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Dear reader,

thank you for your patience while we update our website. We should be back shortly with some more great posts.

In preparation right now: a mini-spectrometer buyer's guide, with terms definition and how to choose the best parameters for your application.

Remember, all of our articles are independent from any manufacturers are we do not receive any form of compensation to write them. To guarantee our future independence and increase the frequency of publication we will shortly introduce a membership system. Any article currently available will still be public.

 

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Wavefront sensors: the ultimate optical diagnostic tool?

Characterizing a laser beam is a growing concern in the industry. A great number of instruments are available on the market, each with their specialities. When it comes to analysing the spatial behaviour of a laser beam, the most common solution is the beam profiler. However another solution starts to be affordable and user-friendly enough to be seriously considered by anyone who wishes to take laser beam characterisation to a whole new dimension. It extends its capabilities far beyond the laser applications, and is of very high interest for astronomy, microscopy, optics caracterisation and more.

Wavefront deformation and poor focus or poor image.

Since light can be modelled as an electromagnetic wave, one can define a surface of constant phase, called wavefront. This is much like the crest of a wave in the water. The little animation below can help understand this concept quite easily.

Optical wavefront curved by a lens

Optical wavefront curved by a lens

Because of imperfections of the media in which the light is going through, the wavefronts are normally deformed and are not a perfectly curved (or "flat") surface anymore. This in turn affects the propagation of the light rays and makes it impossible for them to focus in a single point (one can demonstrate that in any point the ray of light is perpendicular to the wavefront). The result is lower intensity at focus, blur and in general, aberrations. The picture below can give an example, it compares a perfect situation with a case where the wavefront is heavily deformed.

Deformation of the optical wavefront through the eye.

Deformation of the optical wavefront through the eye.

There is a number of very common reasons for this to happen. Heating up of the optical system, atmospheric turbulences, inhomogeneities of the media in which the light propagates, gradient of density in the air (mirage effect), etc...

Wavefront deformation and destructive interferences.

In addition of what we just mentioned above, aberrations in a beam of light will greatly reduce the intensity at focus due to destructive interferences. Once again, the images below will help understand why. First, keep in mind light is an electromagnetic wave. As it goes along, the electrical field \vec{E} varies from +E to -E

Light as an electromagnetic wave

Light as an electromagnetic wave

In the ideal case, when there is no wavefront deformation, the light going through a media or an optical system will arrive at focus at the same time whatever the path it goes through. In this situation (picture below), the electrical fields add up at focus, and the intensity of the light is thus greatly increased.

Ideal lens

Ideal lens

In reality, because of the wavefront is deformed, some of those electrical fields will arrive at the focus point at different "times" (phase). The electrical fields do not have the same values and their addition will be counter-productive, leading to reduced intensity in places.

Lens with aberrations

Lens with aberrations

This leads to the intensity patterns at focus you can see below, and to a reduced Strehl ratio. It creates obvious problem when the aim is to get the highest possible intensity at focus or the best quality image.

Point Spread Functions

Point Spread Functions

Practical consequences of poor wavefront quality.

As a direct result of what has been said above, a poor wavefront will:

  • Reduce intensity at focus. In case of a welding/cutting laser this mean decreased efficiency. Any laser application that focuses the light down would be impacted by poor wavefront, such as welding, cutting, plasma generation, surgery, fluorescence or Raman excitation, etc... It is to be noted that a laser beam can potentially heat up the optics inside itself and create thermal lensing, which in turn will deform the wavefront.
  • Create hotspots. This is particularly crucial in Chirp Pulse Amplification lasers. All the optical components of an amplification chain can induce phase aberrations responsible for spatial intensity modulations. These distortions can generate energy hot-spots and irreversible optical damages of the components, some of which are prohibitively expensive.
  • Lower resolution. Aberrations are the plague of imaging systems, because they create blurry images and effectively reduce the imaging system resolution. This can be caused by the imaging system itself (poor quality lenses, for instance, or mis-alignment), of by the environment (the turbulence in the atmosphere create dynamic aberrations which lower the capabilities of non-adaptive optic telescopes)

What technology is currently available to measure wavefront aberrations?

Shack-Hartman

This is the most wide-spread type of wavefront sensor. A micro-lens array focuses the incident wavefront into a number of small spots on a CCD. Aberrations in the beam will make the spots move away from the place they would occupy in front of the centre of each micro-lens if the wavefront was perfectly flat. The deviation of each spot is directly proportional to the gradient of the wavefront, which can then be reconstructed.

The Shack-Hartman is the most versatile wavefront sensor available at the moment. It can measure wavefront aberrations 1,500 bigger than the wavelength at a precision of one-hundredth of a wavelength. It is the easier to align, the most documented and is already designed-in a number of turnkey solutions for industrial need.

Its main weakness is its poor spatial resolution. With a number of measurement points equal to the number of micro-lenses, it is typically of the order of 1000 to 5000 data points per wavefront.

This instrument is best suited for general measurements, when you need both a good dynamic range and good precision (resolution of the phase), but do not need a high spatial resolution (or transverse precision, helping with high spatial frequency aberrations).

Realistically, this includes most of the cases, since a wavefront reconstructed from 1000 points is able to include aberrations of well past beyond the 10th order.

Hartman

Same as above with a holed mask in place of the micro-lenses array. This could be considered as obsolete technology only interesting when you cannot use lenses (for X-ray wavefront sensing for instance).

Curvature sensors

They measure the intensity profile of the beam in two different planes along the optical axis. By comparing the intensities, the software will compute the axial derivative of the intensity, and then calculate the second derivative of the wavefront using Poisson's equation. This technique gives a very good spatial resolution because one pixel gives one phase data point. One of the main drawbacks of this technique is its limitation in terms of dynamic range, typically limited to a few microns (typically 3 µm). Just as critically, since it is working on the second derivative of the wavefront, it is by nature unable to measure tip-tilt aberrations. Finally, the light beam must be collimated and of reasonable intensity.

This has some uses to measure wavefront with high spatial frequencies of aberration, of low amplitude.

Multi-lateral shearing interferometer

A 2D diffraction grating replicates the incident beam into four beams which are propagated along slightly different directions. The interaction between the beams produces an interference pattern which is imaged on a CCD.

When aberrations are present on the beam, the interference pattern is distorted. The pattern deformations are directly connected to the phase gradients. A spectral analysis using Fourier transforms allows the phase gradient extraction in 2 orthogonal directions. The phase map is finally obtained by integration of these gradients.

Typically you can tune the position of the diffraction grating to change the behaviour of the sensor: either you get high precision measurement of the phase or you get high spatial resolution (to see high spatial frequencies). Also the overall dynamic range of the instrument is limited, so you can tune it either for high precision measurement of the phase or for measurement of a highly aberrated wavefront, but you cannot measure high level of aberration with a good precision. This would probably mean that you need to pay extra care to the alignment when making a precision measurement, otherwise the tip-tilt will bring the wavefront out of the dynamic range.

The wavelength range is the one of the CCD used (generally 350-1100nm), it is insensitive to vibrations. Finally, because the beam is split into 4, you need a reasonable intensity.

This type of instrument is suitable when the measurement you want to make do not tick all the boxes of high aberration amplitude, high precision and high spatial resolution at the same time.

In practice, what help can you expect from a wavefront sensor?

  • Characterise optical aberrations and obtain their projection on Zernike polynomials. This is precious information to understand easily the imperfections of an optical system. Since wavefront sensors are relatively fast (tens of Hertz), they can as well characterise dynamic aberrations such as those induced by thermal effects. High end systems running at a kHz can even measure aberrations due to atmospheric turbulence.
  • Characterise completely the light propagation. The characterisation of a beam of light in terms of intensity and wavefront allows a certain number of its fundamental parameters to be calculated by simply processing the initial measurement data. Using the Fresnel propagation equations, one can reconstruct the phase and intensity profile in any plane along the optical axis.
  • Laser true 3D profile

    Laser true 3D profile

  • Measure the intensity profile at focus of a laser (also known as Point Spread Function or PSF). Or, rather, reconstruct it. This is just a consequence of the point above. A number of people are experiencing issues while trying to measure the intensity of their laser at focus. A Wavefront sensor capable of measuring the intensity as well can reconstructs the PSF while being meters away from the focus, and can then become part of the solution.
  • Characterise an optical system by measuring the Mode Transfer Function MTF. This is only the Fourier transform of the PSF.
  • Measure a number of laser parameters such as M2, beam waist, optical intensity propagation along the optical axis.

Watch out for an exact software description before buying anything! Some of the above features involve advanced data processing and are not proposed by every wavefront sensor manufacturers.

A wavefront sensor will also:

  • Help align the optics of your system. Some of those sensors make it very easy: the idea is to position the first optic of the system, and align it while using the wavefront sensor to find the position that minimizes the aberrations. After having found the right position you would then take a measurement as a reference, introduce a second optic, and subtract the reference to the new wavefront. What you then measure are the aberrations introduced by the latest optic alone, the positioning of which you can now optimise. And so on...
  • Help improve the optical resolution of your system.
  • Help increase the power at focus.
  • Help produce tighter focal spots.

Those last three points all come from the same property. As shown in the point spread functions pictures above, aberrations reduce the quality of the response of an optical system, spreading its PSF (focal spot size) and reducing the intensity at its centre. This results in blurry images and effectively reduce the resolution. By characterising the aberrations introduced by an optical system, a wavefront sensor helps in taking the relevant actions to minimise them (through better alignment or adaptive optic for instance).

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Comparing detector noise specifications

After having explained the causes of optical noise in detectors, I'd like in this post to define the parameters that makes it possible to compare detectors noise specifications.

Signal to noise ratio

Also noted SNR or S/N. This is defined as the ratio between the signal power and the noise power. Hence:

\frac{S}{N}=\frac{\overline{\left|i_{s} \right|^{2}}}{\overline{\left|i_{noise} \right|^{2}}}

Understanding the meaning of this is quite straightforward: the higher this ratio, the best signal you get. At equal input signal, the detector with the highest SNR is the less noisy one. If S/N<1, you cannot see anything, if S/N>>1, the signal is easy to pick up. As such, the signal-to-noise ratio is not a usable figure of merit of a detector. It is rather a measure of how strong your signal is compared to the "sensitivity" of your detector.

However, comparing the optical power needed to get a SNR of 1 is a step in the right direction to compare detector noise. According to the optical noise models explained earlier,

\overline{\left|i_{s} \right|^{2}}=\left(\frac{e\eta}{h\nu}\right)^{2}\overline{P_{opt}^{2}}

\overline{\left|i_{noise} \right|^{2}}=\left[2e\left(\overline{i_{s}}+\overline{i_{0}}\right)+\frac{4k_{B}T}{R} \right]\Delta\nu

Obviously, the noise depends completely on the bandwidth of the detector. This is understandable: to differentiate a true experimental result from random experimental error, you need to repeat the experiment. Translated in detector terminology, to get a better signal you need to increase the integration time of the experiment (= decrease the bandwidth).

To define a good figure of merit, it needs to show the minimum detectable optical power and not to depend on the integration time. Enters the noise equivalent power.

Noise equivalent power

Also noted NEP. This is a slightly confused definition. The initial concept is to define the noise equivalent power as the optical power which will yield a signal to noise ratio of 1. This is then the limit of what can be detected. But with this definition the noise equivalent power can only be given at a specific bandwidth (Δν enters in the expression of S/N).

Since not two detectors have the same integration time, manufacturers tend to call Noise Equivalent Power the minimum detectable power per square root of bandwidth. We will note this noise equivalent power per unit of bandwidth NEP_{\sqrt{\Delta\nu}} to avoid confusion. In this situation we have then:

\left(NEP_{\sqrt{\Delta\nu}}\right)^{2}=\frac{NEP^{2}}{\Delta\nu}=\frac{\overline{P_{opt\mid S/N=1}^{2}}}{\Delta\nu}=\left(\frac{h\nu}{e\eta}\right)^{2}\left[2e\left(\overline{i_{s}}+\overline{i_{0}}\right)+\frac{4k_{B}T}{R} \right]

this normalised NEP_{\sqrt{\Delta\nu}} only depends on the detector itself (and sometime on the ambient temperature!) and is measured in W\cdot Hz^{-1/2}. The smallest the NEP, the better is the detector.

Getting back to the ambient temperature issue, the fluctuations of the ambient temperature are generally too small in comparison of the absolute temperature to introduce a bias in the comparison. However, it is true that the higher the temperature, the more noisy a detector is. For that reason some high quality detector are cooled (generally thermoelectrically but sometime with cryogenic cooling).

Detectivity and Specific detectivity

The detectivity D is defined as the reciprocal of the NEP: D=\frac{1}{NEP}. Since all of parameters we defined depend on the area of the detector, in some cases this introduces a bias in the detector comparison. Thus sometime is specified a specific detectivity D* (D star), defined as:

D^{*}=\frac{\sqrt{A}}{NEP_{spectral}}

In fairness, I have very rarely encountered people using this specific detectivity in optical detectors.

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Noise in photodetectors

After my recent post on power and energy meters, I'd like to speak about the different kind of noises that affect a photodetector and how to model them.

Quantum effect and noise: the shot noise.

Fundamental physics tells us the light is made of particles (photons), which are emitted by the source at random. For that reason, the amount of photons emitted by the source (sun, bulb, laser, etc.) is not constant, but exhibits detectable statistical fluctuations. And this is in a nutshell what shot noise is. Because of its nature, it does not depends on the quality of the detector and is unavoidable. However the shot noise becomes a real issue only when the optical intensity is fairly low: in this case quantum fluctuations become much more noticeable.

The random process of light emission can generally be modelled using a Poisson distribution, the properties of which are very well known. If we note p(n) the probability that n photons arrive on the detector:

p(n)=\frac{\bar{n}^{n}exp(-\bar{n})}{n!}

\bar{n}=\sum_{0}^{+\infty}{np(n)}

\sigma_{n}^{2}=\sum_{0}^{+\infty}{(n-\bar{n})^{2}p(n)=\overline{n^{2}}-\bar{n}^{2}=\bar{n}}

where \sigma_{n} is the standard deviation. What this means is that for 100 photons arriving on the detector, the uncertainty about the number of photon is of ±10 (±10%). If the number of photon is somewhat closer to common levels, e.g. 10^{10}, the uncertainty becomes \pm 10^{5}, which is ±0.000,01%. It then becomes obvious that the shot noise is an issue only at low light level.

Let's find out now what would be the fluctuations of the signal current due to the shot noise. Since each photon induce a free electron with an efficiency \eta , during the time \tau the number of electron produced is then \eta\bar{n}. Every electron contributes to the signal current i_{s} for a charge e, the average value of the signal current \overline{i_{s}} is then:

\overline{i_{s}}=\frac{e}{\tau}\eta\bar{n}

.

Fluctuation in the number of photons create a fluctuation in the signal current, and those fluctuations are characterised by the standard deviation:

\sigma_{i_{s}}^{2}=\left(\frac{e}{\tau}\right)^{2}\sigma_{\eta n}^{2}=\left(\frac{e}{\tau}\right)^{2}\eta \bar{n}

If we note \Delta\nu\simeq 1/2\tau the bandwidth of the detector, we can find the useful formula below:

\sigma_{i_{s}}^{2}=2e\overline{i_{s}}\Delta\nu

And this standard deviation \sigma_{i_{s}} characterise the shot noise current.

Dark current

The dark current is the constant response exhibited by a detector during periods when it is not actively being exposed to light. It is sometime classified as another type of shot noise. It is also referred to as reverse bias leakage current in non optical devices and is present in all diodes. Physically, dark current is due to the random generation of electrons and holes within the depletion region of the device that are then swept by the electric field applied to the diode.

Similarly to the photon noise, it can be modelled by a Poisson distribution, with:

\sigma_{i_{0}}^{2}=2e\overline{i_{0}}\Delta\nu

Let's note that \overline{i_{0}} depends from many parameters, but generally speaking:

  • Si photodiodes : \overline{i_{0}} ranges from 1 to 10 nA
  • Ge photodiodes: \overline{i_{0}} ranges from 50 to 500 nA
  • InGaAs photodiodes: \overline{i_{0}} ranges from 1 to 20 nA

Thermal noise

Thermal noise, also called Johnson noise or Nyquist noise is the electronic noise generated by the thermal agitation of the electrons inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. It was discovered by Johnson in 1927 and explained by Nyquist.

A device (a photodiode for instance) thermal noise can be modelled as a voltage source V_{th}(t) in series with an ideal resistor R. V_{th}(t) has a Gaussian distribution with a mean value of zero. In this case,

\sigma_{V_{th}(t)}^{2}=4k_{B}TR\Delta\nu

Where \sigma_{V_{th}(t)} is the standard deviation of the voltage, k_{B} is the Boltzmann's constant in joules per kelvin and T is the resistor's absolute temperature in kelvins. It can also be modelled a current source i_{th} in parallel with R:

\sigma_{i_{th}(t)}^{2}=\frac{4k_{B}T}{R}\Delta\nu

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Buying a laser power meter: check-list

Because of the wide range of power and energy meter available on the market, and even more because they tend to be not totally versatile, you need to carefully examine your needs against the capabilities of the instrument you are planning to acquire. Here is a little check list to help you decide if a laser power meter or energy meter would fit your application.

  • Is the meter's calibration traceable to internationally recognized standards such as NIST?
  • Is your laser wavelength within the wavelength range of the power meter?
  • What is the power range you expect to measure (highest and lowest limit)? Does it fall within the range the power meter can measure?
  • What is the diameter of your beam at measurement point? Do you have any control on this (using a lens for instance)? Is the power meter aperture big enough?
  • What is your power density (W/cm2) and energy density (J/cm2)? Is it below the damage threshold of the power meter?
  • Is your laser a pulsed femtosecond? If yes you will need a flat spectral response across the laser bandwidth. This may also be the case if your laser is widely tunable and you can't adjust the wavelength setting manually, or simply if you don't know your wavelength.
  • Is your laser pulsed and do you need to measure each pulse's energy or an average power is sufficient? If the average power is enough or if you want to measure a single pulse energy, a thermopile is better. Otherwise you will have to go for a pyroelectric sensor or a specialised photodiode
  • Are there a lot of vibrations in your environment? If so this would rule out a pyroelectric detector.
  • Most power meters are sold nowadays in a set of two separate items: a display and a sensor. Make sure you order both and that they are compatible with each other
  • Assess what type of display you need: do you need computer connectivity, LabView compatibility, is it to go "in the field", do you need it wireless (yes some manufacturer do that now)...
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