Fundamentally, the Gaussian is a solution of the axial Helmholtz equation, the wave equation for an electromagnetic field. Although there exist other solutions, the Gaussian families of solutions are useful for problems involving compact beams.

For a circle of radius r = w(z), the fraction of power transmitted through the circle is P ( z ) P 0 = 1 − e − 2 ≈ 0.865. {\displaystyle {\frac {P(z)}{P_{0}}}=1-e^{-2}\approx 0.865.}

Although there are other modal decompositions, Gaussians are useful for problems involving compact beams, that is, where the optical power is rather closely confined along an axis. Even when a laser is not operating in the fundamental Gaussian mode, its power will generally be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's resonator (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM00) Gaussian mode.

Here λ is the wavelength of the light, n is the index of refraction. At a distance from the waist equal to the Rayleigh range zR, the width w of the beam is √2 larger than it is at the focus where w = w0, the beam waist. That also implies that the on-axis (r = 0) intensity there is one half of the peak intensity (at z = 0). That point along the beam also happens to be where the wavefront curvature (1/R) is greatest.[1]

Substituting this solution into the wave equation above yields the paraxial approximation to the scalar wave equation:[17] ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 2 i k ∂ u ∂ z . {\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=2ik{\frac {\partial u}{\partial z}}.} Writing the wave equations in the light-cone coordinates returns this equation without utilizing any approximation.[18] Gaussian beams of any beam waist w0 satisfy the paraxial approximation to the scalar wave equation; this is most easily verified by expressing the wave at z in terms of the complex beam parameter q(z) as defined above. There are many other solutions. As solutions to a linear system, any combination of solutions (using addition or multiplication by a constant) is also a solution. The fundamental Gaussian happens to be the one that minimizes the product of minimum spot size and far-field divergence, as noted above. In seeking paraxial solutions, and in particular ones that would describe laser radiation that is not in the fundamental Gaussian mode, we will look for families of solutions with gradually increasing products of their divergences and minimum spot sizes. Two important orthogonal decompositions of this sort are the Hermite–Gaussian or Laguerre-Gaussian modes, corresponding to rectangular and circular symmetry respectively, as detailed in the next section. With both of these, the fundamental Gaussian beam we have been considering is the lowest order mode.

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where the combined order of the mode N is defined as N = l + m. While the Gouy phase shift for the fundamental (0,0) Gaussian mode only changes by ±π/2 radians over all of z (and only by ±π/4 radians between ±zR), this is increased by the factor N + 1 for the higher order modes.[10]

where n is the refractive index of the medium the beam propagates through, and λ is the free-space wavelength. The total angular spread of the diverging beam, or apex angle of the above-described cone, is then given by Θ = 2 θ . {\displaystyle \Theta =2\theta \,.}

Lens magnificationChart

It is possible to decompose a coherent paraxial beam using the orthogonal set of so-called Hermite-Gaussian modes, any of which are given by the product of a factor in x and a factor in y. Such a solution is possible due to the separability in x and y in the paraxial Helmholtz equation as written in Cartesian coordinates.[19] Thus given a mode of order (l, m) referring to the x and y directions, the electric field amplitude at x, y, z may be given by: E ( x , y , z ) = u l ( x , z ) u m ( y , z ) exp ⁡ ( − i k z ) , {\displaystyle E(x,y,z)=u_{l}(x,z)\,u_{m}(y,z)\,\exp(-ikz),} where the factors for the x and y dependence are each given by: u J ( x , z ) = ( 2 / π 2 J J ! w 0 ) 1 / 2 ( q 0 q ( z ) ) 1 / 2 ( − q ∗ ( z ) q ( z ) ) J / 2 H J ( 2 x w ( z ) ) exp ⁡ ( − i k x 2 2 q ( z ) ) , {\displaystyle u_{J}(x,z)=\left({\frac {\sqrt {2/\pi }}{2^{J}\,J!\;w_{0}}}\right)^{\!\!1/2}\!\!\left({\frac {{q}_{0}}{{q}(z)}}\right)^{\!\!1/2}\!\!\left(-{\frac {{q}^{\ast }(z)}{{q}(z)}}\right)^{\!\!J/2}\!\!H_{J}\!\left({\frac {{\sqrt {2}}x}{w(z)}}\right)\,\exp \left(\!-i{\frac {kx^{2}}{2{q}(z)}}\right),} where we have employed the complex beam parameter q(z) (as defined above) for a beam of waist w0 at z from the focus. In this form, the first factor is just a normalizing constant to make the set of uJ orthonormal. The second factor is an additional normalization dependent on z which compensates for the expansion of the spatial extent of the mode according to w(z)/w0 (due to the last two factors). It also contains part of the Gouy phase. The third factor is a pure phase which enhances the Gouy phase shift for higher orders J.

Some subfamilies of hypergeometric-Gaussian (HyGG) modes can be listed as the modified Bessel-Gaussian modes, the modified exponential Gaussian modes,[23] and the modified Laguerre–Gaussian modes.

The physical electric field is obtained from the phasor field amplitude given above by taking the real part of the amplitude times a time factor: E phys ( r , z , t ) = Re ⁡ ( E ( r , z ) ⋅ e i ω t ) , {\displaystyle \mathbf {E} _{\text{phys}}(r,z,t)=\operatorname {Re} (\mathbf {E} (r,z)\cdot e^{i\omega t}),} where ω {\textstyle \omega } is the angular frequency of the light and t is time. The time factor involves an arbitrary sign convention, as discussed at Mathematical descriptions of opacity § Complex conjugate ambiguity.

The sign of the Gouy phase depends on the sign convention chosen for the electric field phasor.[10] With eiωt dependence, the Gouy phase changes from -π/2 to +π/2, while with e-iωt dependence it changes from +π/2 to -π/2 along the axis.

The values of hhh and ggg are hidden in the further magnification properties section of our calculator, so if you need to know either of these, just click the button!

A camera is nothing but lenses and a sensor. At least in theory! To understand how it works, we need to explore the world of optics.

The magnification of a lens is an absolute measure of how much the height of a real image differs from the object's height. Remember, that in a camera, the real image forms on the sensor (or on the film, if you're old school).

An elliptical beam will invert its ellipticity ratio as it propagates from the far field to the waist. The dimension which was the larger far from the waist, will be the smaller near the waist.

E l , m ( x , y , z ) = E 0 w 0 w ( z ) H l ( 2 x w ( z ) ) H m ( 2 y w ( z ) ) × exp ⁡ ( − x 2 + y 2 w 2 ( z ) ) exp ⁡ ( − i k ( x 2 + y 2 ) 2 R ( z ) ) × exp ⁡ ( i ψ ( z ) ) exp ⁡ ( − i k z ) . {\displaystyle {\begin{aligned}E_{l,m}(x,y,z)={}&E_{0}{\frac {w_{0}}{w(z)}}\,H_{l}\!{\Bigg (}{\frac {{\sqrt {2}}\,x}{w(z)}}{\Bigg )}\,H_{m}\!{\Bigg (}{\frac {{\sqrt {2}}\,y}{w(z)}}{\Bigg )}\times {}\\&\exp \left({-{\frac {x^{2}+y^{2}}{w^{2}(z)}}}\right)\exp \left({-i{\frac {k(x^{2}+y^{2})}{2R(z)}}}\right)\times {}\\&\exp {\big (}i\psi (z){\big )}\exp(-ikz).\end{aligned}}}

Since that beast would be too dangerous to photograph at a close distance, we suggest you use a 500 mm500\ \text{mm}500 mm telephoto lens. We advise you to keep your distance, let's say 150 meters (but remember that a kangaroo can reach a maximum speed of 70 m/s70\ \text{m/s}70 m/s). Insert these values into our magnification of a lens calculator, which will return:

For a fundamental Gaussian beam, the Gouy phase results in a net phase discrepancy with respect to the speed of light amounting to π radians (thus a phase reversal) as one moves from the far field on one side of the waist to the far field on the other side. This phase variation is not observable in most experiments. It is, however, of theoretical importance and takes on a greater range for higher-order Gaussian modes.[10]

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Beam profiles which are circularly symmetric (or lasers with cavities that are cylindrically symmetric) are often best solved using the Laguerre-Gaussian modal decomposition.[6] These functions are written in cylindrical coordinates using generalized Laguerre polynomials. Each transverse mode is again labelled using two integers, in this case the radial index p ≥ 0 and the azimuthal index l which can be positive or negative (or zero):[20][21]

As a special case of electromagnetic radiation, Gaussian beams (and the higher-order Gaussian modes detailed below) are solutions to the wave equation for an electromagnetic field in free space or in a homogeneous dielectric medium,[17] obtained by combining Maxwell's equations for the curl of E and the curl of H, resulting in: ∇ 2 U = 1 c 2 ∂ 2 U ∂ t 2 , {\displaystyle \nabla ^{2}U={\frac {1}{c^{2}}}{\frac {\partial ^{2}U}{\partial t^{2}}},} where c is the speed of light in the medium, and U could either refer to the electric or magnetic field vector, as any specific solution for either determines the other. The Gaussian beam solution is valid only in the paraxial approximation, that is, where wave propagation is limited to directions within a small angle of an axis. Without loss of generality let us take that direction to be the +z direction in which case the solution U can generally be written in terms of u which has no time dependence and varies relatively smoothly in space, with the main variation spatially corresponding to the wavenumber k in the z direction:[17] U ( x , y , z , t ) = u ( x , y , z ) e − i ( k z − ω t ) x ^ . {\displaystyle U(x,y,z,t)=u(x,y,z)e^{-i(kz-\omega t)}\,{\hat {\mathbf {x} }}\,.}

In optics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse Gaussian mode describes the intended output of many lasers, as such a beam diverges less and can be focused better than any other. When a Gaussian beam is refocused by an ideal lens, a new Gaussian beam is produced. The electric and magnetic field amplitude profiles along a circular Gaussian beam of a given wavelength and polarization are determined by two parameters: the waist w0, which is a measure of the width of the beam at its narrowest point, and the position z relative to the waist.[1]

The shape of a Gaussian beam of a given wavelength λ is governed solely by one parameter, the beam waist w0. This is a measure of the beam size at the point of its focus (z = 0 in the above equations) where the beam width w(z) (as defined above) is the smallest (and likewise where the intensity on-axis (r = 0) is the largest). From this parameter the other parameters describing the beam geometry are determined. This includes the Rayleigh range zR and asymptotic beam divergence θ, as detailed below.

The radius of the wavefront's curvature is largest on either side of the waist, crossing zero curvature (radius = ∞) at the waist itself. The rate of change of the wavefront's curvature is largest at the Rayleigh distance, z = ±zR. Beyond the Rayleigh distance, |z| > zR, it again decreases in magnitude, approaching zero as z → ±∞. The curvature is often expressed in terms of its reciprocal, R, the radius of curvature; for a fundamental Gaussian beam the curvature at position z is given by:

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u ε ( ξ , η , z ) = w 0 w ( z ) C p m ( i ξ , ε ) C p m ( η , ε ) exp ⁡ [ − i k r 2 2 q ( z ) − ( p + 1 ) ζ ( z ) ] , {\displaystyle u_{\varepsilon }\left(\xi ,\eta ,z\right)={\frac {w_{0}}{w\left(z\right)}}\mathrm {C} _{p}^{m}\left(i\xi ,\varepsilon \right)\mathrm {C} _{p}^{m}\left(\eta ,\varepsilon \right)\exp \left[-ik{\frac {r^{2}}{2q\left(z\right)}}-\left(p+1\right)\zeta \left(z\right)\right],} where ξ and η are the radial and angular elliptic coordinates defined by x = ε / 2 w ( z ) cosh ⁡ ξ cos ⁡ η , y = ε / 2 w ( z ) sinh ⁡ ξ sin ⁡ η . {\displaystyle {\begin{aligned}x&={\sqrt {\varepsilon /2}}\;w(z)\cosh \xi \cos \eta ,\\y&={\sqrt {\varepsilon /2}}\;w(z)\sinh \xi \sin \eta .\end{aligned}}} Cmp(η, ε) are the even Ince polynomials of order p and degree m where ε is the ellipticity parameter. The Hermite-Gaussian and Laguerre-Gaussian modes are a special case of the Ince-Gaussian modes for ε = ∞ and ε = 0 respectively.[7]

The radius of the beam w(z), at any position z along the beam, is related to the full width at half maximum (FWHM) of the intensity distribution at that position according to:[4] w ( z ) = FWHM ( z ) 2 ln ⁡ 2 . {\displaystyle w(z)={\frac {{\text{FWHM}}(z)}{\sqrt {2\ln 2}}}.}

In this form, the parameter w0, as before, determines the family of modes, in particular scaling the spatial extent of the fundamental mode's waist and all other mode patterns at z = 0. Given that w0, w(z) and R(z) have the same definitions as for the fundamental Gaussian beam described above. It can be seen that with l = m = 0 we obtain the fundamental Gaussian beam described earlier (since H0 = 1). The only specific difference in the x and y profiles at any z are due to the Hermite polynomial factors for the order numbers l and m. However, there is a change in the evolution of the modes' Gouy phase over z: ψ ( z ) = ( N + 1 ) arctan ⁡ ( z z R ) , {\displaystyle \psi (z)=(N+1)\,\arctan \left({\frac {z}{z_{\mathrm {R} }}}\right),}

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When light hits a surface and is s-polarized, the electric field oscillates in a direction perpendicular to the plane of incidence, and no component of this ...

Arbitrary solutions of the paraxial Helmholtz equation can be decomposed as the sum of Hermite–Gaussian modes (whose amplitude profiles are separable in x and y using Cartesian coordinates), Laguerre–Gaussian modes (whose amplitude profiles are separable in r and θ using cylindrical coordinates) or similarly as combinations of Ince–Gaussian modes (whose amplitude profiles are separable in ξ and η using elliptical coordinates).[5][6][7] At any point along the beam z these modes include the same Gaussian factor as the fundamental Gaussian mode multiplying the additional geometrical factors for the specified mode. However different modes propagate with a different Gouy phase which is why the net transverse profile due to a superposition of modes evolves in z, whereas the propagation of any single Hermite–Gaussian (or Laguerre–Gaussian) mode retains the same form along a beam.

If you'd like to know more, try our others optic calculators dedicated to lenses, like the thin lens equation calculator or the lens maker equation calculator.

Although the tails of a Gaussian function never actually reach zero, for the purposes of the following discussion the "edge" of a beam is considered to be the radius where r = w(z). That is where the intensity has dropped to 1/e2 of its on-axis value. Now, for z ≫ zR the parameter w(z) increases linearly with z. This means that far from the waist, the beam "edge" (in the above sense) is cone-shaped. The angle between that cone (whose r = w(z)) and the beam axis (r = 0) defines the divergence of the beam: θ = lim z → ∞ arctan ⁡ ( w ( z ) z ) . {\displaystyle \theta =\lim _{z\to \infty }\arctan \left({\frac {w(z)}{z}}\right).}

Lens magnificationmicroscope

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u p m ( ρ , ϕ , Z ) = 2 p + | m | + 1 π Γ ( p + | m | + 1 ) Γ ( p 2 + | m | + 1 ) Γ ( | m | + 1 ) i | m | + 1 × Z p 2 ( Z + i ) − ( p 2 + | m | + 1 ) ρ | m | × exp ⁡ ( − i ρ 2 Z + i ) e i m ϕ 1 F 1 ( − p 2 , | m | + 1 ; ρ 2 Z ( Z + i ) ) {\displaystyle {\begin{aligned}u_{{\mathsf {p}}m}(\rho ,\phi ,\mathrm {Z} ){}={}&{\sqrt {\frac {2^{{\mathsf {p}}+|m|+1}}{\pi \Gamma ({\mathsf {p}}+|m|+1)}}}\;{\frac {\Gamma \left({\frac {\mathsf {p}}{2}}+|m|+1\right)}{\Gamma (|m|+1)}}\,i^{|m|+1}\times {}\\&\mathrm {Z} ^{\frac {\mathsf {p}}{2}}\,(\mathrm {Z} +i)^{-\left({\frac {\mathsf {p}}{2}}+|m|+1\right)}\,\rho ^{|m|}\times {}\\&\exp \left(-{\frac {i\rho ^{2}}{\mathrm {Z} +i}}\right)\,e^{im\phi }\,{}_{1}F_{1}\left(-{\frac {\mathsf {p}}{2}},|m|+1;{\frac {\rho ^{2}}{\mathrm {Z} (\mathrm {Z} +i)}}\right)\end{aligned}}}

Expand the further magnification properties section to see the variable extension tube. We set it at 0 mm0\ \text{mm}0 mm by default, but change it according to your needs!

If P0 is the total power of the beam, I 0 = 2 P 0 π w 0 2 . {\displaystyle I_{0}={2P_{0} \over \pi w_{0}^{2}}.}

First thing – the upward facing arrow on the left of the image is the object we are looking at. The rays of light coming from it hit the lens. The one parallel to the optic axis (the topmost line) gets focused and so converges on the focus. The ray passing through the center of the lens meets the focused ray on the other side of the lens, which creates a flipped image called the real image of the object.

The equations below assume a beam with a circular cross-section at all values of z; this can be seen by noting that a single transverse dimension, r, appears. Beams with elliptical cross-sections, or with waists at different positions in z for the two transverse dimensions (astigmatic beams) can also be described as Gaussian beams, but with distinct values of w0 and of the z = 0 location for the two transverse dimensions x and y.

E ( r , z ) = E 0 x ^ w 0 w ( z ) exp ⁡ ( − r 2 w ( z ) 2 ) exp ⁡ ( − i ( k z + k r 2 2 R ( z ) − ψ ( z ) ) ) {\displaystyle {\mathbf {E} (r,z)}=E_{0}\,{\hat {\mathbf {x} }}\,{\frac {w_{0}}{w(z)}}\exp \left({\frac {-r^{2}}{w(z)^{2}}}\right)\exp \left(\!-i\left(kz+k{\frac {r^{2}}{2R(z)}}-\psi (z)\right)\!\right)}

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At a position z along the beam (measured from the focus), the spot size parameter w is given by a hyperbolic relation:[1] w ( z ) = w 0 1 + ( z z R ) 2 , {\displaystyle w(z)=w_{0}\,{\sqrt {1+{\left({\frac {z}{z_{\mathrm {R} }}}\right)}^{2}}},} where[1] z R = π w 0 2 n λ {\displaystyle z_{\mathrm {R} }={\frac {\pi w_{0}^{2}n}{\lambda }}} is called the Rayleigh range as further discussed below, and n {\displaystyle n} is the refractive index of the medium.

The magnification, which depends on w 0 {\displaystyle w_{0}} and z 0 {\displaystyle z_{0}} , is given by

These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. Their intensity profiles are characterized by a single brilliant ring; like Laguerre–Gaussian modes, their intensities fall to zero at the center (on the optical axis) except for the fundamental (0,0) mode. A mode's complex amplitude can be written in terms of the normalized (dimensionless) radial coordinate ρ = r/w0 and the normalized longitudinal coordinate Ζ = z/zR as follows:[23]

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Lenses can focus or "unfocus" light rays. In this tool, we will only consider converging lenses. Their main feature is the ability to focus every ray entering the lens parallel to the optical axis at a specific point, the focus.

🔎 Lenses and their properties have been known by humanity for a long time. However, only in the 13ᵗʰ century did lens-making skills reach a level of refinement that allowed for the construction of glasses, telescopes, and much more!

Hermite-Gaussian modes are typically designated "TEMlm"; the fundamental Gaussian beam may thus be referred to as TEM00 (where TEM is transverse electro-magnetic). Multiplying ul(x, z) and um(y, z) to get the 2-D mode profile, and removing the normalization so that the leading factor is just called E0, we can write the (l, m) mode in the more accessible form:

where u x ( x , z ) = 1 q x ( z ) exp ⁡ ( − i k x 2 2 q x ( z ) ) , u y ( y , z ) = 1 q y ( z ) exp ⁡ ( − i k y 2 2 q y ( z ) ) , {\displaystyle {\begin{aligned}u_{x}(x,z)&={\frac {1}{\sqrt {{q}_{x}(z)}}}\exp \left(-ik{\frac {x^{2}}{2{q}_{x}(z)}}\right),\\u_{y}(y,z)&={\frac {1}{\sqrt {{q}_{y}(z)}}}\exp \left(-ik{\frac {y^{2}}{2{q}_{y}(z)}}\right),\end{aligned}}}

where Lpl are the generalized Laguerre polynomials. CLGlp is a required normalization constant:[22] C l p L G = 2 p ! π ( p + | l | ) ! ⇒ ∫ 0 2 π d ϕ ∫ 0 ∞ d r r | u ( r , ϕ , z ) | 2 = 1 , {\displaystyle C_{lp}^{LG}={\sqrt {\frac {2p!}{\pi (p+|l|)!}}}\Rightarrow \int _{0}^{2\pi }d\phi \int _{0}^{\infty }dr\;r\,|u(r,\phi ,z)|^{2}=1,} .

u ( r , ϕ , z ) = C l p L G 1 w ( z ) ( r 2 w ( z ) ) | l | exp ( − r 2 w 2 ( z ) ) L p | l | ( 2 r 2 w 2 ( z ) ) × exp ( − i k r 2 2 R ( z ) ) exp ⁡ ( − i l ϕ ) exp ⁡ ( i ψ ( z ) ) , {\displaystyle {\begin{aligned}u(r,\phi ,z)={}&C_{lp}^{LG}{\frac {1}{w(z)}}\left({\frac {r{\sqrt {2}}}{w(z)}}\right)^{\!|l|}\exp \!\left(\!-{\frac {r^{2}}{w^{2}(z)}}\right)L_{p}^{|l|}\!\left({\frac {2r^{2}}{w^{2}(z)}}\right)\times {}\\&\exp \!\left(\!-ik{\frac {r^{2}}{2R(z)}}\right)\exp(-il\phi )\,\exp(i\psi (z)),\end{aligned}}}

The Gouy phase results in an increase in the apparent wavelength near the waist (z ≈ 0). Thus the phase velocity in that region formally exceeds the speed of light. That paradoxical behavior must be understood as a near-field phenomenon where the departure from the phase velocity of light (as would apply exactly to a plane wave) is very small except in the case of a beam with large numerical aperture, in which case the wavefronts' curvature (see previous section) changes substantially over the distance of a single wavelength. In all cases the wave equation is satisfied at every position.

The Gouy phase is a phase shift gradually acquired by a beam around the focal region. At position z the Gouy phase of a fundamental Gaussian beam is given by[1] ψ ( z ) = arctan ⁡ ( z z R ) . {\displaystyle \psi (z)=\arctan \left({\frac {z}{z_{\mathrm {R} }}}\right).}

For the common case of a circular beam profile, qx(z) = qy(z) = q(z) and x2 + y2 = r2, which yields[14] u ( r , z ) = 1 q ( z ) exp ⁡ ( − i k r 2 2 q ( z ) ) . {\displaystyle u(r,z)={\frac {1}{q(z)}}\exp \left(-ik{\frac {r^{2}}{2q(z)}}\right).}

When you are snapping a picture, you don't usually know the values of hhh and ggg, but you know the focal length for sure, and you likely know the distance between you and your subject. These two quantities are enough for you to calculate the magnification of your lens!

The set of hypergeometric-Gaussian modes is overcomplete and is not an orthogonal set of modes. In spite of its complicated field profile, HyGG modes have a very simple profile at the beam waist (z = 0): u ( ρ , ϕ , 0 ) ∝ ρ p + | m | e − ρ 2 + i m ϕ . {\displaystyle u(\rho ,\phi ,0)\propto \rho ^{{\mathsf {p}}+|m|}e^{-\rho ^{2}+im\phi }.}

Many laser beams have an elliptical cross-section. Also common are beams with waist positions which are different for the two transverse dimensions, called astigmatic beams. These beams can be dealt with using the above two evolution equations, but with distinct values of each parameter for x and y and distinct definitions of the z = 0 point. The Gouy phase is a single value calculated correctly by summing the contribution from each dimension, with a Gouy phase within the range ±π/4 contributed by each dimension.

With a beam centered on an aperture, the power P passing through a circle of radius r in the transverse plane at position z is[11] P ( r , z ) = P 0 [ 1 − e − 2 r 2 / w 2 ( z ) ] , {\displaystyle P(r,z)=P_{0}\left[1-e^{-2r^{2}/w^{2}(z)}\right],} where P 0 = 1 2 π I 0 w 0 2 {\displaystyle P_{0}={\frac {1}{2}}\pi I_{0}w_{0}^{2}} is the total power transmitted by the beam.

I ( r , z ) = | E ( r , z ) | 2 2 η = I 0 ( w 0 w ( z ) ) 2 exp ⁡ ( − 2 r 2 w ( z ) 2 ) , {\displaystyle I(r,z)={|E(r,z)|^{2} \over 2\eta }=I_{0}\left({\frac {w_{0}}{w(z)}}\right)^{2}\exp \left({\frac {-2r^{2}}{w(z)^{2}}}\right),}

In elliptic coordinates, one can write the higher-order modes using Ince polynomials. The even and odd Ince-Gaussian modes are given by[7]

Holds a Ph.D. degree in mathematics obtained at Jagiellonian University. An active researcher in the field of quantum information and a lecturer with 8+ years of teaching experience. Passionate about everything connected with maths in particular and science in general. Fond of coding, she has a keen eye for detail and perfectionistic leanings. A bookworm, an avid language learner, and a sluggish hiker. She is hopelessly addicted to coffee & chocolate – the darker and bitterer, the better. See full profile

Zoomlens magnification

However, when talking cameras, the magnification is usually a really small number. The number followed by a ×\times× is the zoom.

This last expression makes clear that the ray optics thin lens equation is recovered in the limit that | ( z R z 0 ) ( z R z 0 − f ) | ≪ 1 {\displaystyle \left|\left({\tfrac {z_{R}}{z_{0}}}\right)\left({\tfrac {z_{R}}{z_{0}-f}}\right)\right|\ll 1} . It can also be noted that if | z 0 + z R 2 z 0 − f | ≫ f {\displaystyle \left|z_{0}+{\frac {z_{R}^{2}}{z_{0}-f}}\right|\gg f} then the incoming beam is "well collimated" so that z 0 ′ ≈ f {\displaystyle z_{0}'\approx f} .

Imagine you are taking a picture of a huge kangaroo, let's say two meters tall and weighing 95 kg95\ \text{kg}95 kg, like the one that terrorized Brisbane a few years ago.

The Gaussian beam is a transverse electromagnetic (TEM) mode.[2] The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation.[1] Assuming polarization in the x direction and propagation in the +z direction, the electric field in phasor (complex) notation is given by:

In the case of violent kangaroos, it may be better to go for the second option: that's why camera manufacturers sell extension tubes, short rings to mount between the lens and the body, which end up increasing hhh by some precious millimeters.

w(z) and R(z) have the same definitions as above. As with the higher-order Hermite-Gaussian modes the magnitude of the Laguerre-Gaussian modes' Gouy phase shift is exaggerated by the factor N + 1: ψ ( z ) = ( N + 1 ) arctan ⁡ ( z z R ) , {\displaystyle \psi (z)=(N+1)\,\arctan \left({\frac {z}{z_{\mathrm {R} }}}\right),} where in this case the combined mode number N = |l| + 2p. As before, the transverse amplitude variations are contained in the last two factors on the upper line of the equation, which again includes the basic Gaussian drop off in r but now multiplied by a Laguerre polynomial. The effect of the rotational mode number l, in addition to affecting the Laguerre polynomial, is mainly contained in the phase factor exp(−ilφ), in which the beam profile is advanced (or retarded) by l complete 2π phases in one rotation around the beam (in φ). This is an example of an optical vortex of topological charge l, and can be associated with the orbital angular momentum of light in that mode.

In the paraxial case, as we have been considering, θ (in radians) is then approximately[1] θ = λ π n w 0 {\displaystyle \theta ={\frac {\lambda }{\pi nw_{0}}}}

Since the Gaussian function is infinite in extent, perfect Gaussian beams do not exist in nature, and the edges of any such beam would be cut off by any finite lens or mirror. However, the Gaussian is a useful approximation to a real-world beam for cases where lenses or mirrors in the beam are significantly larger than the spot size w(z) of the beam.

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Since the Gaussian beam model uses the paraxial approximation, it fails when wavefronts are tilted by more than about 30° from the axis of the beam.[9] From the above expression for divergence, this means the Gaussian beam model is only accurate for beams with waists larger than about 2λ/π.

where the constant η is the wave impedance of the medium in which the beam is propagating. For free space, η = η0 ≈ 377 Ω. I0 = |E0|2/2η is the intensity at the center of the beam at its waist.

As you can see, now the rays on the right side of the lens do not converge. We are dealing in terms of virtual images, which originate from the virtual continuations of the rays, creating a non reversed image of the object.

A lens is a device made of a material with a different refraction index to air (there can even be electromagnetic lenses that act on electric currents). This and its shape allows it to bend rays of light as they come into contact with it.

As derived by Saleh and Teich, the relationship between the ingoing and outgoing beams can be found by considering the phase that is added to each point ( x , y ) {\displaystyle (x,y)} of the gaussian beam as it travels through the lens.[15] An alternative approach due to Self is to consider the effect of a thin lens on the gaussian beam wavefronts.[16]

Because the divergence is inversely proportional to the spot size, for a given wavelength λ, a Gaussian beam that is focused to a small spot diverges rapidly as it propagates away from the focus. Conversely, to minimize the divergence of a laser beam in the far field (and increase its peak intensity at large distances) it must have a large cross-section (w0) at the waist (and thus a large diameter where it is launched, since w(z) is never less than w0). This relationship between beam width and divergence is a fundamental characteristic of diffraction, and of the Fourier transform which describes Fraunhofer diffraction. A beam with any specified amplitude profile also obeys this inverse relationship, but the fundamental Gaussian mode is a special case where the product of beam size at focus and far-field divergence is smaller than for any other case.

The perceived magnification of an object, thanks to the use of powerful telephoto lenses, comes from the reduced projection of the object onto relatively small sensors. If that projected image can change the size, let's say by a factor of two, we say that the lens has a 2× zoom.

Macrolens magnification

Maybe you expected the magnification to be a bigger number, something like 10×10\times10× or 20×20\times20×, like the values you see on binoculars or telescopes (we made an entire calculator for that, check out our telescope magnification calculator).

The final two factors account for the spatial variation over x (or y). The fourth factor is the Hermite polynomial of order J ("physicists' form", i.e. H1(x) = 2x), while the fifth accounts for the Gaussian amplitude fall-off exp(−x2/w(z)2), although this isn't obvious using the complex q in the exponent. Expansion of that exponential also produces a phase factor in x which accounts for the wavefront curvature (1/R(z)) at z along the beam.

The limit can be evaluated using L'Hôpital's rule: I ( 0 , z ) = P 0 π lim r → 0 [ − ( − 2 ) ( 2 r ) e − 2 r 2 / w 2 ( z ) ] w 2 ( z ) ( 2 r ) = 2 P 0 π w 2 ( z ) . {\displaystyle I(0,z)={\frac {P_{0}}{\pi }}\lim _{r\to 0}{\frac {\left[-(-2)(2r)e^{-2r^{2}/w^{2}(z)}\right]}{w^{2}(z)(2r)}}={2P_{0} \over \pi w^{2}(z)}.}

In photography, the magnification of a lens is the ratio between the height of the image projected onto the sensor or film of the camera and the height of the real image you are taking a picture of.

Since this solution relies on the paraxial approximation, it is not accurate for very strongly diverging beams. The above form is valid in most practical cases, where w0 ≫ λ/n.

In some applications it is desirable to use a converging lens to focus a laser beam to a very small spot. Mathematically, this implies minimization of the magnification M {\displaystyle M} . If the beam size is constrained by the size of available optics, this is typically best achieved by sending the largest possible collimated beam through a small focal length lens, i.e. by maximizing z R {\displaystyle z_{R}} and minimizing f {\displaystyle f} . In this situation, it is justifiable to make the approximation z R 2 / ( z 0 − f ) 2 ≫ 1 {\displaystyle z_{R}^{2}/(z_{0}-f)^{2}\gg 1} , implying that M ≈ f / z R {\displaystyle M\approx f/z_{R}} and yielding the result w 0 ′ ≈ f w 0 / z R {\displaystyle w_{0}'\approx fw_{0}/z_{R}} . This result is often presented in the form

Our lens magnification calculator will focus on the world of lenses in photography, finally explaining what magnification is, why it is different from zoom, and much more!

The magnification of a lens is an absolute value that depends on the focal length of the lens itself, while the zoom is a relative quantity that describes how much you can change the focal length of a lens by, thus changing its magnification.

Since, in most cases (unless you are using a microscope), the lens shrinks the object, the magnification value is less than 1.

The zoom describes how much the lens's focal length can change by (there are such things as zoom lenses). A typical 18/5518/5518/55 lens will have its zoom defined by:

1 q ( z ) = 1 R ( z ) − i λ n π w 2 ( z ) . {\displaystyle {1 \over q(z)}={1 \over R(z)}-i{\lambda \over n\pi w^{2}(z)}.}

The numerical aperture of a Gaussian beam is defined to be NA = n sin θ, where n is the index of refraction of the medium through which the beam propagates. This means that the Rayleigh range is related to the numerical aperture by z R = n w 0 N A . {\displaystyle z_{\mathrm {R} }={\frac {nw_{0}}{\mathrm {NA} }}.}

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Thanks to the properties of similar triangles, we can compute the magnification of a lens also using the distances between the object/image and the lens:

Their lenses are usually manufactured with a focal length of 25 cm25\ \text{cm}25 cm. If you use the lens to look at an object closer to it than that distance, you create a virtual image of the object.

The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength λ (in the dielectric medium, if not free space) and the following beam parameters, all of which are connected as detailed in the following sections.

where the rotational index m is an integer, and p ≥ − | m | {\displaystyle {\mathsf {p}}\geq -|m|} is real-valued, Γ(x) is the gamma function and 1F1(a, b; x) is a confluent hypergeometric function.

Lens magnificationformula

The magnification of a lens with focal length 55 mm at a distance of 100 m is m = 0.0005506. To calculate it, follow the steps:

The Spectralon™ is a surface of reference with a known spectrum [3] that is used for the calibration of spectral measurements of unknown surface reflectance. It ...

Maximummagnificationcameralens

The complex beam parameter simplifies the mathematical analysis of Gaussian beam propagation, and especially in the analysis of optical resonator cavities using ray transfer matrices.

which is found after assuming that the medium has index of refraction n ≈ 1 {\displaystyle n\approx 1} and substituting z R = π w 0 2 / λ {\displaystyle z_{R}=\pi w_{0}^{2}/\lambda } . The factors of 2 are introduced because of a common preference to represent beam size by the beam waist diameters 2 w 0 ′ {\displaystyle 2w_{0}'} and 2 w 0 {\displaystyle 2w_{0}} , rather than the waist radii w 0 ′ {\displaystyle w_{0}'} and w 0 {\displaystyle w_{0}} .

Then using this form, the earlier equation for the electric (or magnetic) field is greatly simplified. If we call u the relative field strength of an elliptical Gaussian beam (with the elliptical axes in the x and y directions) then it can be separated in x and y according to: u ( x , y , z ) = u x ( x , z ) u y ( y , z ) , {\displaystyle u(x,y,z)=u_{x}(x,z)\,u_{y}(y,z),}

Using this form along with the paraxial approximation, ∂2u/∂z2 can then be essentially neglected. Since solutions of the electromagnetic wave equation only hold for polarizations which are orthogonal to the direction of propagation (z), we have without loss of generality considered the polarization to be in the x direction so that we now solve a scalar equation for u(x, y, z).

1 R ( z ) = z z 2 + z R 2 , {\displaystyle {\frac {1}{R(z)}}={\frac {z}{z^{2}+z_{\mathrm {R} }^{2}}},}

Hermite Gaussian modes, with their rectangular symmetry, are especially suited for the modal analysis of radiation from lasers whose cavity design is asymmetric in a rectangular fashion. On the other hand, lasers and systems with circular symmetry can better be handled using the set of Laguerre-Gaussian modes introduced in the next section.

When a gaussian beam propagates through a thin lens, the outgoing beam is also a (different) gaussian beam, provided that the beam travels along the cylindrical symmetry axis of the lens, and that the lens is larger than the width of the beam. The focal length of the lens f {\displaystyle f} , the beam waist radius w 0 {\displaystyle w_{0}} , and beam waist position z 0 {\displaystyle z_{0}} of the incoming beam can be used to determine the beam waist radius w 0 ′ {\displaystyle w_{0}'} and position z 0 ′ {\displaystyle z_{0}'} of the outgoing beam.

The spot size and curvature of a Gaussian beam as a function of z along the beam can also be encoded in the complex beam parameter q(z)[12][13] given by: q ( z ) = z + i z R . {\displaystyle q(z)=z+iz_{\mathrm {R} }.}

Now consider that the sensor is at most a few centimeters wide, while you can take a picture of the Eiffel Tower, which is 330 m330\ \text{m}330 m tall. Even from afar and with a powerful telephoto lens, you'll always get a magnification that is much smaller than you expect when taking pictures with a camera.

Similarly, about 90% of the beam's power will flow through a circle of radius r = 1.07 × w(z), 95% through a circle of radius r = 1.224 × w(z), and 99% through a circle of radius r = 1.52 × w(z).[11]

The peak intensity at an axial distance z from the beam waist can be calculated as the limit of the enclosed power within a circle of radius r, divided by the area of the circle πr2 as the circle shrinks: I ( 0 , z ) = lim r → 0 P 0 [ 1 − e − 2 r 2 / w 2 ( z ) ] π r 2 . {\displaystyle I(0,z)=\lim _{r\to 0}{\frac {P_{0}\left[1-e^{-2r^{2}/w^{2}(z)}\right]}{\pi r^{2}}}.}

Laser beam quality is quantified by the beam parameter product (BPP). For a Gaussian beam, the BPP is the product of the beam's divergence and waist size w0. The BPP of a real beam is obtained by measuring the beam's minimum diameter and far-field divergence, and taking their product. The ratio of the BPP of the real beam to that of an ideal Gaussian beam at the same wavelength is known as M2 ("M squared"). The M2 for a Gaussian beam is one. All real laser beams have M2 values greater than one, although very high quality beams can have values very close to one.

The reciprocal of q(z) contains the wavefront curvature and relative on-axis intensity in its real and imaginary parts, respectively:[12]

so the radius of curvature R(z) is [1] R ( z ) = z [ 1 + ( z R z ) 2 ] . {\displaystyle R(z)=z\left[{1+{\left({\frac {z_{\mathrm {R} }}{z}}\right)}^{2}}\right].} Being the reciprocal of the curvature, the radius of curvature reverses sign and is infinite at the beam waist where the curvature goes through zero.

Beam Quality. Beam quality refers to the overall energy or wavelength of the beam and its penetrating power. A high-quality beam has a short wavelength, high ...

There is another important class of paraxial wave modes in cylindrical coordinates in which the complex amplitude is proportional to a confluent hypergeometric function.

🔎 The word "focus" comes from Latin for "fireplace". This is because the Romans believed that their ancestral gods were located in the fireplace, or hearth, and so would direct (or focus) their worship towards it.