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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 11 — May. 25, 2009
  • pp: 9347–9356
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Complete modal decomposition for optical fibers using CGH-based correlation filters

Thomas Kaiser, Daniel Flamm, Siegmund Schröter, and Michael Duparré  »View Author Affiliations


Optics Express, Vol. 17, Issue 11, pp. 9347-9356 (2009)
http://dx.doi.org/10.1364/OE.17.009347


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Abstract

The description of optical fields in terms of their eigenmodes is an intuitive approach for beam characterization. However, there is a lack of unambiguous, pure experimental methods in contrast to numerical phase-retrieval routines, mainly because of the difficulty to characterize the phase structure properly, e.g. if it contains singularities. This paper presents novel results for the complete modal decomposition of optical fields by using computer-generated holographic filters. The suitability of this method is proven by reconstructing various fields emerging from a weakly multi-mode fiber (V ≈ 5) with arbitrary mode contents. Advantages of this approach are its mathematical uniqueness and its experimental simplicity. The method constitutes a promising technique for real-time beam characterization, even for singular beam profiles.

© 2009 Optical Society of America

1. Introduction

Although the method itself has been known for more than a decade, serious technological limitations in producing the holograms with high resolution and accuracy as well as the lack of a convincing computational scheme for processing the correlation patterns made it difficult to obtain fast and reliable results. However, a series of improvements on these issues helped to overcome the limitations so that optical correlation analysis might become a more common method for optical field characterization in the future. It can complement established techniques (ISO 11146, WIGNER-distribution, interferometry, HARTMANN-SHACK analysis, etc.) in cases where they either require too much time, are too complicated or do not provide all the needed information.

The simple experimental realization of the method provides an easy and low-cost integration into industrial environments. However, despite this experimental simplicity, the problem of evaluating the occurring correlation patterns correctly has to be considered carefully, as will be shown in this paper.

2. Modal decomposition for step-index fiber modes

The spatial structure of a monochromatic light field is described by HELMHOLTZ’ equation

(Δ+k2)={E(r)H(r)}=0.
(1)

Starting from this equation, the spatial structure of light can be derived under different circumstances, e.g. for laser resonators (HERMITE-/LAGUERRE-GAUSSIAN modes), waveguides or waveguide arrays (supermodes). Our method was demonstrated earlier for the characterization of beams emerging a solid-state laser resonator [10

10. 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, Laurent Mazuray and Rolf Wartmann, eds., Proc. SPIE 5962, p. 59622G (2005). [CrossRef]

], where an incoherent superposition of modes takes place. In the present paper, it will be illustrated for the coherent mode superposition in weakly guiding optical fibers.

In such a case, Eq. (1) can be used in a scalar representation to calculate the (approximately) linear polarized (LP) modes, which yields the eigenvalue equation

[Δ+k02n2(u)β2]U(u)=0,
(2)

where u denotes the transverse coordinates. The spatial dependence of the field then reads as

U(u,z)=U(u)exp(iβz).
(3)

Any arbitrary propagating field U(u) that is normalized to unit power can be regarded as a superposition of eigensolutions ψn(u) of Eq. (2)

U(u)=n=1nmaxcnψn(u),
(4)

due to their orthonormal property

ψn,ψm=ψ∫∫2d2uψn*(u)ψm(u)=δnm.
(5)

cn=ρnexp(iϕn)=ψn,U=∫∫2d2uψn*(u)U(u),
(6)

and fulfill the relation

cn2=ρn2=1.
(7)

In the case of a step-index profile , the transverse mode structure can be factorized and reads as [13

13. A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, 1997).

]

ψ(u)=R(r)Φ(φ),
(8)

where

R(r)=C·{Jl(Ur/a)Jl(U)raKl(Wr/a)Kl(W)r>a,andΦ(φ)={cos()forevenmodessin()foroddmodes,
(8b)

respectively. The symbol a denotes the fiber core radius and C is chosen so that ψ(u) fulfills the normalization condition. The values for U and W can be calculated from the characteristic equations

UJn+1(U)Jn(U)=WKn+1(W)Kn(W)andU2+W2=V2,
(9a)

where

U=ak02ncore2β2
(9b)
W=aβ2k02nclad2,
(9c)

with V = ak 0(n 2 core - n 2 clad)1/2 being the well-known V-parameter of the fiber. Jl means the BESSEL-function of first kind and l th order, whereas Kl means the modified BESSEL-function of second kind and l th order.

3. Optical correlation analysis

It was already shown earlier that the integral relation Eq. (6) can be performed all-optically by using computer generated holograms [8

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

]. Since that method leads to diffraction patterns of cross correlation functions, we refer to this approach as “optical correlation analysis” [11

11. T. Kaiser, B. Lüdge, S. Schröter, D. Kauffmann, and M. Duparré, “Detection of mode conversion effects in passive LMA fibres by means of optical correlation analysis,” in Solid State Lasers and Amplifiers III, J. A. Terry, T. Graf, and H. Jeĺinková, eds., Proc. SPIE 6998, p. 69980J (2008). [CrossRef]

]. The transmission function of such a diffractive mode analyzing element (simply called “MODAN”) can be adapted for the intended measurement task. This approach has been investigated regarding questions of diffraction efficiency, reproducibility, signal-to-noise ratio (SNR) or different encoding schemes. For metrology purposes, it has been found most convenient to design the holograms as amplitude-masks in detour-phase representation. Throughout this paper, all elements were encoded using the method suggested by H.W. LEE [14

14. W. H. Lee, “Sampled Fourier Transform Hologram Generated by Computer,” Appl. Opt. 9, 639–643 (1970). [CrossRef] [PubMed]

] and fabricated via laser lithography.

If the transmission function of such an element is

T(u)=ψn*(u),
(10)

U(u)=nnmaxcnψn(u),
(11)

the field directly behind the MODAN will read as

W0(u)=ψn*(u)U(u).
(12)

The far field diffraction is realized experimentally by using a lens with focal length f. This yields [7

7. J. W. Goodman, Introduction to Fourier Optics ( McGraw-Hill Publishing Company, 1968).

]

Wf(u)=k02πifexp(2ik0f)∫∫2d2uW0(u)exp[ik0fuu]
=2πik0fexp(2ik0f)≗A0W˜0(k0fu),
(13)

with A 0 being a constant factor. By applying the convolution theorem

𝓕={f(x)g(x)}=[f˜*g˜](k)
(14)

to Eq. (13), one obtains

Wf(u)=A0∫∫2d2uψ˜n*(k0fu)U˜(k0f[uu]).
(15)

In the center of the focal plane, we find that according to Eq. (6)

Wf(0)=A0ψn,Ucn.
(16)

Multibranch operation and intermodal phase measurement The transmission function of the holographic element can be modified by using angular multiplexing so that more than one channel or interferometric superpositions of them can be realized simultaneously. If the modified transmission function reads as

T(u)=nnmaxψn*(u)exp(iVnu),
(17)

each of the convolution signals will be separated spatially due to different spatial carrier frequencies V n. In the MODAN plane, we obtain now

W0(u)=nψn*(u)U(u)exp(iVnu),
(18)

By using the FOURIER shifting theorem

𝓕{f(x)exp(iv0x)}=f˜(kv0),
(19)

we find

Wf(u)=A0n[∫∫2d2uψ˜n*(k0fu)U˜(k0f[uu])Vn],
(20)

i.e. the output pattern consists of a superposition of several cross correlations of the incident field U(u) and the mode field distributions ψn. Each of them is shifted transverse by V n, f/k 0. If the combination of V nf/ and k 0 is chosen properly, the contributions from different channels do not overlap due to their strong decay for ∣u∣ ≫ ∣V nf/k 0.

The signal of interest is the intensity at the points u n = V nf/k 0, for which we obtain

Wf(un=Vnfk0)2=A02ψn,W(u)2cn2=ρn2,
(21)

If two further branches per mode ψn are added with transmission functions reading as

Tncos(u)=12[ψ0*(u)+ψn*(u)]exp(iVncosu)
(22a)
Tnsin(u)=12[ψ0*(u)+n*(u)]exp(iVnsinu),
(22b)

the interferometric superposition is convoluted with the beam under investigation in the output focal plane. There, we find

Wf(un=Vncosfk0)2=12A0(c0+cn)2
=12A02[ρ02+ρn2+2ρ0ρncos(ϕnϕ0)],
(23a)
Wf(un=Vnsinfk0)2=12A0(c0+icn)2
=12A02[ρ02+ρn2+2ρ0ρnsin(ϕnϕ0)].
(23b)

This result illustrates, why two additional branches are necessary to obtain an unambiguous solution for the intermodal phase difference. ψ 0(u) denotes a specific mode, which phase is used as reference (most commonly the fundamental mode of the system). Therefore, ϕ 0 is a meaningless total phase and can be chosen as ϕ 0 = 0.

4. Experimental verification

Optical correlation analysis can be realized experimentally by a simple 2f-setup, which is a major advantage of this method. However, accurate adjustment of the optical components relative to each other is required.

The complete experimental setup is shown in Fig. 1. As test system, a weakly multi-mode step-index optical fiber was used.

Due to degeneracy of the modes regarding polarization, the 16 occurring LP modes result in just 8 different field distributions, which can be investigated separately for the two polarization states. For simplicity, we investigated only one polarization state to demonstrate the potential of the method to uniquely decompose a given field into its spatial eigenstates. The most important parameters of the used fiber are given in Table 1.

Fig. 1. Experimental setup for MODAN experiments. The setup is divided into an analytic branch containing the MODAN and a verification branch to check the results. MO1, MO2 - microscope objectives. L1, L2 - imaging lenses. FL - Fourier lens. The combination of MO2 and L1 causes a magnification of the near-field.

The HOE used for the measurements was a LEE–encoded binary element possessing a discretization of 512 × 512 pixels, which was etched into a chromium layer deposited on a SiO2 substrate. The size of a LEE–cell was 4 × 4μm while a minimum feature size of 600 nm was used for microlithography. The hologram contained (3 × 8 - 2) = 22 branches for the measurement of the intermodal weights ρn 2 and phases ϕn of the 8 spatial LP eigenmodes of the fiber.

In addition, four “adjustment channels” were added to include information about the geometry, defining an absolute coordinate system in the plane of the CCD camera. This is necessary to find the required points of interest, where the intensity is read out. For such a branch, the transmission function is

Tnadj(u)=exp(iVnadju).
(24)

Table 1. Most important properties of the test fiber used for verification of the MODAN method.

table-icon
View This Table
Fig. 2. Working principle of optical correlation analysis. The beam emerging the fiber (d) can equivalently be described by the measured correlation pattern (a) which is processed by our software (b) to obtain a reconstructed field (c).

Fig. 3. Modal decomposition of a beam with large amount of higher-order mode content. (a) – near-field of the investigated beam. (b) – reconstruction result. (c) modal excitation statistics for the field.
Fig. 4. Phase resolving character of the MODAN method. (a) – investigated beam. (b) – result of the reconstruction shown with isophase lines. A first order singularity occurs. (c) – 3D-representation of the reconstructed vortex phase front.

Fig. 5. Although the investigated field in (a) seems to be nearly perfect the fundamental mode, the excitation statistics (c) shows that ≈ 15% of the total power is propagating in higher-order modes and leads to a loss in beam quality [5].

5. Conclusions

References and links

1.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

2.

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

3.

A. P. Napartovich and D. V. Vysotsky, “Theory of spatial mode competition in a fiber amplifier,” Phys. Rev. A 76, 063801 (2007). [CrossRef]

4.

R. Schermer and J. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007). [CrossRef]

5.

H. Yoda, P. Polynkin, and M. Mansuripur, “Beam Quality Factor of Higher Order Modes in a Step-Index Fiber,” J. Lightwave Technol. 24, 1350 (2006). [CrossRef]

6.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete Modal Decomposition for Optical Waveguides,” Phys. Rev. Lett. 94, 143902 (2005). [CrossRef] [PubMed]

7.

J. W. Goodman, Introduction to Fourier Optics ( McGraw-Hill Publishing Company, 1968).

8.

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

9.

M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of DOEs,” in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich and S. H. Lee, eds., Proc. SPIE 3291, p. 104–114 (1998). [CrossRef]

10.

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, Laurent Mazuray and Rolf Wartmann, eds., Proc. SPIE 5962, p. 59622G (2005). [CrossRef]

11.

T. Kaiser, B. Lüdge, S. Schröter, D. Kauffmann, and M. Duparré, “Detection of mode conversion effects in passive LMA fibres by means of optical correlation analysis,” in Solid State Lasers and Amplifiers III, J. A. Terry, T. Graf, and H. Jeĺinková, eds., Proc. SPIE 6998, p. 69980J (2008). [CrossRef]

12.

S. J. van Enk and G. Nienhuis, “Photons in polychromatic rotating modes,” Phys. Rev. A 76, 053825 (2007). [CrossRef]

13.

A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, 1997).

14.

W. H. Lee, “Sampled Fourier Transform Hologram Generated by Computer,” Appl. Opt. 9, 639–643 (1970). [CrossRef] [PubMed]

15.

T. Kaiser, S. Schröter, and M. Duparré, “Modal decomposition in step-index fibers by optical correlation analysis,” in Laser Resonators and Beam Control XI, A. V. Kudryashov, A. H. Paxton, V. S. Ilchenko, and L. Aschke, eds., Proc. SPIE 7194, p. 719407 (2009). [CrossRef]

16.

O. A. Schmidt, T. Kaiser, B. Lüdge, S. Schröter, and M. Duparré, “Laser-beam characterization by means of modal decomposition versus M2 method,” in Laser Resonators and Beam Control XI, A. V. Kudryashov, A. H. Paxton, V. S. Ilchenko, and L. Aschke, eds., Proc. SPIE 7194, p. 71940C (2009). [CrossRef]

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(090.1970) Holography : Diffractive optics
(090.2890) Holography : Holographic optical elements
(120.3940) Instrumentation, measurement, and metrology : Metrology
(140.3295) Lasers and laser optics : Laser beam characterization

ToC Category:
Holography

History
Original Manuscript: March 30, 2009
Revised Manuscript: May 13, 2009
Manuscript Accepted: May 13, 2009
Published: May 19, 2009

Citation
Thomas Kaiser, Daniel Flamm, Siegmund Schröter, and Michael Duparré, "Complete modal decomposition for optical fibers using CGH-based correlation filters," Opt. Express 17, 9347-9356 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-11-9347


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References

  1. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).
  2. A. E. Siegman, Lasers (University Science Books, 1986).
  3. A. P. Napartovich and D. V. Vysotsky, "Theory of spatial mode competition in a fiber amplifier," Phys. Rev. A 76, 063801 (2007). [CrossRef]
  4. R. Schermer and J. Cole, "Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment," IEEE J. Quantum Electron. 43, 899-909 (2007). [CrossRef]
  5. H. Yoda, P. Polynkin, and M. Mansuripur, "Beam Quality Factor of Higher Order Modes in a Step-Index Fiber," J. Lightwave Technol. 24, 1350 (2006). [CrossRef]
  6. O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, "Complete Modal Decomposition for Optical Waveguides," Phys. Rev. Lett. 94, 143902 (2005). [CrossRef] [PubMed]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Publishing Company, 1968).
  8. V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).
  9. M. R. Duparre, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, "Generation, superposition, and separation of Gauss-Hermite modes by means of DOEs," in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich and S. H. Lee, eds., Proc. SPIE 3291, 104-114 (1998). [CrossRef]
  10. M. Duparre, B. Ludge and S. Schroter, "On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters," Proc. SPIE 5962, 59622G (2005). [CrossRef]
  11. T. Kaiser, B. Ludge, S. Schroter, D. Kauffmann, and M. Duparre, "Detection of mode conversion effects in passive LMA fibres by means of optical correlation analysis," Proc. SPIE 6998, 69980J (2008). [CrossRef]
  12. S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007). [CrossRef]
  13. A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, 1997).
  14. W. H. Lee, "Sampled Fourier Transform Hologram Generated by Computer," Appl. Opt. 9, 639-643 (1970). [CrossRef] [PubMed]
  15. T. Kaiser, S. Schroter, and M. Duparre, "Modal decomposition in step-index fibers by optical correlation analysis," in Laser Resonators and Beam Control XI, A. V. Kudryashov, A. H. Paxton, V. S. Ilchenko, and L. Aschke, eds., Proc. SPIE 7194, 719407 (2009). [CrossRef]
  16. O. A. Schmidt, T. Kaiser, B. Ludge, S. Schroter, and M. Duparre, "Laser-beam characterization by means of modal decomposition versus M2 method," Proc. SPIE 7194, 71940C (2009). [CrossRef]

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