we don't always use that assumption when solving Maxwell's equations. For example, when we obtain plane wave solutions (which, take note, is a solution that doesn't diverge due to diffraction). $\endgroup$ –  The Photon Commented Apr 2, 2021 at 5:54
• $\begingroup$ @ThePhoton oh, right. But, from what I remember, the plane wave solutions don't produce a Gaussian beam. $\endgroup$ –  The Pointer Commented Apr 2, 2021 at 5:55

5 Answers 5

The key point is that a laser beam is a wave which propagates according to Huygens principle . Once you accept this fact the divergence follows naturally.

In the image we see that the center of the "hole" generates a "flat" wave. The diffraction is evident only in at the edges.

In order to capture the behaviour of the "central part" of a wavefront we use approximation and omit the edges to a certain extend. In the upper picture we might describe the central part as a plane wave. If instead we use spherical mirrors to generate a propagating wave, we end up with the Gaussian beam $$ E \propto exp\left( - \frac{r^2}{w_0^2 (1 + (z/z_R)^2)} \right) $$ If we include the quadratic phase correction for the wavefront and the Gouy phase the approximation improves. However, the Gaussian beam is always an approximation obtained by omitting the edges of the wave (in deriving it, we use the paraxial Helmholz equation).

• $\begingroup$ Thanks for the answer. So, if I'm understanding this correctly, the reasoning is that, since the presence of any aperture/opening will cause diffraction, it is the presence of an aperture in lasers that causes the diffraction? Furthermore, is it necessary to use spherical mirrors for lasers to work? $\endgroup$ –  The Pointer Commented Apr 4, 2021 at 16:37
• $\begingroup$ I'm sorry, but no. The reasoning is: (1) A laser behaves like a wave. (2) Huygens principle describes its propagation. (3) Huygens principle uses the superposition of spherical waves. (5) If we have a superposition of spherical waves, diffraction is inevitable and expected/understandable. The finite size of the laser beam is only important, because diffraction is detectable only in the "wings" of the beam. After all, diffraction is the deviation from the linear propagation -- see Sommerfeld. $\endgroup$ –  Semoi Commented Apr 4, 2021 at 18:15
• $\begingroup$ en.wikipedia.org/wiki/Huygens –Fresnel_principle "It states that every point on a wavefront is itself the source of spherical wavelets, and the secondary wavelets emanating from different points mutually interfere." So are the gaps that we see between the waves due to destructive interference? $\endgroup$ –  The Pointer Commented Apr 4, 2021 at 19:15
• $\begingroup$ No, they are not. These are "cosine waves", but in 3D. Since we are unable to draw a cosine in 3D, we only draw the points, where the phase is zero, $\phi = 0$. These are the lines in my picture. $\endgroup$ –  Semoi Commented Apr 4, 2021 at 19:43
• $\begingroup$ Ok, I understand. Thank you for taking the time to explain. $\endgroup$ –  The Pointer Commented Apr 5, 2021 at 3:21

The simplest description of a laser beam uses ray optics. Often it is a good approximation. In it light is a ray that follows a straight line. According to this description, there need be no divergence. This description is too simple.

A better description is light as a wave. To get the true beam, you must solve the classical Maxwell's equations with a boundary condition. The optical cavity of a laser must have curved mirrors to be stable. The wave solution for a cavity bounded by spherical mirrors is a Gaussian Beam . Wavefronts are spherical. "Rays" are not quite straight, but follow hyperbolic paths. The beam cross section is Gaussian. The intensity is maximum at the beam axis and falls off smoothly away from the axis.

Image from https://www.rp-photonics.com/gaussian_beams.html

There is also a quantum mechanical explanation. The simplest quantum mechanical explanation invokes the Uncertainty Principle.

Imagine a beam with a uniform amplitude across the cross section. The beam consists of photons. The photons pass through a circular aperture, which confines the beam cross section to a limited $\Delta x$ . Because $\Delta x \Delta p \ge \hbar$ , the photon must have a non-zero momentum in the direction perpendicular to the beam. The beam cannot be perfectly collimated.

In practice, the solution to Maxwell's equations has a Gaussian cross section. Apertures are carefully chosen large enough to not significantly distort the beam by truncating the edge. Even though not physically confined, the beam cross section is confined because of the Gaussian profile. The beam cannot be perfectly collimated because of the Uncertainty Principle.

This is enough to tell you that a small diameter beam will have a large divergence. If you focus a beam to a small spot, it will have a very small waist. Therefore it must have a large divergence angle.

The image is from optique-ingenieur

A better quantum mechanical explanation shows the classical explanation is the same thing in disguise. See Interesting relationship between diffraction and Heisenberg’s uncertainty principle?

A photon has a wave function that is a solution of the Schrodinger equation. Like Maxwell's equations, this is a wave equation. A photon in a cavity has the same boundary conditions as the electromagnetic wave in the same cavity. The photon's wave function is also a radially symmetric function with spherical wavefronts and a Gaussian profile.

The wave function is in the position basis. You take the Fourier Transform to convert to the momentum basis. The Fourier Transform of the Gaussian cross section is a Gaussian cross section. The transverse momentum of the beam is a superposition of non-zero momentum states. The beam cannot be perfectly collimated. It has the same divergence as the electromagnetic wave.

• $\begingroup$ Thanks for the answer. But how does this answer my question about diffraction causing beam divergence? This isn't clear to me. The only time I see the word 'diffraction' used is when you link to the other question regarding Heisenberg's uncertainty principle. $\endgroup$ –  The Pointer Commented Apr 2, 2021 at 2:23
• $\begingroup$ Good point. Loosely, diffraction is the difference between ray optics and wave descriptions of light. Diffraction is the bending of light caused by its wave nature. So if you include diffraction in your description of how light propagates by choosing a wave description, you get beam divergence. $\endgroup$ –  mmesser314 Commented Apr 2, 2021 at 2:32
• $\begingroup$ Historically, lens design was done by ray tracing. The goal is to get as close to the perfect lens as possible by minimizing lens aberrations. Spherical surfaces are not the ideal shape, but they are much easier to manufacture. They do not direct rays exactly where you want. But you can choose radii and separations between elements to minimize aberrations. If you do a very good job, you get a lens so good that the biggest error is diffraction. Such a lens is diffraction limited. Diffraction is not predicted by ray tracing, but it is easy to calculate from the size of lens apertures. $\endgroup$ –  mmesser314 Commented Apr 2, 2021 at 2:34
• $\begingroup$ Your answer is interesting and informative, but all of this dances around my question. You're repeating that diffraction causes beam divergence, which I understand, but there is no explanation here for how it does so, and why a beam will always diverge, due to diffraction. The point of my question was to get a direct and clear answer to this. $\endgroup$ –  The Pointer Commented Apr 2, 2021 at 2:38
• $\begingroup$ Diffraction isn't exactly the cause of beam divergence. The Uncertainty Principle, or quantum mechanics, is the cause of beam divergence and of diffraction. That is the Uncertainty Principle is the cause of light not traveling in straight lines as predicted by ray tracing. $\endgroup$ –  mmesser314 Commented Apr 2, 2021 at 2:43

I will answer this only in terms of diffraction since that is the fundamental limit of laser beam divergence. Diffraction is the spreading of the beam because of the finite width of the beam. It is a fundamental of physics. Even the uncertainty principle is an aspect of the same thing. I.e, the more you confine the location of a particle, the less you know about its direction. In a way, this all comes down to wave mechanics.

Consider that each point in a wave propagates out in a circular fashion from that point (like a water ripple out from what a stone falls into the water). If a point next to that point also propagates out with the same phase (the peaks and valleys oscillate together), the two circular waves will add together to give a composite wave. As the line of “emitters” increases, the wave starts to look like a planar wave, but the edges will still propagate outwards. As the “beam” of the wave gets wider and wider from the emitters, the net effect is that the spread is less and less. It doesn’t matter if this is a light wave from a laser, a water wave, or a slit in a quantum experiment, the result is the same.

So, we can say a laser beam diverges because of fundamental physics and the nature of the spatial superposition of the coherently emitted photons from a laser.

Here are some resources where you could read further:

https://www.gentec-eo.com/blog/laser-beam-divergence-measurement#:~:text=Simply%20put%2C%20it%20tells%20you,beam%20on%20an%20infinite%20distance .

https://qr.ae/pG8mU9

• $\begingroup$ oh and do read the answer to a similar question on physics stack exchange: physics.stackexchange.com/a/79469/290525 $\endgroup$ –  Raghavendra Singh Commented Apr 4, 2021 at 13:50

I am no expert on this particular topic. I could be wrong.

To my knowledge, all electromagnetic waves diffract. Since laser is a highly coherent monochromatic light created from stimulated emission, the wavelengths are all the same and the troughes overlap with the troughes and crests with the crests. This means that it would follow a perfect path of diffraction through a gap without the waves themselves cancelling each other out.

The laser beam source is always of finite dimensions, thus it cannot provide a truly parallel beam. The laser beam is sligtly divergent by its origin.

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Experiment-4-Laser Beam Divergence-Simulation

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.

Prepared by Dr. A. R. Khalkar Engineering Physics Page 1 of 9 GES’s R. H. Sapat College of Engineering, Management Studies & Research, Nashik-422005, Maharashtra

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

Prepared by Dr. A. R. Khalkar Engineering Physics Page 2 of 9 GES’s R. H. Sapat College of Engineering, Management Studies & Research, Nashik-422005, Maharashtra

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.

Prepared by Dr. A. R. Khalkar Engineering Physics Page 3 of 9 GES’s R. H. Sapat College of Engineering, Management Studies & Research, Nashik-422005, Maharashtra

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

Prepared by Dr. A. R. Khalkar Engineering Physics Page 4 of 9 GES’s R. H. Sapat College of Engineering, Management Studies & Research, Nashik-422005, Maharashtra

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.

Prepared by Dr. A. R. Khalkar Engineering Physics Page 5 of 9 GES’s R. H. Sapat College of Engineering, Management Studies & Research, Nashik-422005, Maharashtra

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

Prepared by Dr. A. R. Khalkar Engineering Physics Page 6 of 9 GES’s R. H. Sapat College of Engineering, Management Studies & Research, Nashik-422005, Maharashtra

Graphs: A) For z1 = 50 cm

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

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

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

Prepared by Dr. A. R. Khalkar Engineering Physics Page 7 of 9 GES’s R. H. Sapat College of Engineering, Management Studies & Research, Nashik-422005, Maharashtra

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.

Prepared by Dr. A. R. Khalkar Engineering Physics Page 8 of 9 GES’s R. H. Sapat College of Engineering, Management Studies & Research, Nashik-422005, Maharashtra

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

For at 0mm detector distance. z2 = 100cm

GES’s RHSCOEMSR, Nashik

Prepared by Dr. A. R. Khalkar Engineering Physics Page 9 of 9

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June 26, 2024

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Experiment captures atoms in free fall to look for gravitational anomalies caused by dark energy

by University of California - Berkeley

Dark energy—a mysterious force pushing the universe apart at an ever-increasing rate—was discovered 26 years ago, and ever since, scientists have been searching for a new and exotic particle causing the expansion.

Pushing the boundaries of this search, University of California, Berkeley physicists have now built the most precise experiment yet to look for minor deviations from the accepted theory of gravity that could be evidence for such a particle, which theorists have dubbed a chameleon or symmetron. The results are published in the June 11, 2024, issue of Nature Physics .

The experiment, which combines an atom interferometer for precise gravity measurements with an optical lattice to hold the atoms in place, allowed the researchers to immobilize free-falling atoms for seconds instead of milliseconds to look for gravitational effects, besting the current most precise measurement by a factor of five.

Though the researchers found no deviation from what is predicted by the theory spelled out by Isaac Newton 400 years ago, expected improvements in the precision of the experiment could eventually turn up evidence that supports or disproves theories of a hypothetical fifth force mediated by chameleons or symmetrons.

The ability of the lattice atom interferometer to hold atoms for up to 70 seconds—and potentially 10 times longer—also opens up the possibility of probing gravity at the quantum level , said Holger Müller, UC Berkeley professor of physics. While physicists have well-tested theories describing the quantum nature of three of the four forces of nature—electromagnetism and the strong and weak forces—the quantum nature of gravity has never been demonstrated.

"Most theorists probably agree that gravity is quantum. But nobody has ever seen an experimental signature of that," Müller said.

"It's very hard to even know whether gravity is quantum, but if we could hold our atoms 20- or 30-times longer than anyone else, because our sensitivity increases exponentially, we could have a 400 to 800,000 times better chance of finding experimental proof that gravity is indeed quantum mechanical."

Aside from precision measurements of gravity, other applications of the lattice atom interferometer include quantum sensing.

"Atom interferometry is particularly sensitive to gravity or inertial effects. You can build gyroscopes and accelerometers," said UC Berkeley postdoctoral fellow Cristian Panda, who is first author of the paper. "But this gives a new direction in atom interferometry, where quantum sensing of gravity, acceleration and rotation could be done with atoms held in optical lattices in a compact package that is resilient to environmental imperfections or noise."

Because the optical lattice holds atoms rigidly in place, the lattice atom interferometer could even operate at sea, where sensitive gravity measurements are employed to map the geology of the ocean floor.

Screened forces can hide in plain sight

Dark energy was discovered in 1998 by two teams of scientists: a group of physicists based at Lawrence Berkeley National Laboratory, led by Saul Perlmutter, now a UC Berkeley professor of physics, and a group of astronomers that included UC Berkeley postdoctoral fellow Adam Riess. The two shared the 2011 Nobel Prize in Physics for the discovery.

The realization that the universe was expanding more rapidly than it should came from tracking distant supernovas and using them to measure cosmic distances. Despite much speculation by theorists about what's actually pushing space apart, dark energy remains an enigma—a large enigma, since about 70% of the entire matter and energy of the universe is in the form of dark energy.

One theory is that dark energy is merely the vacuum energy of space. Another is that it is an energy field called quintessence, which varies over time and space.

Another proposal is that dark energy is a fifth force much weaker than gravity and mediated by a particle that exerts a repulsive force that varies with the density of surrounding matter. In the emptiness of space, it would exert a repulsive force over long distances, able to push space apart. In a laboratory on Earth, with matter all around to shield it, the particle would have an extremely small reach.

This particle has been dubbed a chameleon, as if it's hiding in plain sight.

In 2015, Müller adapted an atom interferometer to search for evidence of chameleons using cesium atoms launched into a vacuum chamber , which mimics the emptiness of space.

During the 10 to 20 milliseconds it took the atoms to rise and fall above a heavy aluminum sphere, he and his team detected no deviation from what would be expected from the normal gravitational attraction of the sphere and Earth.

The key to using free-falling atoms to test gravity is the ability to excite each atom into a quantum superposition of two states, each with a slightly different momentum that carries them different distances from a heavy tungsten weight hanging overhead. The higher momentum, higher elevation state experiences more gravitational attraction to the tungsten, changing its phase.

When the atom's wave function collapses, the phase difference between the two parts of the matter wave reveals the difference in gravitational attraction between them.

"Atom interferometry is the art and science of using the quantum properties of a particle, that is, the fact that it's both a particle and a wave. We split the wave up so that the particle is taking two paths at the same time and then interfere them at the end," Müller said.

"The waves can either be in phase and add up, or the waves can be out of phase and cancel each other out. The trick is that whether they are in phase or out of phase depends very sensitively on some quantities that you might want to measure, such as acceleration, gravity, rotation or fundamental constants."

In 2019, Müller and his colleagues added an optical lattice to keep the atoms close to the tungsten weight for a much longer time—an astounding 20 seconds—to increase the effect of gravity on the phase. The optical lattice employs two crossed laser beams that create a lattice-like array of stable places for atoms to congregate, levitating in the vacuum. But was 20 seconds the limit, he wondered?

During the height of the COVID-19 pandemic, Panda worked tirelessly to extend the hold time, systematically fixing a list of 40 possible roadblocks until establishing that the wiggling tilt of the laser beam, caused by vibrations, was a major limitation.

By stabilizing the beam within a resonant chamber and tweaking the temperature to be a bit colder—in this case less than a millionth of a Kelvin above absolute zero, or a billion times colder than room temperature—he was able to extend the hold time to 70 seconds.

Gravitational entanglement

In the newly reported gravity experiment, Panda and Müller traded a shorter time, 2 seconds, for a greater separation of the wave packets to several microns, or several thousandths of a millimeter. There are about 10,000 cesium atoms in the vacuum chamber for each experiment—too sparsely distributed to interact with one another—dispersed by the optical lattice into clouds of about 10 atoms each.

"Gravity is trying to push them down with a force a billion times stronger than their attraction to the tungsten mass, but you have the restoring force from the optical lattice that's holding them, kind of like a shelf," Panda said.

"We then take each atom and split it into two wave packets, so now it's in a superposition of two heights. And then we take each one of those two wave packets and load them in a separate lattice site, a separate shelf, so it looks like a cupboard. When we turn off the lattice, the wave packets recombine, and all the quantum information that was acquired during the hold can be read out."

Panda plans to build his own lattice atom interferometer at the University of Arizona, where he was just appointed an assistant professor of physics. He hopes to use it to, among other things, more precisely measure the gravitational constant that links the force of gravity with mass.

Meanwhile, Müller and his team are building from scratch a new lattice atom interferometer with better vibration control and a lower temperature. The new device could produce results that are 100 times better than the current experiment, sensitive enough to detect the quantum properties of gravity .

The planned experiment to detect gravitational entanglement, if successful, would be akin to the first demonstration of quantum entanglement of photons performed at UC Berkeley in 1972 by the late Stuart Freedman and former postdoctoral fellow John Clauser. Clauser shared the 2022 Nobel Prize in Physics for that work.

Other co-authors of the gravity paper are graduate student Matthew Tao and former undergraduate student Miguel Ceja of UC Berkeley, Justin Khoury of the University of Pennsylvania in Philadelphia and Guglielmo Tino of the University of Florence in Italy.

Journal information: Nature , Nature Physics

Provided by University of California - Berkeley

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