M²-Sensor for the Adaptive Optical System


When adaptive optical system is applied for laser beam control it is very important to know the beam parameters such as beam width, divergence angle, beam quality factor M2 etc. before and after correction. That is why the sensor making such estimations should be included in any laser adaptive optical system. This paper describes the sensor design, possibilities, the principals of measurements and it’s place in the whole adaptive optical system.

Keywords: beam widths, beam quality factor M2, adaptive system.


A good laser beam quality is necessary for many scientific and industrial applications for example for material processing. Different active medium inhomogeneities and turbulence of the air at the beam delivery path cause laser beam phase and intensity distortions and power instability. To improve laser beam parameters adaptive optical system could be used.

In our group of Adaptive Optics for Industry and Medicine, ILIT, Russian Academy of Sciences we design and make commercially available adaptive optical system for correction of the wavefront aberrations and for formation of a specified laser beam intensity distribution. The main element of our adaptive system is a bimorph flexible mirror well described in Ref.1-5. For the wavefront analysis we use the Shack-Hartmann sensor. For the estimations of laser beam widths, divergence angle, beam quality factor M2, power instability we designed so-called M2-Sensor. The place of the M2-Sensor in the whole adaptive system infrastructure is shown in Fig.1.

Adaptive system with the M²-Sensor
Fig.1. Adaptive system with the M²-Sensor: 1 – laser; 2 – beam splitter, 3 – deformable mirror, 4 – mirror control unit, 5 – semi transparent mirror, 6 – lenslet array, 7 – CCD camera, 8 – computer, 9 – focusing lens

Methods and equiations for beam parameters calculation

According to the International standard ISO11146 different methods of beam parameters determination could be applied6-10. In our case we use the direct method of beam parameters measurement.

The more detailed scheme of the M²-Sensor and it’s photo are presented in Fig.2.

M²-Sensor design

M²-Sensor photo
Fig.2. M²-Sensor design: 1 –laser beam, 2 – neutral density variable filter, 3 – focusing lens, 4 – CCD camera, 5 – moving stage, 6 – computer with framegrabber , (b) The photo of M²-Sensor

To estimate divergence angle, beam quality factor M2 the beam diameter value is used 6. In order to calculate beam diameter intensity distribution is registered with the CCD. The beam diameters along x and y directions are defined as:

, (1)

where the second moments σx, σy of the intensity distribution I(x,y,z) at the location z are given by:

, (2)

Here are the first moments of the intensity distribution giving coordinates of the beam centre:

. (3)

For elliptical beams the software includes the calculation of the azimuthal angle φ between the beam principal axes x', y' and the axis of laboratory coordinate system x, y (Fig.3). Then the definition of all the main beam parameters takes place in the principal beams axis direction.

Elliptical beam
Fig.3. Elliptical beam

In order to reduce an influence of different expose signals it is possible to select an area of interest for calculations from the whole surface of the CCD window (Fig.4). To increase the accuracy of calculations the dark current of camera is subtracted from the image.

Selecting of an area of interest
Fig.4. Selecting of an area of interest

Our sensor allows to define the beam quality factor M2 and divergence angle by two methods described in ISO11146 6.

The first one is based on a multiple measurements of laser beam diameters in several cross-sections before and after the beam waist. At least 10 measurements have to be taken - half of them within one Rayleigh length on either side of the beam waist. The rest measurements have to be taken beyond double Rayleigh distance on either side of the beam waist. If the beam waist is accessible for direct measurements the beam parameters are determined by a hyperbolic fit to the measured beam diameters di along the beam propagation axis:

, (4)

where di is a beam diameter at the location zi, A,B,C are the hyperbola parameters. Then the divergence angle and beam propagation factor can be calculated using:

, (5)

. (6)

If the beam waist is not accessible for direct measurements the artificial waist should be created with the help of a focusing element and then the same measurements should be made around the artificial beam waist (fig.2).

Fig.5 shows the realization of multiple measurements method by the software of our M2-Sensor.

Realization of multiple measurements for determination M<SUP>2</SUP>-parameter
Fig.5. Realization of multiple measurements for determination M2-parameter

The second method to define beam parameters is so-called single measurement method. To use this method first of all the near-field laser beam diameter d0 is determined. Then the far-field diameter of the beam d is measured in the focal plane of the lens. M2-factor is calculated according to the formula:

, (7)

where f is a focal length of the lens. The divergence angle is determined using the relationship:

. (8)

Gaussian and flat-top fit and oscilloscope options of M2-Sensor

For laser applications sometimes it is useful to know gaussian or flat-top fit to laser beam intensity. For example, the last one is important during formation of the super-gaussian intensity distribution.

Gaussian fit to the beam intensity and option of oscilloscope
Fig.6. Gaussian fit to the beam intensity and option of oscilloscope

To compare real intensity distribution with gaussian one the following formula is used:

, (9)

where I(x,y) is the intensity value of the point with transversal coordinates (x,y), A,B are coefficients of gaussian function. Then the RMS error σ² is analysed:

. (10)

Here Ireal(x,y) is the real intensity value, Igauss(x,y) is the calculated gaussian one, N is the number of pairs (x,y) used in fitting. If σ=0 then our beam becomes the gaussian. The more σ, the less the real intensity distribution function could be described by gaussian function.

While forming super-gaussian intensity distribution in some area (x,y)≤(x0,y0) it is convenient to use for fitting rectangular function:

, (11)

We minimize the RMS-error of deviation of power of real beam () and flat-top one ():

. (12)

In order to be sure that during the experiments of the formation of the laser intensity distribution the important parameters of a laser beam remains as good as before correction the software includes options for evaluation of the short-term and long-term power stability. The short-term stability defined as:

, (13)

where W is the power distribution in one frame, is the averaged power of all frames measured per short time, N is the number of frames.

The long-term power stability of a laser beam defined as:

, (14)

Here is the short-term averaged power measured for a few seconds, is the averaged one measured for the compatible long time, N is the number of counts of .

Fig.6 illustrates the oscilloscope option of M2-Sensor. The frequency of the measurements is 10 Hz.

Accuracy of the measurements

Let’s consider reasons and corresponding errors influenced on the accuracy of measurements. The main reasons are:

  1. finite CCD resolution - σ1;
  2. error of CCD position measurements - σ2;
  3. dark current of the CCD - σ3;
  4. laser intensity instability - σ4.

So the whole error is:

. (15)

Estimations shows that in our case:




Error caused by laser intensity instability depends on the own laser characteristic. For example for LGN-302 He-Ne laser this error is σ4=0.01%.

So the total RMS-error is:


According to International Standard ISO11146 the admissible mistake for beam parameters calculation is σtotal=10%.

Two He-Ne lasers were taken for testing of the M2-Sensor: LGN-302 and LGN-207А. The obtained results are presented in Table 1. All parameters were determined along x and y laser beam directions.

Parameter θx, mrad θy, mrad σlong-term, % σshort-term, %
LGN-302 1.17 1.36 1.116 1.044 0.01 0.2
LGN-207A 2.06 1.59 2.07 1.6 0.15 0.4
Table 1. Main results


Our M2-Sensor is designed as a part of adaptive system for simple measurements of laser beam parameters. The software and hardware allow to define such laser beam characteristics as:

All the measurements correspond to the International Standard ISO11146.


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