## Abstract

We propose an improvement of the interferometric method used up to now to measure the chromatic dispersion in single mode optical fibers, which enables dispersion measurements in higher-order modes over a wide spectral range. To selectively excite a specific mode, a spatial light modulator was used in the reflective configuration to generate an appropriate phase distribution across an input supercontinuum beam. We demonstrate the feasibility of the proposed approach using chromatic dispersion measurements of the six lowest order spatial modes supported by an optical fiber in the spectral range from 450 to 1600 nm. Moreover, we present the results of numerical simulations that confirm sufficient selectivity of higher-order mode excitation.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Selective excitation of higher-order modes (HOMs) in optical fibers is required in numerous applications, including multiplexing techniques based on spatial modes [1,2] and intermodal nonlinear phenomena [3], which have been intensively researched in recent years. HOMs are also used in specific fiber-based devices, such as dispersion compensators [4–6], spatial polarization filters [7], high-power lasers [8], and sensors [9]. The chromatic dispersion of HOMs over a wide spectral range is of particular importance for the control of various nonlinear phenomena [3,10–15], particularly supercontinuum generation [10], intermodal four-wave mixing [11,12], soliton propagation [13], and intermodal cross-phase modulation [14].

Spectrally broad and accurate measurements of chromatic dispersion in the fundamental mode are typically conducted using different modifications of white-light interferometric methods, such as zero-order interference fringe tracking [16] or phase retrieval from spectral interference patterns [17–21]. These approaches can potentially be adopted for measurements of chromatic dispersion in HOMs if the required modes are excited over a wide spectral range sequentially with sufficient selectivity, as all modifications of the broadband interferometric method require well-resolved spectral interference patterns at the interferometer output.

Previous literature has reported numerous methods for the excitation of HOMs in optical fibers that were developed for specific applications. These include computer-generated phase masks [22] or phase masks imprinted on the fiber end-face [23,24], modal couplers [25–27], mechanical stress [28,29], fiber gratings [6,30], side coupling [31], specially designed microstructure in the fiber [32] and excitation with offset [10,33,34] or tilted beams [34,35]. However, all of these methods have limitations that exclude them from applications in white-light interferometric dispersion measurements. The most critical are the impossibility of dynamic switching between different HOMs and lack of wavelength tunability. Therefore, chromatic dispersion measurements in HOMs that have been reported thus far were performed usually in narrow spectral ranges [33,36–39].

Liquid crystal spatial light modulators (SLMs) offer the possibility of the selective excitation of HOMs combined with fast switching between different modes, which has been successfully exploited in several applications [40–46]. Typically, a light wavefront diffracted in the first order is shaped by the SLM using various dynamic holographic techniques, thereby enabling effective coupling to the desired fiber mode. However, because the diffraction angle is wavelength sensitive, it is not possible to utilize an SLM as a dynamic hologram for the broadband excitation of HOMs which is required in white-light interferometric dispersion measurements.

In this study, we propose a method to mitigate this limitation by operating the SLM as a reflective-type dynamic phase filter. We demonstrate that for the optimal phase distribution introduced by the SLM, such an approach enables the successive excitation of the desired HOMs in the spectral range exceeding 200 nm in the visible region and even more in the infrared region. Therefore, this method can be effectively applied to white-light interferometric dispersion measurements in HOMs, which, up to our knowledge, has not been demonstrated thus far. We present example measurements of chromatic dispersion in all HOMs supported by a Corning SMF-28e optical fiber in the spectral range from 450 to 1600 nm. Moreover, to confirm sufficient selectivity of the HOMs excitation with the proposed method, we present the results of numerical simulations of the excitation efficiency for all possible combinations of illumination beams and fiber modes, taking into account typical alignment errors.

## 2. Selective HOM excitation and chromatic dispersion measurements

To excite a specific HOM, we appropriately modulated the phase distribution across an input supercontinuum beam using the SLM as a dynamic phase mask in a reflective configuration. A scheme of the experimental setup for HOM chromatic dispersion measurements is shown in Fig. 1. A collimated linearly polarized (electric vector parallel to the extraordinary axes of the nematic liquid crystal molecules) supercontinuum beam was expanded to a diameter of 6 mm by a pair of microscope objectives (40× and 10×) and subsequently directed onto the SLM at a normal angle. The SLM surface was divided into azimuthally and radially distributed sectors that introduced alternate phase shifts of 0 or π radians, similar to the target mode. The back-reflected phase-modulated beam was divided into the measurement and reference arms of a Mach–Zehnder interferometer by beam splitters BS1 and BS2, respectively. In the measurement arm, the beam was coupled/decoupled to/from the respective mode of the fiber under test by a pair of microscope objectives (20×). In the reference arm (marked with the red dashed frame in Fig. 1), we used a pair of identical microscope objectives to balance the optical path delays introduced by these elements in both interferometer arms. Mirrors M2 and M3, placed on a moving stage (marked by a dashed black frame in Fig. 1), were used to vary the length of the reference arm until the optical path delay that was introduced by the tested fiber was compensated at successive wavelengths. Finally, the reference and measurement beams were recombined by beam splitter BS3 and directed to spectrometer S and camera C.

To generate the phase masks we used a Holoeye PLUTO 020 VIS SLM capable of phase shift modulation up to 6π at *λ* = 632 nm and more than 3π at *λ* = 1064 nm. The required phase shift distribution was obtained by applying the appropriate voltage to each SLM liquid crystal cell with an 8-bit resolution. The dependence between the applied voltage Δ*U* and the introduced phase shift Δ*φ* was first calibrated experimentally using an optical vortex with a topological charge of *q* = 1 as a phase shift marker. In this experiment, the intensity distribution across the beam that is reflected from the SLM was monitored using a camera placed directly after beam splitter BS1. The spiral phase mask responsible for optical vortex generation was displayed by the SLM. If the range of the applied voltage Δ*U* to the SLM cells caused a 2π phase shift, a symmetric optical vortex with a topological charge of *q* = 1 was observed on the camera. Otherwise, if the voltage range Δ*U* was too small or too large and did not match the 2π phase change, the topological charge *q* became fractional, which led to the creation of a radial phase discontinuity [47], as shown in Figs. 2(a) and (c), respectively. This procedure was repeated for different wavelengths, which were selected by narrowband interference filters (FWHM < 20 nm) in the supercontinuum beam. Figure 3 shows the measured (circles) and linearly approximated (line) spectral dependence of the voltage Δ*U*_{2π} that generates the 2π phase shift. We estimated the precision of Δ*U*_{2π} to be approximately 10% at every wavelength. The spectral range of measurement was limited to 400–1100 nm by the sensitivity of the camera. To generate the 0–π spatial phase distribution that is required to excite a specific HOM in the tested fiber, the voltage applied to the appropriate sectors of the SLM matrix was varied by Δ*U*_{2π}/2.

Assuming that the supercontinuum beam incident on the SLM is Gaussian and the phase shift pattern generated on the SLM corresponds to the fiber mode with an amplitude distribution of $E_{lm}^f(r,\theta )$, the amplitude $E_{lm}^i(r,\theta )$ that is back-reflected and imaged on the fiber input facet is represented as:

*w*is the Gaussian beam waist, and δ

_{0}*φ*is the wavelength-dependent phase error related to the fact that the applied voltage Δ

*U*

_{2π}/2 introduces a phase shift equal to π only at one wavelength denoted by

*λ*

_{opt}. The phase error

*δφ*related to the wavelength deviation δ

*λ*from optimality is given by:

The term $\pi [{{\mathop{\rm sgn}} [{E_{lm}^f(r,\theta )} ]+ 1} ]/2$ represents the phase distribution on the SLM that enables optimal excitation of the fiber mode $E_{lm}^f(r,\theta )$ at *λ _{opt}*. The phase patterns generated in this way, shown in Fig. 4 (right column), are responsible for the excitation of specific HOMs in the investigated fiber. To prove that the proposed method can be effectively used for the selective excitation of HOMs, we show all modes excited in the Corning SMF-28e fiber in two scenarios in Fig. 4 (left and middle columns). First, the HOMs were excited with the phase masks optimal for

*λ*

_{opt}= 650 nm and registered at the fiber output without any spectral filter. Second, the same modes were excited with the phase masks optimal for

*λ*

_{opt}= 475 nm and registered with the 475 nm filter. The spectral range of operation of the color camera used in this experiment was 400–800 nm, whereas the supercontinuum (NKT Photonics SuperK Compact, pulse duration t = 2 ns, repetition rate R > 2.5 kHz) spectrum extended from 450 to more than 1600 nm. The intensity patterns characteristic for the LP

_{11}, LP

_{21}, and LP

_{02}modes, which are clearly seen in the polychromatic light, indirectly prove that they are excited over a wide spectral range. The other HOMs (LP

_{12}and LP

_{31}) are only visible in the green light, owing to the cut-off effect. The measured cut-off wavelengths for each mode are indicated below the mode order. The cut-off wavelength of the LP

_{41}mode was 490 nm; therefore, this mode was not visible in polychromatic light with optimal excitation at 650 nm.

To confirm that the proposed excitation method is only minimally wavelength sensitive, in Fig. 5 we show the intensity distributions registered at the fiber output with different interference filters for the LP_{11} and LP_{02} modes, which were excited in the full visible range with the phase masks displayed on the SLM optimized for *λ*_{opt} = 650 nm. The registered photographs prove that both modes are selectively excited in the range of 550–750 nm. Outside this range, the modes begin to fade, owing to the increasing phase error δ*φ*, which is related to an excessively large deviation from the optimal wavelength. The LP_{11} mode is still excited more efficiently than the fundamental mode up to 850 nm. However, LP_{02} cuts off at approximately 800 nm; therefore, it is only visible below this wavelength. This experiment proves that the proposed method enables the excitation of HOMs in optical fiber with sufficient selectivity to conduct white-light dispersion measurements in the spectral range exceeding 200 nm. The small wavelength sensitivity of the proposed excitation method is also confirmed by the results of detailed numerical simulations presented in the following section. The optimal wavelength is shifted step-by-step by changing the steering voltage applied to the SLM cells to cover a broader measurement range.

To demonstrate the usability of the proposed HOM excitation method in chromatic dispersion measurements, we performed measurements on a 116 cm long Corning SMF-28e fiber, which supports the LP_{01}, LP_{11}, LP_{21}, LP_{02}, LP_{31}, and LP_{12} modes. The measurements were conducted in the range from 450 nm up to the cut-off wavelengths of the respective modes and up to 1600 nm for the fundamental mode. To cover so wide spectral range we used two Ocean Optics spectrometers: USB4000 (200-1100 nm, resolution of 0.2 nm) and NIR512 (900-1700nm, resolution of 2.6 nm). The dispersion measurement for the LP_{41} mode using the spectral interferometry method was not possible, owing to a very short cut-off wavelength of this mode (*λ*_{c} = 490 nm). In Fig. 6, we show the example interference patterns for the LP_{01}, LP_{11}, LP_{02}, and LP_{21} modes, which are selectively excited by switching the phase mask generated on the SLM (optimized for *λ*_{opt} = 700 nm for each mode) with an unchanged length of the reference arm. To maximize the contrast of the interference pattern associated with the excited HOM, the position of the fiber output was adjusted until the phase distributions in the measurement and the phase-modulated reference beams were azimuthally and laterally aligned. To suppress the interference patterns generated by other modes that were residually excited in the tested fiber, both beams were slightly defocused on the spectrometer slit, and only a part of the two beams corresponding to the maximum intensity in the interrogated HOM was collected and spectrally analyzed. Variable neutral density filters in the reference and measurement arms (not shown in Fig. 1) were tuned for each mode to match the intensities of the two interfering beams and to maximize the contrast of the interference pattern. This alignment was not changed during wavelength scanning. Such a procedure resulted in the appearance of spectrograms, shown in Fig. 6, with visible patterns associated with one spatial mode selected by the phase mask. In case of the LP_{02} mode, a weak zero-order interference fringe associated with the fundamental mode can be observed next to the strong pattern corresponding to the LP_{02} mode (small oscillations at approximately 744 nm, yellow line in Fig. 6). The interferograms registered for other modes are almost undisturbed by parasitic modes, which makes it possible to determine the spectral position of the zero-order interference fringe for every excited mode with a precision of approximately 1 nm. By varying the length of the reference arm and switching the phase masks generated on the SLM, we could determine the spectral dependence of the optical path delay *G*_{lm}(λ) that was introduced by the tested fiber for every spatial mode with precision to a constant value. Finally, by numerically differentiating *G*_{lm}(λ) (approximated by a modified Cauchy polynomial [21]) with respect to wavelength, we determined the chromatic dispersion for every spatial mode according to the following formula:

*c*is the velocity of light, and

*L*is the fiber length. The optical path delay and chromatic dispersion curves measured in this way for all HOMs supported by the Corning SMF-28e fiber in the spectral range from 450 to 1600 nm are shown in Figs. 7 and 8, respectively.

These results prove that the proposed HOM excitation method is sufficiently selective and can effectively be exploited for broadband interferometric measurements of chromatic dispersion in optical fibers that support multiple spatial modes.

The measured values of dispersion for the fundamental mode were compared with the data given by the manufacturer for dispersion slope *S _{0}* = 0.089 ps/(km×nm2) and zero dispersion wavelength

*λ*= 1322 nm [48]. The differences between the measured and calculated values do not exceed 10 ps/(km×nm) in the spectral range from 500 nm to 1600 nm. The highest discrepancy occurs at the edges of the spectral window where the results are the most affected by the polynomial approximation.

_{0}## 3. Simulations of HOM excitation selectivity

In this section, we present the results of numerical simulations showing that the proposed HOM excitation method has a relatively small sensitivity to a deviation in the operation wavelength from the optimal value and to other misalignments. These include the lateral offset and tilt of the incident beam with respect to the core facet, which results in the appearance of the non-zero phase error δ*φ* in Eq. (1). The field distributions $E_{lm}^f(r,\theta )$ in the excited HOMs were calculated numerically, assuming the geometrical parameters of the Corning SMF-28e fiber used in the experiments, i.e., the radius of the fiber core and cladding *a*_{core} = 4.1 μm and *a*_{clad} = 62.5 μm, respectively, and the GeO_{2} dopant concentration in the core *d* = 3.8 mol%. The spectral dependence of the refractive indices in the core *n*_{core} and cladding *n*_{clad} was calculated using Sellmeier’s formula [49]. Because the field distributions in the illuminating beams are not identical to the fiber modes, one may expect the residual excitation of undesired modes $E_{pq}^f(r,\theta )$ for each illuminating beam $E_{lm}^i(r,\theta )$, which can be quantitatively represented by the overlap integrals:

*A*.

To analyze the cross-excitation effect, we first determined the optimal values of the illuminating Gaussian beam waist *w _{0}* for each excited mode for

*λ*= 500 nm by maximizing the overlap integral

_{opt}*η*

_{lm/lm}with respect to

*w*

_{0}. We assumed that the modes are excited with a perfectly aligned incident beam $E_{lm}^i(r,\theta )$ with waist

*w*and a phase shift pattern both optimal at

_{0}*λ*

_{opt}= 500 nm. We then calculated the

*η*

_{lm/pq}coefficients for all combinations of the illuminating beams and fiber modes at

*λ*

_{opt}. In Table 1, we show only those values of the calculated coefficients

*η*

_{lm/pq}that are greater than 10

^{−2}. The obtained results prove that a noticeable cross-excitation appears only between the LP

_{11}, LP

_{12}, and LP

_{31}beams/modes and between the beams/modes from the LP

_{0m}group with different radial modal numbers

*m*, however, in all cases, the maximum excitation of the undesired modes is below 0.1. None of the other combinations of the illuminating beams and fiber modes cause the cross-excitation of undesired modes. It is worth mentioning that the excitation efficiencies measured at 475 nm were around 30% smaller than the calculated values for both the fundamental mode and HOMs.

We then analyzed the effect of the lateral shift of the illuminating beam on the excitation efficiency for all possible combinations of the illuminating beams and fiber modes at a fixed wavelength of *λ _{opt}* = 500 nm. In this case, the excitation coefficients

*η*

_{lm/pq}are dependent on the amount and direction of the offset δ

*r*of the illuminating beam with respect to the core center. The simulations of

*η*

_{lm/pq}were performed for different directions of

*δr*, assuming that the illuminating beams were optimal for

*λ*

_{opt}= 500 nm, and the offset was |δ

*r*| = 0.5 μm. In Table 2, we show only the maximum values of the coefficients

*η*

_{lm/pq}corresponding to the worst-case scenario. The strongest cross-excitation occurs between the LP

_{02}and LP

_{12}illuminating beam/fiber modes, for which the maximum value of the η

_{02/12}coefficient reached 0.08 for the offset along the y-axis (

*θ*= 90°).

For all combinations of the illuminating beams/fiber modes, the cross-excitation coefficients were lower than 0.1. This indicates that the targeted modes are always predominantly excited, and the 0.5 μm offset does not critically disturb the white-light interferometric measurements.

We also numerically analyzed the effect of the tilt of the illuminating beams on the excitation coefficients *η*_{lm/pq}. In these simulations, we considered a tilt angle α in the range from 0° to 5° and different azimuthal orientations of the tilt plane θ from 0° to 90°, where *θ* = 0° corresponds to the tilt plane defined by the fiber symmetry axis and the x-axis of the Cartesian coordinate system. Example excitation coefficients between the $E_{11}^i(r,\theta )$ illuminating beam and the LP_{11} and LP_{21} fiber modes are shown in Fig. 9. These results prove that the excitation coefficients are very sensitive to the wavefront tilt. As shown in Fig. 8(a), a tilt of the illuminating beam $E_{11}^i(r,\theta )$ by *α* = 2° causes a significant drop in the coupling efficiency from 70% to 30%–45%, depending on the orientation of the tilt plane. Moreover, for the tilt angle *α* = 3°, the excitation coefficient for the undesired mode LP_{21} becomes greater than that for the targeted mode LP_{11}. In Table 3, we show the maximum values of the cross-excitation coefficients for all possible combinations of the fiber modes and tilted illuminating beams (*λ*_{opt} = 500 nm), which were calculated using the tilt direction for a fixed tilt angle *α* = 1° and *λ*_{opt} = 500 nm. These results emphasize the importance of proper fiber cleaving and the precise angular adjustment of the illuminating beam for the selectivity of HOM excitation. It should be noted that the cross-excitation coefficient for the LP_{02} and LP_{12} illuminating beams/fiber modes takes significant values for a relatively small tilt angle *α* = 1°. However, in the experiment it is possible to reduce the effect of parasitic undesired modes on the measurement precision by collecting light only from these regions in the output beam where the intensity distribution in the targeted modes takes maximum values.

Because interferometric dispersion measurements require broadband light, it is necessary that the targeted HOM is excited with sufficient selectivity over a wide spectral range for every phase mask displayed on the SLM, which is optimal at only one wavelength. The experimental observations reported in Section 2 demonstrated the selective excitation of HOMs in the spectral range of approximately 200 nm without the need to correct the voltage applied to the SLM cells. Additionally, we have analyzed this problem numerically. To evaluate the effect of the wavelength change by δ*λ* from the optimal value *λ*_{opt}, we calculated the cross-excitation coefficients *η*_{lm/pq} for all combinations of illuminating beams/fiber modes with respect to the deviation of the phase modulation depth δ*φ* from the optimal value equal to π, which is given by Eq. (2). The simulations show that cross-excitation is always the strongest in the fundamental mode LP_{01}, regardless of the order of the targeted mode. In Fig. 10, we show the example results for the illuminating beam $E_{12}^i(r,\theta )$ optimized for *λ*_{opt} = 500 nm and all fiber modes. The cross-excitation coefficient of the fundamental mode reaches the experimentally tolerable level of approximately 0.1 for δ*φ* = π/4, which corresponds to a wavelength deviation from *λ*_{opt} by approximately ±100 nm in the visible spectral range. Moreover, the spectral window of our excitation method does not vary significantly with the order of the targeted mode, which is in good agreement with the experimental observations.

The simulations show that the cross-excitation caused by the phase mismatch is independent of the mode order. However, in case of highly multimode fiber, the precision of spatial and angular alignment of the exciting beam will become more critical. One can expect that the offset well below 1 μm and tilt much smaller than 1° will cause strong cross-excitation between different HOMs making the interferometric measurements more difficult or impossible.

## 4. Conclusions

We proposed and experimentally verified an effective method for HOM excitation in an optical fiber employing a liquid crystal SLM as a reconfigurable reflective-type phase mask. In contrast to the SLM operating in a diffractive configuration or the resonant intermodal coupling that has been reported in previous literature, the method based on a reconfigurable phase mask can be applied to a broadband excitation of the targeted modes, which covers the spectral range from 0.8λ_{opt} to 1.2λ_{opt}. Owing to the non-orthogonality of the illuminating beams and the fiber modes, the proposed method is not perfectly selective, i.e., in addition to the targeted mode defined by the phase distribution in the illuminating beam, other undesired modes are residually excited. The cross-excitation effect can be further increased by fiber misalignment, such as lateral shift and tilt, which were analyzed numerically. None of the analyzed cross-excitations to undesired modes were significant in the dispersion measurements. Particularly, the parasitic interference patterns that are visible in the output spectrum can be further suppressed by the spatial filtering of the output beam on the spectrometer slit.

The developed excitation method enabled interferometric dispersion measurements of HOMs over a wide spectral range, which is important for intermodal nonlinear phenomena and spatial division multiplexing based on HOMs. The feasibility of our approach was confirmed by measurements of chromatic dispersion of the LP_{01}, LP_{11}, LP_{21}, LP_{02}, LP_{31}, and LP_{12} modes in a Corning SMF-28e fiber in the spectral range from 450 nm to the cut-off wavelength of each HOM and up to 1600 nm for the fundamental mode. Because wavelength sweeping only requires a change of the voltage applied to the SLM, the method can be effectively applied in automated setups.

## Funding

Narodowe Centrum Nauki (Maestro 8, DEC-016/22/A/ST7/00089); Smart Growth Operational Programme 2014-2020 (POIR.04.02.00-00-B003/18-00).

## Disclosures

The authors declare no conflicts of interest.

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