Some people talk about a 'collimation distance' of a laser output -– what's that? Is that referring to distance where the beam waist located from the laser output?

The goal of spectral density estimation is to estimate the spectral density of a random signal from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involve parametric or non-parametric approaches, and may be based on time-domain or frequency-domain analysis. For example, a common parametric technique involves fitting the observations to an autoregressive model. A common non-parametric technique is the periodogram.

power densityw/m2

To achieve the smallest possible divergence for a laser diode beam, is it better to directly use a suitable lens (e.g. an aspheric doublet) in front of the laser first couple the beam into a single-mode fiber and then collimate what emerges from that?

That is a bit tricky, partly because you need sufficient data concerning chromatic aberrations, i.e., essentially the wavelength dependence of the focal length. The output beam can be perfectly collimated only for one of the involved wavelengths, but you can minimize chromatic effect by choosing a correspondingly optimized achromatic collimation lens.

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I am trying to estimate the divergence of a multi-color laser source that is fiber-coupled. How do you calculate results approximation (mainly collimated beam size and divergence) given fiber size, NA, wavelength, and focal length of collimator?

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No, that's not possible because of the inevitable effect of diffraction. A beam always has some confinement in the transverse direction, and that causes diffraction to make it expand.

This is and unwanted slight curvature of the emission pattern of a diode bar. Ideally, the emitters of such an array would be exactly in a straight line. If they deviate from that, it is more difficult to collimate the output such that one obtains a high beam quality.

That depends on the optical wavelength and on the propagation distance over which it needs to be collimated. For example, if you need a 1064-nm beam to be collimated over a length of 1 m, you want its Rayleigh length to be of the order of 1 m (or longer), which implies a Gaussian beam diameter of 1.2 mm (or larger).

A PSD can be either a one-sided function of only positive frequencies or a two-sided function of both positive and negative frequencies but with only half the amplitude. Noise PSDs are generally one-sided in engineering and two-sided in physics.[5]

Power densityformula

Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain S x y ( f ) = lim T → ∞ 1 T [ x ^ T ∗ ( f ) y ^ T ( f ) ] S y x ( f ) = lim T → ∞ 1 T [ y ^ T ∗ ( f ) x ^ T ( f ) ] {\displaystyle {\begin{aligned}S_{xy}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {x}}_{T}^{*}(f){\hat {y}}_{T}(f)\right]&S_{yx}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {y}}_{T}^{*}(f){\hat {x}}_{T}(f)\right]\end{aligned}}} where, again, the contributions of S x x ( f ) {\displaystyle S_{xx}(f)} and S y y ( f ) {\displaystyle S_{yy}(f)} are already understood. Note that S x y ∗ ( f ) = S y x ( f ) {\displaystyle S_{xy}^{*}(f)=S_{yx}(f)} , so the full contribution to the cross power is, generally, from twice the real part of either individual CPSD. Just as before, from here we recast these products as the Fourier transform of a time convolution, which when divided by the period and taken to the limit T → ∞ {\displaystyle T\to \infty } becomes the Fourier transform of a cross-correlation function.[16] S x y ( f ) = ∫ − ∞ ∞ [ lim T → ∞ 1 T ∫ − ∞ ∞ x T ∗ ( t − τ ) y T ( t ) d t ] e − i 2 π f τ d τ = ∫ − ∞ ∞ R x y ( τ ) e − i 2 π f τ d τ S y x ( f ) = ∫ − ∞ ∞ [ lim T → ∞ 1 T ∫ − ∞ ∞ y T ∗ ( t − τ ) x T ( t ) d t ] e − i 2 π f τ d τ = ∫ − ∞ ∞ R y x ( τ ) e − i 2 π f τ d τ , {\displaystyle {\begin{aligned}S_{xy}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )y_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{xy}(\tau )e^{-i2\pi f\tau }d\tau \\S_{yx}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }y_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{yx}(\tau )e^{-i2\pi f\tau }d\tau ,\end{aligned}}} where R x y ( τ ) {\displaystyle R_{xy}(\tau )} is the cross-correlation of x ( t ) {\displaystyle x(t)} with y ( t ) {\displaystyle y(t)} and R y x ( τ ) {\displaystyle R_{yx}(\tau )} is the cross-correlation of y ( t ) {\displaystyle y(t)} with x ( t ) {\displaystyle x(t)} . In light of this, the PSD is seen to be a special case of the CSD for x ( t ) = y ( t ) {\displaystyle x(t)=y(t)} . If x ( t ) {\displaystyle x(t)} and y ( t ) {\displaystyle y(t)} are real signals (e.g. voltage or current), their Fourier transforms x ^ ( f ) {\displaystyle {\hat {x}}(f)} and y ^ ( f ) {\displaystyle {\hat {y}}(f)} are usually restricted to positive frequencies by convention. Therefore, in typical signal processing, the full CPSD is just one of the CPSDs scaled by a factor of two. CPSD Full = 2 S x y ( f ) = 2 S y x ( f ) {\displaystyle \operatorname {CPSD} _{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)}

Collimated beams are very useful in laboratory setups because the beam radius stays approximately constant, so that the distances between optical components may be easily varied without applying extra optics, and excessive beam radii are avoided. Most solid-state lasers naturally emit collimated beams; a flat output coupler enforces flat wavefronts (i.e., a beam waist) at the output, and the beam waist is usually large enough to avoid excessive divergence. Edge-emitting laser diodes, however, emit strongly diverging beams, and are therefore often equipped with collimation optics – at least with a fast axis collimator, largely reducing the strong divergence in the “fast” direction. For fibers, a simple optical lens may often suffice for collimation, although the beam quality can be better preserved with an aspheric lens, particularly for single-mode fibers with a large numerical aperture.

Why isn't it possible to have a fully collimated beam (meaning no divergence at all)? Would we have to be infinitely precise with the placement of the lens or is there something more to it?

Power densityRF

For beams with non-ideal beam quality, the Rayleigh length is effectively reduced by the so-called M2 factor, so that the beam waist radius needs to be larger for a beam to be collimated.

In analyzing the frequency content of the signal x ( t ) {\displaystyle x(t)} , one might like to compute the ordinary Fourier transform x ^ ( f ) {\displaystyle {\hat {x}}(f)} ; however, for many signals of interest the Fourier transform does not formally exist.[nb 1] Regardless, Parseval's theorem tells us that we can re-write the average power as follows. P = lim T → ∞ 1 T ∫ − ∞ ∞ | x ^ T ( f ) | 2 d f {\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|{\hat {x}}_{T}(f)|^{2}\,df}

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Power densitylaser

From here we see, again assuming the ergodicity of x ( t ) {\displaystyle x(t)} , that the power spectral density can be found as the Fourier transform of the autocorrelation function (Wiener–Khinchin theorem).[11]

The Model 02-M010 is a three-element, air-spaced anastigmat designed specifically for collimating the output of large diameter silica fibers used in high power medical and industrial applications. It is equally suitable for collimating the output of Large Mode Area (LMA) or Photonic Crystal (PC) fibers with smaller numerical apertures. The mechanical assembly allows a precise translation of the lens (without rotation) relative to the fiber face.

However this article concentrates on situations in which the time series is known (at least in a statistical sense) or directly measured (such as by a microphone sampled by a computer). The power spectrum is important in statistical signal processing and in the statistical study of stochastic processes, as well as in many other branches of physics and engineering. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of spatial frequency.[1]

Shanghai Optics provide many different types of standard collimating lenses, including aspheric and achromatic lenses for many different light sources such as highly divergent laser diodes. Our standard collimating lenses can convert divergent laser beams to well-collimated laser beams that enter beam expanders for interferometry, laser material processing and laser scanning applications.

This definition generalizes in a straightforward manner to a discrete signal with a countably infinite number of values x n {\displaystyle x_{n}} such as a signal sampled at discrete times t n = t 0 + ( n Δ t ) {\displaystyle t_{n}=t_{0}+(n\,\Delta t)} : S ¯ x x ( f ) = lim N → ∞ ( Δ t ) 2 | ∑ n = − N N x n e − i 2 π f n Δ t | 2 ⏟ | x ^ d ( f ) | 2 , {\displaystyle {\bar {S}}_{xx}(f)=\lim _{N\to \infty }(\Delta t)^{2}\underbrace {\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}} _{\left|{\hat {x}}_{d}(f)\right|^{2}},} where x ^ d ( f ) {\displaystyle {\hat {x}}_{d}(f)} is the discrete-time Fourier transform of x n . {\displaystyle x_{n}.}   The sampling interval Δ t {\displaystyle \Delta t} is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit Δ t → 0. {\displaystyle \Delta t\to 0.}   But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (also see normalized frequency)

What restrictions apply to how long the light from an extended, non-laser source like a tungsten lamp can stay collimated?

Collimating astigmatic beams usually requires a separate treatment in both transverse directions, e.g. with two different cylindrical lenses or curved laser mirrors. Special challenges arise for general astigmatic beams, where a simple separation of <$x$> and <$y$> direction is not possible, but such cases are rare in practice.

As a physical example of how one might measure the energy spectral density of a signal, suppose V ( t ) {\displaystyle V(t)} represents the potential (in volts) of an electrical pulse propagating along a transmission line of impedance Z {\displaystyle Z} , and suppose the line is terminated with a matched resistor (so that all of the pulse energy is delivered to the resistor and none is reflected back). By Ohm's law, the power delivered to the resistor at time t {\displaystyle t} is equal to V ( t ) 2 / Z {\displaystyle V(t)^{2}/Z} , so the total energy is found by integrating V ( t ) 2 / Z {\displaystyle V(t)^{2}/Z} with respect to time over the duration of the pulse. To find the value of the energy spectral density S ¯ x x ( f ) {\displaystyle {\bar {S}}_{xx}(f)} at frequency f {\displaystyle f} , one could insert between the transmission line and the resistor a bandpass filter which passes only a narrow range of frequencies ( Δ f {\displaystyle \Delta f} , say) near the frequency of interest and then measure the total energy E ( f ) {\displaystyle E(f)} dissipated across the resistor. The value of the energy spectral density at f {\displaystyle f} is then estimated to be E ( f ) / Δ f {\displaystyle E(f)/\Delta f} . In this example, since the power V ( t ) 2 / Z {\displaystyle V(t)^{2}/Z} has units of V2 Ω−1, the energy E ( f ) {\displaystyle E(f)} has units of V2 s Ω−1 = J, and hence the estimate E ( f ) / Δ f {\displaystyle E(f)/\Delta f} of the energy spectral density has units of J Hz−1, as required. In many situations, it is common to forget the step of dividing by Z {\displaystyle Z} so that the energy spectral density instead has units of V2 Hz−1.

Power densityunit

The FiberOut fiber collimator transforms the divergent beam emitted at the end of an optical fiber into a collimated one. It can be equipped with a variety of lenses, matching different fiber mode-field diameters and output beam sizes. The rugged, inexpensive collimator can be used for both FC/PC and FC/APC-type connectors. It can be easily mounted on post or into optical mounts (25 mm diameter).

When a signal is defined in terms only of a voltage, for instance, there is no unique power associated with the stated amplitude. In this case "power" is simply reckoned in terms of the square of the signal, as this would always be proportional to the actual power delivered by that signal into a given impedance. So one might use units of V2 Hz−1 for the PSD. Energy spectral density (ESD) would have units of V2 s Hz−1, since energy has units of power multiplied by time (e.g., watt-hour).[3]

Just as with the energy spectral density, the definition of the power spectral density can be generalized to discrete time variables x n {\displaystyle x_{n}} . As before, we can consider a window of − N ≤ n ≤ N {\displaystyle -N\leq n\leq N} with the signal sampled at discrete times t n = t 0 + ( n Δ t ) {\displaystyle t_{n}=t_{0}+(n\,\Delta t)} for a total measurement period T = ( 2 N + 1 ) Δ t {\displaystyle T=(2N+1)\,\Delta t} . S x x ( f ) = lim N → ∞ ( Δ t ) 2 T | ∑ n = − N N x n e − i 2 π f n Δ t | 2 {\displaystyle S_{xx}(f)=\lim _{N\to \infty }{\frac {(\Delta t)^{2}}{T}}\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}} Note that a single estimate of the PSD can be obtained through a finite number of samplings. As before, the actual PSD is achieved when N {\displaystyle N} (and thus T {\displaystyle T} ) approaches infinity and the expected value is formally applied. In a real-world application, one would typically average a finite-measurement PSD over many trials to obtain a more accurate estimate of the theoretical PSD of the physical process underlying the individual measurements. This computed PSD is sometimes called a periodogram. This periodogram converges to the true PSD as the number of estimates as well as the averaging time interval T {\displaystyle T} approach infinity.[13]

Avantier offers a wide range of standard collimating lenses, which includes aspheric and achromatic lenses suitable for various light sources such as laser diodes with high divergence. These standard collimating lenses have the ability to convert divergent laser beams into well-collimated laser beams. These collimated beams can then be utilized for laser material processing, laser scanning applications, and interferometry by entering beam expanders.

Given two signals x ( t ) {\displaystyle x(t)} and y ( t ) {\displaystyle y(t)} , each of which possess power spectral densities S x x ( f ) {\displaystyle S_{xx}(f)} and S y y ( f ) {\displaystyle S_{yy}(f)} , it is possible to define a cross power spectral density (CPSD) or cross spectral density (CSD). To begin, let us consider the average power of such a combined signal. P = lim T → ∞ 1 T ∫ − ∞ ∞ [ x T ( t ) + y T ( t ) ] ∗ [ x T ( t ) + y T ( t ) ] d t = lim T → ∞ 1 T ∫ − ∞ ∞ | x T ( t ) | 2 + x T ∗ ( t ) y T ( t ) + y T ∗ ( t ) x T ( t ) + | y T ( t ) | 2 d t {\displaystyle {\begin{aligned}P&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left[x_{T}(t)+y_{T}(t)\right]^{*}\left[x_{T}(t)+y_{T}(t)\right]dt\\&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|x_{T}(t)|^{2}+x_{T}^{*}(t)y_{T}(t)+y_{T}^{*}(t)x_{T}(t)+|y_{T}(t)|^{2}dt\\\end{aligned}}}

For discrete signals xn and yn, the relationship between the cross-spectral density and the cross-covariance is S x y ( f ) = ∑ n = − ∞ ∞ R x y ( τ n ) e − i 2 π f τ n Δ τ {\displaystyle S_{xy}(f)=\sum _{n=-\infty }^{\infty }R_{xy}(\tau _{n})e^{-i2\pi f\tau _{n}}\,\Delta \tau }

Power densityof battery

Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. This includes familiar entities such as visible light (perceived as color), musical notes (perceived as pitch), radio/TV (specified by their frequency, or sometimes wavelength) and even the regular rotation of the earth. When these signals are viewed in the form of a frequency spectrum, certain aspects of the received signals or the underlying processes producing them are revealed. In some cases the frequency spectrum may include a distinct peak corresponding to a sine wave component. And additionally there may be peaks corresponding to harmonics of a fundamental peak, indicating a periodic signal which is not simply sinusoidal. Or a continuous spectrum may show narrow frequency intervals which are strongly enhanced corresponding to resonances, or frequency intervals containing almost zero power as would be produced by a notch filter.

We also provide custom collimating lenses for projecting a source at infinity for infinite conjugate testing of optical systems. The collimating lenses can consist of several optical elements. The selection of optical materials and optical configuration depends on the entrance pupil diameter, wavelength, focal length, and field of view of the optical system under test.

In signal processing, the power spectrum S x x ( f ) {\displaystyle S_{xx}(f)} of a continuous time signal x ( t ) {\displaystyle x(t)} describes the distribution of power into frequency components f {\displaystyle f} composing that signal.[1] According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of any sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum.

Primordial fluctuations, density variations in the early universe, are quantified by a power spectrum which gives the power of the variations as a function of spatial scale.

When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (PSD, or simply power spectrum), which applies to signals existing over all time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The PSD then refers to the spectral energy distribution that would be found per unit time, since the total energy of such a signal over all time would generally be infinite. Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process), identical to what would be obtained by integrating x 2 ( t ) {\displaystyle x^{2}(t)} over the time domain, as dictated by Parseval's theorem.[1]

Now, if we divide the time convolution above by the period T {\displaystyle T} and take the limit as T → ∞ {\displaystyle T\rightarrow \infty } , it becomes the autocorrelation function of the non-windowed signal x ( t ) {\displaystyle x(t)} , which is denoted as R x x ( τ ) {\displaystyle R_{xx}(\tau )} , provided that x ( t ) {\displaystyle x(t)} is ergodic, which is true in most, but not all, practical cases.[nb 2] lim T → ∞ 1 T | x ^ T ( f ) | 2 = ∫ − ∞ ∞ [ lim T → ∞ 1 T ∫ − ∞ ∞ x T ∗ ( t − τ ) x T ( t ) d t ] e − i 2 π f τ   d τ = ∫ − ∞ ∞ R x x ( τ ) e − i 2 π f τ d τ {\displaystyle \lim _{T\to \infty }{\frac {1}{T}}\left|{\hat {x}}_{T}(f)\right|^{2}=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }d\tau }

The spectral density is usually estimated using Fourier transform methods (such as the Welch method), but other techniques such as the maximum entropy method can also be used.

Power densityvs energydensity

The unique design of the Model 02-M010 prevents retroreflections near the fiber face or within the core material. All elements are fused silica (the exception being the 1800–2000 nm collimator optics that are Infrasil) with either V-type or broadband coatings, depending on the operating wavelength range. When used for imaging purposes, the three-element design ensures the output mode from the fiber is preserved, without distortion, even at high throughput powers.

Energy spectral density describes how the energy of a signal or a time series is distributed with frequency. Here, the term energy is used in the generalized sense of signal processing;[6] that is, the energy E {\displaystyle E} of a signal x ( t ) {\displaystyle x(t)} is: E ≜ ∫ − ∞ ∞ | x ( t ) | 2   d t . {\displaystyle E\triangleq \int _{-\infty }^{\infty }\left|x(t)\right|^{2}\ dt.}

You can estimate that with the beam parameter product, using the source size (like a beam waist) and the divergence of outgoing light (light a beam divergence). You can reduce that product e.g. with one or more optical apertures, but at the expense of losing part of the optical power.

Power densitysymbol

In principle, one can use lenses with very different focal lengths to collimate a diverging beam. The longer the focal length, the larger will be the resulting diameter of the collimated beam. Assuming a tight focus to start with (and subsequent beam expansion over a distance far beyond the Rayleigh length, the required distance between focus and collimation lens will equal the focal length. From that, one can obtain the collimated beam radius as the product of beam divergence half-angle (or precisely speaking its tangent) and the distance. And for a Gaussian beam, the beam divergence half-angle is <$\lambda / (\pi w_0)$>. In total, we obtain (within the paraxial approximation):

Collimation of single mode fibres can be made simple with the use of a PowerPhotonic fiber collimating micro lens array. We design and manufacture standard and custom in 1D and 2D arrays. All products are made in high grade fused silica and capable of both high efficiency and high power handling and our unique process minimises channel cross talk due to extremely low scatter. Lenses can spheric, aspheric or freeform due to our unique manufacturing process.

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The length over which the diameter of the formed beam stays roughly constant can be estimated as the effective Rayleigh length, which can be calculated using the beam parameter product.

Mathematically, it is not necessary to assign physical dimensions to the signal or to the independent variable. In the following discussion the meaning of x(t) will remain unspecified, but the independent variable will be assumed to be that of time.

A collimated beam of light is a beam (typically a laser beam) propagating in a homogeneous medium (e.g. in air) with a low beam divergence, so that the beam radius does not undergo significant changes within moderate propagation distances. In the simple (and frequently encountered) case of Gaussian beams, this means that the Rayleigh length must be long compared with the envisaged propagation distance. For example, a 1064-nm beam with a 1-mm beam radius at its beam waist has a Rayleigh length of ≈ 3 m in air, so that it can be considered as being collimated within a normal laboratory setup. Note that the Rayleigh length scales with the square of the beam waist radius, so that large beam radii are essential for long propagation distances.

When describing a collimated beam with light rays (geometrical optics), it consists of essentially parallel rays only. However, the ray picture cannot account for the phenomenon of beam divergence and is therefore of limited value.

The energy spectral density is most suitable for transients—that is, pulse-like signals—having a finite total energy. Finite or not, Parseval's theorem (or Plancherel's theorem) gives us an alternate expression for the energy of the signal:[7] ∫ − ∞ ∞ | x ( t ) | 2 d t = ∫ − ∞ ∞ | x ^ ( f ) | 2 d f , {\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }\left|{\hat {x}}(f)\right|^{2}\,df,} where: x ^ ( f ) ≜ ∫ − ∞ ∞ e − i 2 π f t x ( t )   d t {\displaystyle {\hat {x}}(f)\triangleq \int _{-\infty }^{\infty }e^{-i2\pi ft}x(t)\ dt} is the value of the Fourier transform of x ( t ) {\displaystyle x(t)} at frequency f {\displaystyle f} (in Hz). The theorem also holds true in the discrete-time cases. Since the integral on the left-hand side is the energy of the signal, the value of | x ^ ( f ) | 2 d f {\displaystyle \left|{\hat {x}}(f)\right|^{2}df} can be interpreted as a density function multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency f {\displaystyle f} in the frequency interval f + d f {\displaystyle f+df} .

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The power of the signal in a given frequency band [ f 1 , f 2 ] {\displaystyle [f_{1},f_{2}]} , where 0 < f 1 < f 2 {\displaystyle 0

CSRayzer provides different kinds of sing mode or polarization-maintaining fiber pigtail collimators, large beam collimators, and fixed focus collimators.

From here, due to the convolution theorem, we can also view | x ^ T ( f ) | 2 {\displaystyle |{\hat {x}}_{T}(f)|^{2}} as the Fourier transform of the time convolution of x T ∗ ( − t ) {\displaystyle x_{T}^{*}(-t)} and x T ( t ) {\displaystyle x_{T}(t)} , where * represents the complex conjugate. Taking into account that F { x T ∗ ( − t ) } = ∫ − ∞ ∞ x T ∗ ( − t ) e − i 2 π f t d t = ∫ − ∞ ∞ x T ∗ ( t ) e i 2 π f t d t = ∫ − ∞ ∞ x T ∗ ( t ) [ e − i 2 π f t ] ∗ d t = [ ∫ − ∞ ∞ x T ( t ) e − i 2 π f t d t ] ∗ = [ F { x T ( t ) } ] ∗ = [ x ^ T ( f ) ] ∗ {\displaystyle {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)[e^{-i2\pi ft}]^{*}dt\\&=\left[\int _{-\infty }^{\infty }x_{T}(t)e^{-i2\pi ft}dt\right]^{*}\\&=\left[{\mathcal {F}}\left\{x_{T}(t)\right\}\right]^{*}\\&=\left[{\hat {x}}_{T}(f)\right]^{*}\end{aligned}}} and making, u ( t ) = x T ∗ ( − t ) {\displaystyle u(t)=x_{T}^{*}(-t)} , we have: | x ^ T ( f ) | 2 = [ x ^ T ( f ) ] ∗ ⋅ x ^ T ( f ) = F { x T ∗ ( − t ) } ⋅ F { x T ( t ) } = F { u ( t ) } ⋅ F { x T ( t ) } = F { u ( t ) ∗ x T ( t ) } = ∫ − ∞ ∞ [ ∫ − ∞ ∞ u ( τ − t ) x T ( t ) d t ] e − i 2 π f τ d τ = ∫ − ∞ ∞ [ ∫ − ∞ ∞ x T ∗ ( t − τ ) x T ( t ) d t ] e − i 2 π f τ   d τ , {\displaystyle {\begin{aligned}\left|{\hat {x}}_{T}(f)\right|^{2}&=[{\hat {x}}_{T}(f)]^{*}\cdot {\hat {x}}_{T}(f)\\&={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\mathbin {\mathbf {*} } x_{T}(t)\right\}\\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }u(\tau -t)x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau \\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau ,\end{aligned}}} where the convolution theorem has been used when passing from the 3rd to the 4th line.

Probably not – I would think it is the distance over which a laser beam stays collimated, i.e., maintains an approximately constant beam radius. That would be something like the (effective) Rayleigh length. But I think the term is not generally defined.

We also offer a complete range of aspheric collimators with excellent performance, small and light design, and with fewer components in the optical system. Manufactured using glass replication technology, the lenses are a cost effective solution for a wide range of application and are available in a wide range of specification.

If two signals both possess power spectral densities, then the cross-spectral density can similarly be calculated; as the PSD is related to the autocorrelation, so is the cross-spectral density related to the cross-correlation.

The function S ¯ x x ( f ) {\displaystyle {\bar {S}}_{xx}(f)} and the autocorrelation of x ( t ) {\displaystyle x(t)} form a Fourier transform pair, a result also known as the Wiener–Khinchin theorem (see also Periodogram).

In fiber optics, one often uses fiber collimators. These are available both for bare optical fibers and for connectorized fibers, i.e., for mating with fiber connectors.

The encyclopedia covers this topic well: with a long overview article on laser material processing and specialized articles on

The spectrum of a physical process x ( t ) {\displaystyle x(t)} often contains essential information about the nature of x {\displaystyle x} . For instance, the pitch and timbre of a musical instrument are immediately determined from a spectral analysis. The color of a light source is determined by the spectrum of the electromagnetic wave's electric field E ( t ) {\displaystyle E(t)} as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the Fourier transform, and generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a dispersive prism is used to obtain a spectrum of light in a spectrograph, or when a sound is perceived through its effect on the auditory receptors of the inner ear, each of which is sensitive to a particular frequency.

The average power P {\displaystyle P} of a signal x ( t ) {\displaystyle x(t)} over all time is therefore given by the following time average, where the period T {\displaystyle T} is centered about some arbitrary time t = t 0 {\displaystyle t=t_{0}} : P = lim T → ∞ 1 T ∫ t 0 − T / 2 t 0 + T / 2 | x ( t ) | 2 d t {\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{t_{0}-T/2}^{t_{0}+T/2}\left|x(t)\right|^{2}\,dt}

That depends basically on whether the near-field profile from the laser diode or the fiber mode is closer to a Gaussian. That is hard to predict without specific information on the two.

A divergent beam can be collimated with a beam collimator device, which in simple case is essentially a lens or a curved mirror, where the focal length or curvature radius is chosen such that the originally curved wavefronts become flat. (Of course, the beam radius at the position of the lens or mirror should be large enough to obtain a low divergence.) Any residual divergence can be fine adjusted via the position of the lens or mirror along the beam direction. The collimation can be checked, for example, by measuring the evolution of beam radius over some distance in free space, via a Shack–Hartmann wavefront sensor, or with certain kinds of interferometers.

Edmund Optics offers a wide range of laser accessories, including different kinds of beam collimators and expanders. In particular, we have fiber-coupled collimators which are suitable for FC/PC, FC/APC and SMA connectors.

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The above definition of energy spectral density is suitable for transients (pulse-like signals) whose energy is concentrated around one time window; then the Fourier transforms of the signals generally exist. For continuous signals over all time, one must rather define the power spectral density (PSD) which exists for stationary processes; this describes how the power of a signal or time series is distributed over frequency, as in the simple example given previously. Here, power can be the actual physical power, or more often, for convenience with abstract signals, is simply identified with the squared value of the signal. For example, statisticians study the variance of a function over time x ( t ) {\displaystyle x(t)} (or over another independent variable), and using an analogy with electrical signals (among other physical processes), it is customary to refer to it as the power spectrum even when there is no physical power involved. If one were to create a physical voltage source which followed x ( t ) {\displaystyle x(t)} and applied it to the terminals of a one ohm resistor, then indeed the instantaneous power dissipated in that resistor would be given by x 2 ( t ) {\displaystyle x^{2}(t)} watts.

The spectrum analyzer measures the magnitude of the short-time Fourier transform (STFT) of an input signal. If the signal being analyzed can be considered a stationary process, the STFT is a good smoothed estimate of its power spectral density.

In physics, the signal might be a wave, such as an electromagnetic wave, an acoustic wave, or the vibration of a mechanism. The power spectral density (PSD) of the signal describes the power present in the signal as a function of frequency, per unit frequency. Power spectral density is commonly expressed in SI units of watts per hertz (abbreviated as W/Hz).[2]

The concept and use of the power spectrum of a signal is fundamental in electrical engineering, especially in electronic communication systems, including radio communications, radars, and related systems, plus passive remote sensing technology. Electronic instruments called spectrum analyzers are used to observe and measure the power spectra of signals.

In the general case, the units of PSD will be the ratio of units of variance per unit of frequency; so, for example, a series of displacement values (in meters) over time (in seconds) will have PSD in units of meters squared per hertz, m2/Hz. In the analysis of random vibrations, units of g2 Hz−1 are frequently used for the PSD of acceleration, where g denotes the g-force.[4]

However, for the sake of dealing with the math that follows, it is more convenient to deal with time limits in the signal itself rather than time limits in the bounds of the integral. As such, we have an alternative representation of the average power, where x T ( t ) = x ( t ) w T ( t ) {\displaystyle x_{T}(t)=x(t)w_{T}(t)} and w T ( t ) {\displaystyle w_{T}(t)} is unity within the arbitrary period and zero elsewhere. P = lim T → ∞ 1 T ∫ − ∞ ∞ | x T ( t ) | 2 d t . {\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left|x_{T}(t)\right|^{2}\,dt.} Clearly, in cases where the above expression for P is non-zero, the integral must grow without bound as T grows without bound. That is the reason why we cannot use the energy of the signal, which is that diverging integral, in such cases.