## Real-time determination of laser beam quality by modal decomposition |

Optics Express, Vol. 19, Issue 7, pp. 6741-6748 (2011)

http://dx.doi.org/10.1364/OE.19.006741

Acrobat PDF (767 KB)

### Abstract

We present a real-time method to determine the beam propagation ratio *M*^{2} of laser beams. The all-optical measurement of modal amplitudes yields *M*^{2} parameters conform to the ISO standard method. The experimental technique is simple and fast, which allows to investigate laser beams under conditions inaccessible to other methods.

© 2011 OSA

## 1. Introduction

2. International Organization for Standardization, *ISO 11146-1/2/3 Test methods for laser beam widths, divergence angles and beam propagation ratios – Part 1: Stigmatic and simple astigmatic beams / Part 2: General astigmatic beams / Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods* (ISO, Geneva, 2005). [PubMed]

*M*

^{2}parameter, the intrinsic astigmatism, and the twist parameter [2

2. International Organization for Standardization, *ISO 11146-1/2/3 Test methods for laser beam widths, divergence angles and beam propagation ratios – Part 1: Stigmatic and simple astigmatic beams / Part 2: General astigmatic beams / Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods* (ISO, Geneva, 2005). [PubMed]

*M*

^{2}factor has become a well accepted measure for beam quality in the laser community by now—although it has to be used with care.

*M*

^{2}, which are not comparable to each other [1]. This is an important fact since other techniques as the aforementioned knife-edge method are still in use for one reason: their simplicity. As stated above, a measurement of a general astigmatic beam, which is fully conform to the ISO standard, requires the measurement of ten second order moments of the Wigner distribution function and, thus, is experimentally cumbersome and slow.

*M*

^{2}determination is based on a caustic measurement that includes the determination of beam width as a function of propagation distance. To do this, the beam width has to be determined at least at 10 positions along the propagation axis, completed by a hyperbolic fit. According to ISO 11146-1, the beam width determination has to be carried out using the three spatial second order moments of the intensity distribution [2

2. International Organization for Standardization, *ISO 11146-1/2/3 Test methods for laser beam widths, divergence angles and beam propagation ratios – Part 1: Stigmatic and simple astigmatic beams / Part 2: General astigmatic beams / Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods* (ISO, Geneva, 2005). [PubMed]

3. B. Schäfer and K. Mann, “Determination of beam parameters and coherence properties of laser radiation by use of an extended Hartmann-Shack wave-front sensor,” Appl. Opt. **41**, 2809–2817 (2002). [CrossRef] [PubMed]

5. R. W. Lambert, R. Cortés-Martínez, A. J. Waddie, J. D. Shephard, M. R. Taghizadeh, A. H. Greenaway, and D. P. Hand, “Compact optical system for pulse-to-pulse laser beam quality measurement and applications in laser machining,” Appl. Opt. **43**, 5037–5046 (2004). [CrossRef] [PubMed]

*M*

^{2}= 1. Any deviation from the ideal diffraction-limited Gaussian beam profile can be attributed to the contribution of higher order modes, leading to

*M*

^{2}

*>*1. Here, any excited higher order mode contributes with a certain value to the beam propagation parameter. This value can be readily calculated for every mode in a way which is conform to the ISO standard [7

7. S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. **153**, 207–210 (1998). [CrossRef]

*M*

^{2}which is compatible to the one resulting from a caustic measurement.

8. E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B **49**, 409–414 (1989). [CrossRef]

9. A. Cutolo, T. Isernia, I. Izzo, R. Pierri, and L. Zeni, “Transverse mode analysis of a laser beam by near-and far-field intensity measurements,” Appl. Opt. **34**, 7974–7978 (1995). [CrossRef] [PubMed]

11. X. Xue, H. Wei, and A. G. Kirk, “Intensity-based modal decomposition of optical beams in terms of Hermite-Gaussian functions,” J. Opt. Soc. Am. A **17**, 1086–1091 (2000). [CrossRef]

12. N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B **91**, 353–357 (2008). [CrossRef]

15. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express **17**, 9347–9356 (2009). [CrossRef] [PubMed]

16. D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distributionof multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. **35**, 3429–3431 (2010). [CrossRef] [PubMed]

*M*

^{2}values conform to the ISO standard but with diverse advantages. The experimental realization is simple and yields information about the beam quality in real-time in contrast to other approaches. It relies on a decomposition of the electromagnetic field into the spatial modes of the resonator generating the laser beam. Since our investigated laser cavity possesses rectangular symmetry, Hermite-Gaussian modes are used in this work, but represent no restriction of the method. Any complete set of suitable modes may be used in other cases.

*M*

^{2}and compare our results to values obtained from traditional caustic measurements as defined by the ISO standard. Moreover, we demonstrate the on-line monitoring of the modal spectrum as well as the

*M*

^{2}factor while realigning the resonator in real-time.

## 2. Modal decomposition

### 2.1. Expansion into Hermite-Gaussian modes

17. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. **5**, 1550–1567 (1966). [CrossRef] [PubMed]

*w*

_{0}is the waist radius of the fundamental mode HG

_{00}and H

*denotes the Hermite polynomial of order*

_{l}*l*.

*U*can be expanded into a superposition of HG modes [6] where the asterisk denotes complex conjugation. The complex-valued expansion coefficients

*c*=

_{mn}*ρ*e

_{mn}^{i}

*include modal amplitudes*

^{ϕmn}*ρ*=

_{mn}*|c*and phases

_{mn}|*ϕ*= arg(

_{mn}*c*). Using normalized fields with unit power

_{mn}*mode since*

_{mn}### 2.2. Beam propagation ratio

*d*is the beam diameter and

*θ*denotes the divergence angle. These parameters are based on second order moments σ

^{2}. For instance, the beam diameter in

*x*-direction reads as [2

*M*

^{2}factor.

_{m}_{0}modes with higher order modes in one direction [18

18. H. Laabs and B. Ozygus, “Excitation of Hermite-Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. **28**, 213–214 (1996). [CrossRef]

17. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. **5**, 1550–1567 (1966). [CrossRef] [PubMed]

*p*=

*m*+

*n*=

*const*belonging to one

*mode group*have the same frequency and, thus, temporally stable intermodal phase differences. Due to the huge camera integration time compared to the beating period, the interference terms between modes of different mode groups vanish, whereas modes of one and the same mode group contribute coherently to a recorded intensity distribution. Furthermore, a rotation of the principal axis of one mode towards the laboratory system can be described by a superposition of HG modes of the associated mode group—similar to the expansion of Laguerre-Gaussian modes into HG modes [6]. Consequently, the total intensity distribution is the sum of intensity contributions from single mode groups where

*p*denotes the

*p*

^{th}mode group and

*I*is the intensity formed by a superposition of the modes from mode group

_{p}*p*.

*have been used.*

_{i j}## 3. Experiments

### 3.1. Measurement setup

*M*

^{2}parameter of the emerging beam. The Nd:YAG laser can produce different HG

*modes at*

_{mn}*λ*= 1064nm. The excited HG

*modes are restricted to*

_{mn}*n*= 0 since the pump light (

*λ*= 808nm) is coupled into the plane-concave resonator by a horizontally movable fiber [18

18. H. Laabs and B. Ozygus, “Excitation of Hermite-Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. **28**, 213–214 (1996). [CrossRef]

*f*-setup is recorded by a CCD camera. This intensity pattern contains the information about the weights

*correlation filter*[15

15. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express **17**, 9347–9356 (2009). [CrossRef] [PubMed]

*M*

^{2}factor in real-time using a computer-aided calculation of Eq. (11).

*modes with (*

_{mn}*m*+

*n*) ≤ 5 are implemented taking into account a possible rotation of the resonator coordinate system. This correlation filter configuration corresponds to a truncation of all sums in the equations of section 2.

### 3.2. Measurement results

_{20}and HG

_{30}is depicted in Fig. 2. Using Eq. (6) the intensity distribution in the plane of the CGH can be reconstructed. Since the method will only be as good as it is able to reconstruct the investigated beam, the measured intensity distribution is compared to the reconstructed one by calculating their two-dimensional cross-correlation coefficient. A value of 1.0 denotes perfect match. Cross-correlation coefficients greater than 0.9 in all investigated cases indicate the excellent functionality of the method. In particular, the cross-correlation coefficient for the beam investigated in Fig. 2 is 0.98. For the determination of

*M*

^{2}, the reconstruction of the intensity profile is not necessary as a matter of course and just shown here to illustrate the functionality of the method.

*M*

^{2}factors. For instance, the beam in Fig. 2 is denoted as Mix 6. As the bar charts in Fig. 3 show, the results of both measurement techniques are in very good agreement. The

_{m}_{0}modes consistent with the theory. The slight deviations in the

*m*+

*n*near the truncation limit. Due to the finite number of implemented modes in the CGH, modal weights of HG

*modes with (*

_{mn}*m*+

*n*)

*>*5, which would increase the beam quality

*M*

^{2}measurement on a time scale of 30 s, where the cavity was initially aligned to produce the fundamental HG

_{00}mode. Then we changed the transverse position of the end-pumping fiber [18

18. H. Laabs and B. Ozygus, “Excitation of Hermite-Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. **28**, 213–214 (1996). [CrossRef]

*M*

^{2}factor increased. After 15 s we reversed the process to return to the fundamental Gaussian mode after 30 s. The reachable speed of the method is only limited by the used hardware.

## 4. Conclusion

*M*

^{2}. Using the correlation filter technique, it is possible to determine the modal spectrum in real-time. We demonstrated the working capabilities by analyzing the beam emerging from an end-pumped Nd:YAG laser. By changing the alignment of its cavity, several Hermite-Gaussian modes of higher order could be excited. We showed that a correlation analysis involving the first 21 Hermite-Gaussian modes yields values for a derived

*M*

^{2}parameter, which are in good agreement with the caustic measurements proposed by the ISO standard. The advantage of the method is the ability for real-time analysis of the beam that was demonstrated by continuously changing the alignment of the laser cavity and monitoring the beam propagation ratio

*M*

^{2}as a function of time.

*M*

^{2}, which is compatible to the ISO standard in cases where caustic measurements are too time-consuming or even impossible to perform. The additional information about the modal spectrum of the beam is highly advantageous for diverse laser applications.

## References and links

1. | A. E. Siegman, “How to (maybe) measure laser beam quality,” in |

2. | International Organization for Standardization, |

3. | B. Schäfer and K. Mann, “Determination of beam parameters and coherence properties of laser radiation by use of an extended Hartmann-Shack wave-front sensor,” Appl. Opt. |

4. | B. Eppich, G. Mann, and H. Weber, “Measurement of the four-dimensional Wigner distribution of paraxial light sources,” in |

5. | R. W. Lambert, R. Cortés-Martínez, A. J. Waddie, J. D. Shephard, M. R. Taghizadeh, A. H. Greenaway, and D. P. Hand, “Compact optical system for pulse-to-pulse laser beam quality measurement and applications in laser machining,” Appl. Opt. |

6. | A. E. Siegman, |

7. | S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. |

8. | E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B |

9. | A. Cutolo, T. Isernia, I. Izzo, R. Pierri, and L. Zeni, “Transverse mode analysis of a laser beam by near-and far-field intensity measurements,” Appl. Opt. |

10. | M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite-Gaussian modes,” Appl. Opt. |

11. | X. Xue, H. Wei, and A. G. Kirk, “Intensity-based modal decomposition of optical beams in terms of Hermite-Gaussian functions,” J. Opt. Soc. Am. A |

12. | N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B |

13. | V. A. Soifer and M. Golub, |

14. | M. Duparré, B. Lüdge, and S. Schröter, “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters,” in |

15. | T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express |

16. | D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distributionof multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. |

17. | H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. |

18. | H. Laabs and B. Ozygus, “Excitation of Hermite-Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. |

19. | G. Szegö, |

**OCIS Codes**

(030.4070) Coherence and statistical optics : Modes

(090.1760) Holography : Computer holography

(140.3295) Lasers and laser optics : Laser beam characterization

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: January 18, 2011

Revised Manuscript: March 11, 2011

Manuscript Accepted: March 13, 2011

Published: March 24, 2011

**Citation**

Oliver A. Schmidt, Christian Schulze, Daniel Flamm, Robert Brüning, Thomas Kaiser, Siegmund Schröter, and Michael Duparré, "Real-time determination of laser beam quality by modal decomposition," Opt. Express **19**, 6741-6748 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6741

Sort: Year | Journal | Reset

### References

- A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues , M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.
- International Organization for Standardization, ISO 11146-1/2/3 Test methods for laser beam widths, divergence angles and beam propagation ratios – Part 1: Stigmatic and simple astigmatic beams / Part 2: General astigmatic beams / Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods (ISO, Geneva, 2005). [PubMed]
- B. Schäfer and K. Mann, “Determination of beam parameters and coherence properties of laser radiation by use of an extended Hartmann-Shack wave-front sensor,” Appl. Opt. 41, 2809–2817 (2002). [CrossRef] [PubMed]
- B. Eppich, G. Mann, and H. Weber, “Measurement of the four-dimensional Wigner distribution of paraxial light sources,” in Optical Design and Engineering II , L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622D (2005).
- R. W. Lambert, R. Cortés-Martínez, A. J. Waddie, J. D. Shephard, M. R. Taghizadeh, A. H. Greenaway, and D. P. Hand, “Compact optical system for pulse-to-pulse laser beam quality measurement and applications in laser machining,” Appl. Opt. 43, 5037–5046 (2004). [CrossRef] [PubMed]
- A. E. Siegman, Lasers (University Science Books, 1986).
- S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998). [CrossRef]
- E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989). [CrossRef]
- A. Cutolo, T. Isernia, I. Izzo, R. Pierri, and L. Zeni, “Transverse mode analysis of a laser beam by near-and far-field intensity measurements,” Appl. Opt. 34, 7974–7978 (1995). [CrossRef] [PubMed]
- M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite-Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999). [CrossRef]
- X. Xue, H. Wei, and A. G. Kirk, “Intensity-based modal decomposition of optical beams in terms of Hermite-Gaussian functions,” J. Opt. Soc. Am. A 17, 1086–1091 (2000). [CrossRef]
- N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008). [CrossRef]
- V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).
- M. Duparré, B. Lüdge, and S. Schröter, “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters,” in Optical Design and Engineering II , L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622G (2005).
- T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17, 9347–9356 (2009). [CrossRef] [PubMed]
- D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distributionof multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010). [CrossRef] [PubMed]
- H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966). [CrossRef] [PubMed]
- H. Laabs and B. Ozygus, “Excitation of Hermite-Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996). [CrossRef]
- G. Szegö, Orthogonal Polynomials (Amer. Math. Soc., 1975).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.