A clear reciprocal relationship exists between the spot size (2W0) and the divergence. In addition, for a Gaussian beam at a given wavelength, the product of the spot size and the divergence is a constant at 4λ/π. This is important in defining how much a beam deviates from a perfect Gaussian beam.

Gaussianbeamdivergence

As a rule of thumb, the higher the magnification of the lens is, the smaller the angle of view. If you manage to fit the Moon in your picture, the vertical angle of view would be around half a degree, but this feat requires a pretty high magnification. On the other hand, a 55 mm55\ \text{mm}55 mm lens on an SLR (single-lens reflex) camera will give you a horizontal angle of view of about 20°20\degree20°.

🙋 To find the size of your sensor, search on Google "[your camera model] sensor size": you will easily find the correct values!

🙋 You can use our camera field of view calculator to find the values of the angles and fields of view and to calculate the needed focal length of the lens you need to mount to obtain a particular field of view. We locked the variables associated with the sensor's size: it's unlikely you will change it instead of the lens!

A rectilinear lens is a lens that preserves orthogonality: two straight, perpendicular lines in the real world are depicted as straight, perpendicular eyes by a rectilinear lens. A fisheye lens, on the other hand, distorts them — a small price to pay for an extremely wide field of view!

The two first ones define the width and height of the rectangle corresponding to our sensor. Notice that these quantities are absolute: you can place the "sphere" as far as you like (even at an infinite distance), and the angle of view will still be the same.

Gaussianbeamprofile

For additional insights into photonics topics like this, download our free MKS Instruments Handbook: Principles & Applications in Photonics Technologies

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Laser beams are often propagated through an optical system consisting of lenses, mirrors, dielectric interfaces, or other optical elements. Fortunately, under most conditions, a Gaussian beam remains a Gaussian beam after encountering these elements, ensuring that the propagation equations used above remain valid. An optical element modifies the input beam by changing the position and size of its beam waist. Knowledge of the input beam parameters and the properties of the optical element can be used to determine the new values using a variety of propagation methods. A thin focusing lens  is probably the most important optical element that impacts a laser beam. Figure 2 illustrates beam propagation through a thin lens resulting in a repositioning and resizing of the beam waist (as well as zR and θ). When a collimated beam (i.e., 2.zR >> f) encounters a lens, the resulting beam waist is simply the product of the focal length and original divergence (see full angular width divergence equation) with the waist positioned at the focal point of the lens.

You already know what the iii means and ddd is the distance at which the field of view is measured. Remember to use the correct dimension of your sensor: you want to use its length to calculate the horizontal field of view of your camera, not the vertical one... unless you are taking a portrait picture.

Why the tiny iii, you ask? This formula holds for all the three possible directions on the sensor: horizontal, vertical, and diagonal.

The concept of field of view is not unique to photography: check how it differs for astronomers at our telescope field of view calculator. 🔭

The sensor size of your camera significantly affects the quality of your pictures. A camera with a larger sensor will give you a wider field of view for the same lens, maintaining the same magnification: your subject will be surrounded by more background. The advantages are relative, though: when printing the photograph in the same format, a larger field of view will necessarily translate to a lower magnification.

As for the angle of view, you can identify three quantities associated with the field of view: a horizontal length, a vertical length, and a diagonal length. To calculate the field of view fovi\text{fov}_ifovi​ of a camera in each of the three possible directions, use the following formula:

The Canon EOS 550D has a sensor size 22.3×14.9 mm22.3\times14.9\ \text{mm}22.3×14.9 mm, which allows us to calculate both the horizontal and the vertical angle of view. Let's not be extreme and use the 24 mm24\ \text{mm}24 mm focal length.

Gaussianbeamradius

Do you want to learn more about the fundamental of photography with a slight technical twist? We made the right calculators for you: the aspect ratio calculator and the crop factor calculator!

The evolution of a Gaussian beam as it propagates along its axial direction (z) is shown in Figure 1. The irradiance has a radial (r) distribution that is circularly symmetric in any plane orthogonal to the axis and the beam's power (φ) is concentrated close to the axis. The functional form of this irradiance distribution (E) is given by:

A camera captures a rectangular portion of the real world, projecting it onto its sensor. We identified two possible ways to measure the size of that portion:

When we use angles to define the dimensions of a picture, we talk of the angle of view. The angle of view is pretty easy to visualize: your camera lies at the center of a sphere, and connecting the angles of the scene you are capturing to its center gives you a set of three angles:

⚠️ There is quite a bit of confusion online (and not only) on the matter, with various definitions and interpretations. Here we gave the one that makes the most sense for us, but feel free to disagree and let us know!

Gaussianbeamintensity formula

This illustrates that the beam shape remains Gaussian at any point along the axis and changes only in its width and amplitude. The beam radius W is defined as the radius at which the irradiance decreases to 1/e2 or 0.135 of the peak on-axis (r = 0) value. This is sometimes referred to as the half-width 1/e2 (HW1/e2) value. As shown in Figure 1, W gradually increases as the distance from the minimum beam radius (known as the beam waist, W0 gets larger. Since E is the power per unit area, as indicated by above equation, the irradiance decreases as one moves away from the beam waist. Integration of the irradiance over the entire radial plane (at any axial position) results in the total optical power. In other words, φ remains constant along the axis. From a practical standpoint, integrating the irradiance within a circle of radius 1.5.W results in 99% of the total power. This is relevant when measuring the optical power of a Gaussian beam.

We define both quantities for three spatial directions, which allows us to calculate a vertical, diagonal, and horizontal field of view for a camera set-up, alongside the respective angles of view.

zR is known as the Rayleigh range and represents the distance from the waist where the radius increases by a factor of . If above equation is substituted into irradiance distribution equation, then zR is the distance at which the irradiance has decreased by a factor of √2 from its peak value at the waist. Twice the Rayleigh range is called the confocal parameter (or depth-of-focus) and is a rough estimate of the collimation of a beam. As shown in the above equation, the beam size increases slowly with axial distance from the waist. As z >> zR, the beam size then increases linearly with z with a slope of W0/zR. This slope represents the full angular width divergence (θ) of the beam given by:

Gaussianbeamsoftware

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📷 Cameras are our way to create memories — instants preserved forever — from what we can see. However, their electronic eyes have some limitations when it comes to how much of those memories they record. We may need to think beforehand about what we want to include in our pictures, and many photographers find it helpful to know their cameras' field of view.

Take the diverging lines from the sphere's center to the corner of the scene, and stop them at a certain distance ddd. Now, draw the corresponding rectangle: you will obtain a set of measurements we call the field of view at a distance ddd.

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That is an extremely wide angle of view: it covers 170017001700 square degrees! However, you need almost 20 of these fields to capture a true 260°260\degree260° picture.

Think about the Moon for a second more. You managed to fit it into your picture — your sensor! We are talking of a 3,500 km3,500\ \text{km}3,500 km body. You can take a picture of a plane passing between you and our satellite with some luck and good timing. A B747 flying at 6.5 km6.5\ \text{km}6.5 km would almost eclipse the Moon, fitting perfectly in the picture together. Does it mean the jumbo jet is really that jumbo, or that the field of view can be a relative concept, too?

What can you capture with this lens? For a distance d=200 md=200\ \text{m}d=200 m, the angle of view converts into the respective linear field of views:

Whether you are planning a photoshoot or just a photography enthusiast, our camera field of view calculator will help you learn the whole picture.

Using angles, it is 26.5° × 17.7°, vertical and horizontal angle of view, respectively. To calculate these values, input them in the angle of view formula:

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Refine your photographic knowledge with our dedicated tools like the depth of field calculator and our magnification of a lens calculator. Cheese! 📸

Gaussianbeamcalculator

The solid angle covered by this set-up is about 272727 square degrees for the vertical one! If you want to take a 360°360\degree360° picture, you'll need more than 150015001500 shots! But at a distance of 200 m200\ \text{m}200 m you would be able to picture an area of 45 m×30 m45\ \text{m}\times 30\ \text{m}45 m×30 m: good enough to take some exciting wildlife pictures without disturbing anyone!

The basic concept underlying the entire matter is that cameras can capture a single, defined portion of the real-world at once. How significant this portion depends on the camera set-up, particularly on the type of lens and camera body used by the photographer.

It's impossible to define a typical field of view of a camera: it depends on the lens you are mounting at the moment. However, we can give you some practical examples.

Gaussianbeampdf

The concepts are used almost interchangeably, as they define the same concept. However, their definitions are different. Let's discover them in detail.

A commonly-available helium-neon laser emits a near-ideal Gaussian beam with a value of M2 < 1.1. For many solid-state lasers, M. is in the range of 1.1-1.3. Collimated laser diodes that emit fundamental TEM00 modes possess M. of 1.1 to 1.7, whereas high-energy multimode lasers can generate M2 factors as large as 10 to 100. Spatial filtering can improve beam quality using Fourier optics. Finally, additional beam quality metrics include the Beam Propagation Factor (K) where K = 1/M2 as well as the Beam Parameter Product (BPP), which is defined as BPP = M2λ/π = 2WM.M2

One practical consequence of this definition is that an ideal Gaussian beam (i.e., M2 = 1) can be focused to a minimum spot diameter, whereas beams of higher M. values focus to larger spot diameters in proportion to the M2 value. As a result, M2 provides meaningful information about lasers, particularly if their application involves small focused spot sizes. While WM and θM are sufficient for determining M2, these values often cannot be measured directly. By focusing the beam with a lens of known focal length (like Figure 2), the characteristics of the artificially created beam waist and divergence can be measured. To provide an accurate calculation of M2, the International Organization for Standards (ISO) requires at least 5 measurements in the focused beam waist region and at least 5 measurements in the far fields (two Rayleigh ranges away from the waist area), as shown in Figure 3. These multiple measurements ensure that the minimum beam width is found while a "curve fit" improves the accuracy by minimizing measurement error at any single point.

While many laser systems operate with near-Gaussian beams, other laser systems possess non-Gaussian beams that propagate differently and exhibit significantly different spatial distributions (see Figure 4 for examples). In some cases, a laser resonator emits a beam with a higher-order TEMmn mode. Depending on the resonator geometry, these modes can be cylindrical in nature and are called Laguerre-Gaussian beams or rectangular and are called Hermite-Gaussian beams. In other cases, a laser beam is modified by an optical system to such an extent that its profile and propagation can no longer be approximated using the Gaussian beam analysis. Flat-top beams are one such example where a beam exhibits a nearly constant irradiance over its beam width (see Figure 4). Given the steep edges of the beam profile, the diameters of these beams are often characterized by their full-width at half-maximum (FWHM) values as opposed to the HW1/e2 radius values used for Gaussian beams. Such flat-top beams are important for laser-based material processing where a constant irradiance provides more uniform material modification. The propagation of these beams can be quite complicated and is often encountered when a laser beam overfills a focusing objective in order to generate a very small spot size in high-resolution microscopy.

The irradiance distribution of a laser beam is determined by the transverse modes that exit the laser cavity. Typically, the lowest-order transverse mode (TEM00) is selected for emission since it propagates with the least beam divergence and can be focused to the tightest spot. The irradiance distribution of this TEM00 mode is described by a Gaussian function and therefore much of this section details a Gaussian beam's profile as well as its evolution with distance. Gaussian beam propagation is well understood and even laser beams that do not possess a TEM00 mode are often described using a modified Gaussian mode analysis using its M-Squared value. Finally, a brief description of non-Gaussian beams is also given.

Here is the lack of absoluteness of the field of view: its value varies with the distance between the camera and the subject. That's why in the same angle of view, you can fit both the Moon and a passenger airplane.

Canon produces a rectilinear 11-24 mm\text{11-24 mm}11-24 mm lens — which is insane — that allows capturing sharp wide-angle images. How wide? Let's mount the lens on a Canon EOS 550D and calculate this camera field of view!

On the other hand, let's calculate the camera field of view for a typical telephoto lens set-up. We keep the same camera but mount a 200 mm200\ \text{mm}200 mm lens. Insert the values in the appropriate fields of our camera field of view calculator and calculate the angles of view in this case:

Gaussianbeamformula

The TEM00 mode of even a well-designed laser system is not a perfect Gaussian beam. The M2 ("M-Squared") analysis was developed to characterize the quality of a laser beam. That is, how close it is to an ideal Gaussian beam. M2 is defined as the product of the spot size (2WM) and divergence (θM) of the real beam divided by the spot-size-divergence product of an ideal Gaussian beam:

With our tool, you will be able to calculate the camera's field of view (any camera!), and not only that, there is more to it. Keep reading to discover:

That is more than enough to fit the whole Colosseum in Rome in a single picture. And all of this standing barely more than the diameter of the arena itself away!

Even if angles are easy to visualize, they can be hard to estimate (how wide is 5°5\degree5°?). The field of view comes in handy to complement the idea of the angle of view.

The equation describing the evolution of the beam radius along the axis where z is the distance from the beam waist is given by:

The angle of view of a camera is an absolute measure of the horizontal and vertical angles captured by a combination of camera and lens. The field of view measures the same concept but uses lengths. Since the angles don't change, the field of view depends on the distance: specifically, it increases alongside it.