**M²-Sensor for the Adaptive Optical System**

**Abstract**

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 *M ^{2}* 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 M^{2}, adaptive system.

**Introduction**

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 M^{2}, power instability we designed
so-called *M ^{2}-Sensor*. The place of the

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 applied^{6-10}. 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.

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 M^{2} 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 σ_{x}, σ_{y} 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.

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.

Fig.4. Selecting of an area of interest

Our sensor allows to define the beam quality factor M^{2} 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 *d _{i}* along
the beam propagation axis:

, (4)

where *d _{i}* is a beam diameter at the location

, (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 *M ^{2}-Sensor*.

Fig.5. Realization of multiple measurements for determination M

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 *d _{0}* is determined. Then the far-field
diameter of the beam

, (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 M ^{2}-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.

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 *I _{real}(x,y)* is the real intensity value,

While forming super-gaussian intensity distribution in some area *(x,y)≤(x _{0},y_{0})* 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 *M ^{2}-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:

- finite CCD resolution - σ
_{1}; - error of CCD position measurements - σ
_{2}; - dark current of the CCD - σ
_{3}; - laser intensity instability - σ
_{4}.

So the whole error is:

. (15)

Estimations shows that in our case:

σ_{1}=0.7%,

σ_{2}≈0,

σ_{3}=6%.

Error caused by laser intensity instability depends on the own laser characteristic. For example
for *LGN-302 He-Ne* laser this error is σ_{4}=0.01%.

So the total RMS-error is:

.

According to International Standard ISO11146 the admissible mistake for beam parameters
calculation is *σ _{total}*=10%.

Two *He-Ne* lasers were taken for testing of the *M ^{2}-Sensor*: LGN-302 and
LGN-207А. The obtained results are presented in Table 1. All parameters were determined along

Parameter | θ, mrad
_{x} | θ, mrad
_{y} | σ, %
_{long-term} | σ, %
_{short-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 |

**Conclusion**

Our *M ^{2}-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:

- Beam Centroid location;
- Beam Diameter or Widths;
- Divergence Angle;
- Beam Quality Factor M
^{2}; - Elliptisity of the beam;
- Gaussian and Flat-top fit to the intensity;
- Short-term and Long-term stability of the beam intensity.

All the measurements correspond to the International Standard ISO11146.

**References**

- A.V. Kudryashov, V.V. Samarkin, “Control of high power
*CO*laser beam by adaptive optical elements”,_{2}*Optics Communications***118**, pp.317-322, 1995. - T.Yu. Cherezova, S.S. Chesnokov, L.N. Kaptsov, A.V. Kudryashov, “Doughnut-like laser beam intensity output formation by means of adaptive optics”,
*Optics Express***155**, pp.99-106, 1998. - T.Yu. Cherezova, L.N. Kaptsov, A.V. Kudryashov, “
*CW*industrial rod*YAG:Nd*laser with an intracavity active bimorph mirror”,^{3+}*Applied Optics***35**, pp.2554-2561, 1996. - A.V. Kudryashov, V.I. Shmalhausen, “Semipassive bimproph flexible mirrors for atmospheric adaptive optics applications”,
*Optical Engineering***35(**11), pp.3064-3073, 1996. - T.Yu. Cherezova, S.S. Chesnokov, L.N. Kaptsov, A.V. Kudryashov, “Super-Gaussian output laser beam formation by bimorph adaptive mirror”,
*Optics Communications***155**, pp.99-106, 1998. - Test method for laser beam parameters: Beam width, divergence angle and beam propagation factor, Document ISO/DIS 11146,
*International Organization for Standardization*, 1996. - P. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams”,
*Optic Communications***105**, pp.233-242, 1994. - T. Jonston, “Beam propagation (M
^{2}) measurement made as easy as it gets: the four-cuts method”,*Applied Optics***37**(21), July 1998. - A.E. Siegman,
*Solid State Lasers: New Developments and Applications*, pp.13-28, Plenum Press, New York, 1993. - K. Roundy, “Propagation factor quantifies laser beam performance”,
*Laser Focus World***12**, pp.119-122, 1999.

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