## 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

For decades, the characterization of optical fields has become an important task, forming the basis for applications of lasers and optical components. A variety of measurement techniques has been developed, each providing information about different field properties [1]. Among the beam characterization approaches, direct modal description promises useful insights into the *physics* of the light field and an easy integration into the theoretical framework, e.g. laser mode competition and oscillations [2, 3], bend loss [4] and beam quality of optical fibers [5]. However, it has already been stated by SHAPIRA *et al*. [6] that a direct experimental evaluation of the modal content of an optical field remains difficult. For this reason, numerical techniques are used most commonly.

In FOURIER Optics, filters performing integral operations on optical fields, like the Van-DerLugt-correlator, have been known for a long time [7]. In this field, they are mainly used for pattern recognition purposes. It was demonstrated earlier that such holographic optical elements (HOEs) are able to provide qualitative and quantitative information about the modal content of laser beams [8, 9, 10], if a suitable transmission function can be found and implemented as computer-generated hologram (CGH). Such an approach allows to analyze the modal behavior of the radiation in *real-time*, e.g. just when a perturbation of the waveguiding device (fiber, resonator, etc.) takes place [11].

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.

In this paper, the advantages and limitations of laser beam characterization by means of HOEs are presented as well as experimental verifications. As a test system, we investigated the mode content of a weakly multi-mode (*V* ≈ 5) LMA fiber, operated at *λ* = 1064nm. To change the modal content, we intentionally applied different perturbations to the system. The possibility to reconstruct fields with various mode contents is proven as well as the ability to resolve phase singularities correctly, which is particularly useful in applications involving the orbital momentum of light [12].

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

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], 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

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

Any arbitrary propagating field *U*(**u**) that is normalized to unit power can be regarded as a superposition of eigensolutions *ψ _{n}*(

**u**) of Eq. (2)

due to their orthonormal property

The modal coefficients *c _{n}* are given by

and fulfill the relation

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

where

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

where

with *V* = *ak*
_{0}(*n*
^{2}
_{core} - *n*
^{2}
_{clad})^{1/2} being the well-known *V*-parameter of the fiber. *J _{l}* means the BESSEL-function of first kind and

*l*

^{th}order, whereas

*K*means the modified BESSEL-function of second kind and

_{l}*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]. Since that method leads to diffraction patterns of cross correlation functions, we refer to this approach as “optical correlation analysis” [11]. 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] and fabricated via laser lithography.

If the transmission function of such an element is

and the illuminating field has a modal composition according to Eq. (4)

the field directly behind the MODAN will read as

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

$$=\underset{{\mathrm{\circeq A}}_{0}}{\underbrace{-2\mathrm{\pi i}\genfrac{}{}{0.1ex}{}{{k}_{0}}{f}\mathrm{exp}\left(2i{k}_{0}f\right)}}{\tilde{W}}_{0}\left(\genfrac{}{}{0.1ex}{}{{k}_{0}}{f}\mathbf{u}\right),$$

with *A*
_{0} being a constant factor. By applying the convolution theorem

to Eq. (13), one obtains

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

Thus, a hologram with the transmission function Eq. (10) produces a diffraction pattern containing information about the modal coefficient *c _{n}*. A detector, placed in the center of the output focal plane, measures a signal which is proportional to ∣

*c*∣

_{n}^{2}=

*ρ*

_{n}^{2}.

**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

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

By using the FOURIER shifting theorem

we find

i.e. the output pattern consists of a superposition of several cross correlations of the incident field *U*(**u**) and the mode field distributions *ψ _{n}*. Each of them is shifted transverse by

**V**

*/*

_{n}, f*k*

_{0}. If the combination of

**V**

*/ and*

_{n}f*k*

_{0}is chosen properly, the contributions from different channels do not overlap due to their strong decay for ∣

**u**∣ ≫ ∣

**V**

_{n}∣

*f*/

*k*

_{0}.

The signal of interest is the intensity at the points **u**
* _{n}* =

**V**

*/*

_{n}f*k*

_{0}, for which we obtain

which is the modal weight of the mode *ψ _{n}* in the investigated beam. We refer to the signals encoded on different spatial carrier frequencies as the “branches” of the correlation pattern. Note that the information about

*all*modes is obtained

*simultaneously*by an experimentally simple optical 2

*f*-setup.

This shows the potential of the method for real-time beam characterization by means of modal decomposition. Changes of the modal content of a light beam can be investigated instantaneously, e.g. just when perturbations are applied to the system, such as coiling the fiber or changing the alignment to the coupled laser source.

With a MODAN having a transmission function given by Eq. (17), the modal power distribution of the light field under investigation can be measured. With this information about all *ρ _{n}*, a field reconstruction is possible by applying numerical phase retrieval algorithms, to obtain the correct intermodal phases [15]. However, the phases can also be

*measured directly*by introducing an interferometric superposition of channels within a branch of the MODAN. This is of particular importance, since many laser beam parameters like the beam propagation ratio M

^{2}and the beam pointing stability depend strongly on the intermodal phase [5].

If two further branches per mode *ψ _{n}* are added with transmission functions reading as

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

$$\phantom{\rule[-0ex]{3em}{0ex}}=\genfrac{}{}{0.1ex}{}{1}{2}{\mid {A}_{0}\mid}^{2}\left[{\rho}_{0}^{2}+{\rho}_{n}^{2}+2{\rho}_{0}{\rho}_{n}\mathrm{cos}\left({\varphi}_{n}-{\varphi}_{0}\right)\right],$$

$$\phantom{\rule[-0ex]{3em}{0ex}}=\genfrac{}{}{0.1ex}{}{1}{2}{\mid {A}_{0}\mid}^{2}\left[{\rho}_{0}^{2}+{\rho}_{n}^{2}+2{\rho}_{0}{\rho}_{n}\mathrm{sin}\left({\varphi}_{n}-{\varphi}_{0}\right)\right].$$

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.

In summary, (3*n* - 2) branches must be comprised in the hologram to obtain the complete modal decomposition for a system with *n* modes on a purely experimental basis. The number of branches would be limited basically by the SNR of angular multiplexing. In our case, the practical limit was given by the dimensions and resolution of the CCD chip. Here, one needs to find an optimum where the extends of the correlation signals are not to small but still separated from each other. So far, we have investigated systems with up to 26 angle-multiplexed branches in one hologram, which allows the investigation of a weakly multi-modal fiber with *V* ≈ 5, as described in this paper.

## 4. Experimental verification

Optical correlation analysis can be realized experimentally by a simple 2*f*-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.

The LMA fiber was excited by a Nd:YAG laser (*λ*
_{0} = 1064 nm), using free-space coupling between the incident beam and the fiber. Behind the end facet, a polarizer was used to select the intended polarization state. An imaging system consisting of a suitable microscope objective MO2 and a lens L1 magnified the imaged near-field on the HOE by a factor of 37.6. The diffracted light in the first diffraction order passed a further lens (FL) to carry out the FOURIER transform. After the image was recorded by a CCD camera, it was evaluated by a specific software created at our institute. The modal content was changed by slightly misaligning the adjustment between laser and fiber to obtain a variety of different multimodal excitations. In a second branch behind a beam splitter, the near- and far-field was recorded by a second CCD camera.

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 × 4*μ*m 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 *ρ _{n}*

^{2}and phases

*ϕ*of the 8 spatial LP eigenmodes of the fiber.

_{n}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

Figure 2 illustrates the working principle of the MODAN method. The correlation pattern recorded by a CCD camera is analyzed by our software. Figure 2(b) shows the basic assembly of the signals in the pattern. The basic idea is that the known spatial carrier frequencies **V**
_{n},**V**
^{cos}
_{n} and **V**
^{sin}
_{n} define the points of interest in the FOURIER domain. The remaining problem is to find the absolute coordinate system, where these coordinates hold. This can be achieved by measuring the first order moments of the adjustment channels. The four points define a rectangular region. Within this region, the location of the points of interest relative to the corners is also known from the spatial frequencies and so, the coordinate system is fixed. There, the intensity has to be read out to obtain the information about the modal coefficients *c _{n}* =

*ρ*exp(

_{n}*iϕ*). With this information, it is possible to reconstruct the field emerging the fiber, including amplitude and phase distribution. The result of the reconstruction shown in Fig. 2(c) can be compared to the directly measured near-field intensity in Fig. 2(d) to verify the correct functionality of the system.

_{n}An exemplary result is shown in Fig. 3. In this case, a high relative percentage of power propagated in higher-order modes, as indicated in Fig. 3(c). The result of reconstruction is in very good agreement with the measured near-field. Deviations occur mainly due to errors in finding the correct coordinates of the correlation signals. We are working on a further improvement of the algorithm. Using the MODAN method, a detailed analysis of the modal weights *ρ _{n}*

^{2}and intermodal phases

*ϕ*in real-time becomes possible.

_{n}It should be emphasized that the method can resolve even singular beam profiles in an unambiguous way. It is known that higher-order spatial modes relate to the orbital momentum of light [12]. Using optical correlation analysis, it is possible to measure the occurrence and order of phase singularities in the beam. This is due to the fact that it directly involves the structures which are responsible for their appearance – the higher-order spatial modes itself. They possess the singularities intrinsically. A method which is able to perform an unambiguous modal decomposition therefore automatically resolves any singularities which might occur in the investigated field. For demonstration, Fig. 4 shows another reconstruction result, this time printed with additional isophase lines. The intersection of these lines at a point with vanishing intensity clearly proofs the occurrence of a 1^{st}-order phase singularity. The vortex phase front is shown in 3D-representation in Fig. 4(c).

For photonic structures, an operation in fundamental mode is often desired for reasons of beam quality, even in cases where the device itself is multi-modal in principle. Here, an accurate measurement of the higher-order mode content is essential. Figure 5(a) shows a measured beam profile which could be expected to be the fundamental mode of the fiber due to its radial symmetry. In fact, the modal excitation statistics in Fig. 5(c) shows that ≈ 15% of the total power is propagating in higher-order modes. Only the radial symmetric LP_{02} mode and an equal excitation of LP^{e}
_{11} and LP^{o}
_{11} account for the beam profile, which explains the radial symmetry of the beam itself. A comparison between optical correlation analysis and standard *M*
^{2} method is given in Ref. [16].

## 5. Conclusions

In this paper we presented optical correlation analysis as a method to obtain an unambiguous experimental solution to the modal decomposition problem. The benefits of this approach are its experimental simplicity and the capability to perform measurements in real-time, without time-consuming numerical analysis.

The method uses a simple FOURIER-optical setup. An HOE with a suitable transmission function is designed to perform the integral relation Eq. (6) all-optically. The modal weights *ρ _{n}*

^{2}and the intermodal phases

*ϕ*can be measured simultaneously in different angle-multiplexed branches of the correlation pattern. Naturally, one would choose the fundamental LP

_{n}_{01}mode as reference for the phase measurement. However, a problem might occur if this reference mode is not excited sufficiently, although this situation is unlikely. By reviewing Eq. (23), it is obvious that the intermodal phase becomes immeasurable in this case. However, this can be overcome by introducing a second reference mode on additional branches of the hologram.

The obtained reconstruction results are in very good agreement with the measured near-field distributions. This confirms the capability of performing accurate measurements of the modal content, even if the phase distribution contains singularities. The method was proven to work well also in cases with a large amount of higher-order mode content. In addition, it is possible to measure the higher-order mode suppression in systems with a high fraction of power in the fundamental mode.

The results presented here can help to improve the characterization of multi-modal beams. They are particularly useful for the investigation of passive as well as active devices, since parameters like beam quality, beam pointing stability or phase distribution can be derived directly from the measured modal content.

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