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Optics Express

Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 19, Iss. 7 — Mar. 28, 2011
  • pp: 6741–6748
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Real-time determination of laser beam quality by modal decomposition

Oliver A. Schmidt, Christian Schulze, Daniel Flamm, Robert Brüning, Thomas Kaiser, Siegmund Schröter, and Michael Duparré  »View Author Affiliations


Optics Express, Vol. 19, Issue 7, pp. 6741-6748 (2011)
http://dx.doi.org/10.1364/OE.19.006741


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Abstract

We present a real-time method to determine the beam propagation ratio M2 of laser beams. The all-optical measurement of modal amplitudes yields M2 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

The characterization of laser beams is traditionally a basic task in optical science and performed for decades now. However, the existence of different archaic approaches like the variable aperture or moving knife-edge method, which are known to produce deviant results, has shown the need for a standardization of the definition. Especially the question how to reproducibly and reliably measure beam quality, which is very important from an user-oriented point of view, stimulated a lot of discussions [1

1. 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.

]. The ISO standard 11146-1/2/3 has brought a great unification by defining all relevant quantities of laser beams including instructions how to perform the measurements conform to the ISO standard [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]

]. In the general case of so-called general astigmatic beams this approach relies on the determination of ten independent parameters, which are the second order moments of the Wigner distribution function. From these, three quantities can easily be derived according to ISO 11146-2: the beam propagation ratio simply called 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]

]. Especially the 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.

Siegman pointed out that the use of measurements not conform to the ISO standard will result in values for M 2, which are not comparable to each other [1

1. 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.

]. 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.

Fortunately, due to their high symmetry most laser beams of practical interest require less than these ten independent parameters to be measured for a complete characterization: Most lasers emit beams that are simple astigmatic or even stigmatic because of their resonator design. In this case the 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]

].

Despite its experimental simplicity, the caustic measurement still is quite time-consuming and requires a careful treatment of background intensity, measuring area, and noise, which makes high demands on the temporal stability of a cw laser or the pulse-to-pulse stability of a pulsed source. Caustic measurements are therefore unsuitable to characterize the fast dynamics of a laser system. To react on the need for a faster and more detailed analysis, several other methods for laser beam characterization were presented such as Shack-Hartmann wavefront analysis [3

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]

], measurement of the complete 4D Wigner distribution function [4

4. 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).

], or the use of diffraction gratings [5

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]

].

Here we will show that the correlation filter method yields 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.

2. Modal decomposition

2.1. Expansion into Hermite-Gaussian modes

Laser resonators with rectangular geometry generate superpositions of nearly Hermite-Gaussian (HG) modes with the field distribution [17

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

]
HGmn(x,y)=1w022m+nπm!n!Hm(2w0y)Hn(2w0y)exp(x2+y2w02),
(1)
where w 0 is the waist radius of the fundamental mode HG00 and Hl denotes the Hermite polynomial of order l.

2.2. Beam propagation ratio

According to the ISO standard [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]

], the beam propagation ratio for simple astigmatic beams is defined by
Mx,y2=πλdx,yθx,y4,
(3)
where d is the beam diameter and θ denotes the divergence angle. These parameters are based on second order moments σ2. For instance, the beam diameter in x-direction reads as [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]

]
dx=8[σx2+σy2+γ(σx2σy2)2+4(σxy2)2]1/2,γ=σx2σy2|σx2σy2|.
(4)

Using HG modes, the beam quality from Eq. (3) simplifies to
Mx,y2=dx,y24w02.
(5)
In other words, the ratio of the beam waist diameter to the one of the fundamental Gaussian beam determines the M 2 factor.

The laser used for experimental demonstration (manufactured by Smart Laser Systems) generates simple astigmatic beams composed of different HGm 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]

]. In general, different modes of a resonator possess slightly dissonant frequencies [17

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

]. Only modes with 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

6. A. E. Siegman, Lasers (University Science Books, 1986).

]. Consequently, the total intensity distribution is the sum of intensity contributions from single mode groups
I=|U|2=p=0IpwithIp=k,mρk,pkρm,pmHGk,pkHGm,pm,
(6)
where p denotes the p th mode group and Ip is the intensity formed by a superposition of the modes from mode group p.

Using the intensity profile from Eq. (6), the first order moments of the beam vanish and, thus, express that the beam centroid is on the optical axis. Hence, the spatial second order moments read as
σx2=x2I(x,y)dxdy=w024m,nρmn2(2m+1),
(7)
σxy2=xyI(x,y)dxdy=w022m,nρm+1,nρm,n+1m+1n+1,
(8)
and analogously for σy2, where the identities [19

19. G. Szegö, Orthogonal Polynomials (Amer. Math. Soc., 1975).

]
x2HGklHGmndxdy=w024δln[(2m+1)δkm+max(k,m)max(k,m)1δ|km|,2],
(9)
xyHGklHGmndxdy=w024max(k,m)δ|km|,1max(l,n)δ|ln|,1,
(10)
and the Kronecker symbol δi j have been used.

3. Experiments

3.1. Measurement setup

The optical setup in Fig. 1 consists of a laser source and two branches that enable two simultaneous but independent measurements of the M 2 parameter of the emerging beam. The Nd:YAG laser can produce different HGmn modes at λ = 1064nm. The excited HGmn modes are restricted to 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]

]. A beam splitter divides the beam and guides the replicas into the two branches for analysis.

Fig. 1 Scheme of experimental setup. Upper branch: M 2 determination using a CGH. Lower branch: M 2 determination by a caustic measurement conform to the ISO standard.

In the first branch a lens images the laser beam waist onto an adapted CGH, which is an amplitude hologram consisting of 601 × 601 Lee cells with cell widths of 16 μm. The far field intensity behind the CGH realized by a subsequent 2 f-setup is recorded by a CCD camera. This intensity pattern contains the information about the weights ρmn2 of the HG modes since the complex conjugated modes are implemented in the CGH called 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]

]. Hence, the correlation filter technique enables the determination of the M 2 factor in real-time using a computer-aided calculation of Eq. (11).

The CGH for this experiment is designed to simultaneously analyze the amplitudes of 21 HG modes, i.e., all HGmn modes with (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.

In the second branch of the setup in Fig. 1, an additional lens is used to analyze the caustic of the beam conform to the ISO standard [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]

], which serves as a reference measurement.

3.2. Measurement results

Fig. 2 Modal spectrum of an arbitrarily chosen mode mixture (left) with corresponding measured and reconstructed near field intensity distributions (right).

Fig. 3 Comparison of the Mx2 and My2 factors determined by the CGH technique and the ISO standard method, respectively.

In all analyzed cases, the maximum measured deviation is 13% for Mx2 and 5% for My2. This result demonstrates the functionality of the method, especially when beams with high beam quality are considered.

To demonstrate the real-time capability of the all-optical CGH technique, we continuously varied the alignment of the laser cavity and monitored the output. Figure 4 illustrates an M 2 measurement on a time scale of 30 s, where the cavity was initially aligned to produce the fundamental HG00 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]

] and the 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.

Fig. 4 Mx2 and My2 on a time scale of 30 s taken with a rate of 4 Hz. To vary the beam quality the laser resonator was tuned continuously. The insets depict the waist intensities at selected points in time.

4. Conclusion

References and links

1.

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.

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]

4.

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).

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]

6.

A. E. Siegman, Lasers (University Science Books, 1986).

7.

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

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]

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. 38, 5272–5281 (1999). [CrossRef]

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]

13.

V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).

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 Optical Design and Engineering II, L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622G (2005).

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]

17.

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

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]

19.

G. Szegö, Orthogonal Polynomials (Amer. Math. Soc., 1975).

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


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References

  1. 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.
  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]
  4. 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).
  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]
  6. A. E. Siegman, Lasers (University Science Books, 1986).
  7. S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998). [CrossRef]
  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]
  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. 38, 5272–5281 (1999). [CrossRef]
  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]
  13. V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).
  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 Optical Design and Engineering II , L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622G (2005).
  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]
  17. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966). [CrossRef] [PubMed]
  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]
  19. G. Szegö, Orthogonal Polynomials (Amer. Math. Soc., 1975).

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