Quantitative mode quality characterization of fibers with extremely large mode areas by matched white-light interferometry

Fanting Kong; Guancheng Gu; Thomas W. Hawkins; Joshua Parsons; Maxwell Jones; Christopher Dunn; Monica T. Kalichevsky-Dong; Stephen P. Palese; Eric Cheung; Liang Dong

doi:10.1364/OE.22.014657

1. Introduction

Despite significant developments in fiber lasers in recent years, there are still great needs to scale powers in both CW and pulsed fiber lasers for use in a wide range of industrial, scientific and defense applications. In order to overcome the limitations in power scaling caused by optical nonlinear effects, such as stimulated Brillouin scattering (SBS), stimulated Raman scattering (SRS), self-phase modulation (SPM) and four-wave-mixing (FWM), a large number of approaches have been studied to increase the fiber core diameter while mitigating the waveguide’s tendency to support an increasing number of modes at large core diameters. One of the most effective methods is to exploit mode-dependent loss in leaky waveguides. Some recent designs such as leakage channel fibers (LCF) [1–3], chirally-coupled-core fibers [4] and all-solid photonic bandgap fibers (ASPBF) [5–7] have built-in mode-dependent losses in the fiber designs. Recently, it has been reported that mode instability can develop in fiber amplifiers supporting few modes due to nonlinear interaction between mode interference and quantum defect heating [8–12]. One method to mitigate this is to make higher-order modes (HOM) significantly more leaky. These highly leaky HOMs are no longer confined to the core and have, therefore, significantly reduced overlap with the active area of the fiber, minimizing the impact of quantum heating.

LCFs are particularly useful for providing high differential losses among modes, benefitting from the discontinuity of core-cladding boundary which makes all modes leaky, allowing the fiber to be engineered to have high transmission loss for all HOMs while maintaining negligible loss of fundamental mode (FM). In addition, the cladding features can be designed to cause resonance coupling between the lowest-loss HOM and the cladding, pulling the HOM further out into the cladding and leading to much improved HOM suppression. These LCFs are referred to as resonantly-enhanced leakage channel fibers (Re-LCF). The performance of Re-LCF with ~50µm core diameters was recently reported [13,14]. Moreover, a more recent report demonstrates that the design of the fiber outer boundaries can have a significant impact on leaky mode losses in a LCF [15]. A deviation from circular fiber boundary is very effective in mitigating coherent reflection from the outer boundary in a fiber, leading to much higher HOM losses, even in a fiber coated with low-index polymer.

In order to verify HOM suppression, precisely quantitative mode characterization of a fiber is required. Several methods have been demonstrated recently, including spatially and spectrally resolved (S²) imaging [16] and cross-correlated (C²) imaging [17], which have demonstrated mode characterization of large-mode-area fibers with up to 50 µm core diameter. As the core diameter becomes much larger, modes become increasingly more densely packed in effective mode refractive index. When the mode spacing is too close, such as in a 100 µm core fiber, the intermodal group delay is much less than the spread in time of the various frequency components due to fiber dispersion. This spread due to chromatic dispersion is ~30fs/m/nm at ~1050nm in silica, comparing to the LP₀₁-LP₁₁ intermodal delay of ~50fs/m in the 100µm-core Re-LCF used in this work. Temporal spread from chromatic dispersion of a >2nm wide source would be larger than the intermodal delay, making the modes inseparable. The C² method can no longer resolve modes in this case. The S² method would also require a very wide-bandwidth optical source to be able to cover multiple periods of the widely spaced frequency beating.

In this work, we demonstrate quantitative mode characterization in a 1m long straight passive Re-LCF with 100µm core diameter using a matched white-light interferometer (MWI) in order to cancel out the effect of fiber dispersion. This is the first quantitative mode characterization in a 100µm-core fiber to our knowledge. The Re-LCF has two non-identical layers in the cladding, optimized for a simulated HOM loss of 60dB/m. The measured relative group delay between the FM and the second-order mode is 55fs/m, very close to the expected 56fs/m. The LP11 mode content after the straight 1m-long Re-LCF with low-index coating was measured to be ~-29dB relative to that of the FM. A separate M² measurement was also performed on the same fiber, giving a M² = 1.01.

2. Experiments

Guidance properties in LCFs have been well understood [1–3]. These fibers have a background silica glass and a cladding lattice of low index nodes with typically identical size. The missing node in the center forms the core of the fiber. The design makes a very leaky waveguide. By tailoring the index contrast and node-pitch ratio of the cladding, strong suppression of HOMs and acceptable confinement loss for the FM can be realized for a given core diameter.

However, the multiple layers of cladding nodes do not need to be identical. In fact, the cladding lattice can use non-identical layers as shown in Fig. 1(a). Under certain conditions, resonant coupling between the second-order core mode and the cladding lattice can be created, which pulls the HOM even further into the cladding and significantly improves HOM suppression. Figure 1(b) shows the optimized node-to-pitch ratios to achieve the highest mode loss for 2nd higher order mode, whose loss is the minimum among HOMs. With a node index of 8 × 10⁻⁴ below that of the silica, a minimum HOM loss of 62dB/m and FM loss of 0.2dB/m can be achieved in a design when the first and second cladding layers have node-to-pitch ratio of 0.45 and 0.40 respectively. The simulated mode field diameter is ~84µm in this design.

Fig. 1 (a) Re-LCF design with non-identical cladding layer structure, (b) simulated 2nd HOM loss versus variable node-pitch ratio. R2 = d₂/Λ.

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The Re-LCF fiber shown in Fig. 2 was fabricated according to the design. The nodes in the cladding lattice are made from fluorine-doped silica which has a slightly lower refractive index than the silica background glass by 8x10⁻⁴. The sizes of the nodes in the outer and inner layers are 33.2µm and 29.6µm respectively with the same pitch Ʌ = 70µm. The diameter of the core in the center is ~100 µm. The shape of the fiber is intentionally kept non-circular to reduce the coherent reflection from the outer boundary, which lowers HOM losses [15]. This passive fiber is coated with low-index coating to mimic a double-clad fiber.

Fig. 2 Fabricated Re-LCF with 100 µm core diameter.

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White light is generally considered as incoherent and it is only temporally coherent over femtoseconds. This unique property enables white light to have a very fine temporal resolution, which is exploited in optical coherence tomography (OCT) for a fine spatial resolution. White-light interferometry composed of a white light source and a Michelson interferometer has been reported previously to perform dispersion measurements on thin optical samples [18]. The same capability, in principle, can also be used to resolve modes in large-core optical fibers with very small intermodal group delays.

This high temporal resolution of MWI is a result of the short coherent length (CL) of the broad-spectrum source and the balanced arms of the interferometer, as illustrated in Fig. 3(a). The filled and open parallelograms represent coherent lights at the detector from the measurement and reference arms respectively. As the length of the reference arm is adjusted, coherent beating can only be observed over a short distance of translation, i.e. coherent beating length (CBL). Figure 3(b) illustrates an interferometry with a narrow-spectrum source and two balanced arms, showing the poor temporal resolution due to the long coherent length of the source. The interferometer with two un-balanced arms shown in Fig. 3(c) also leads to a deterioration of temporal resolution due the spread of coherent light over arrival time at the detector. The broader the light spectrum, the more critical it is to balance the arms in order to obtain high temporal resolution. A MWI is demonstrated here to perform the quantitative mode characterization in our Re-LCF fiber. The principle is displayed in Fig. 4.

Fig. 3 Illustration of interferometry resolution: (a) broad spectrum and balanced dispersion; (b) narrow spectrum and balanced dispersion; (c) broad spectrum and un-balanced dispersion. Filled and open parallelogram represents coherent light from measurement and reference arms respectively.

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Fig. 4 Principle of mode characterization in optical fiber with matched white-light interferometry (MWI).

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A white light beam is equally split into two beams and coupled into two identical fibers under test. The purpose of using the identical fibers in both optical paths is to precisely balance the fiber dispersions. As shown in Fig. 4, the FM ${\tilde{E}}_{10}$ and two HOMs are excited by the white light simultaneously at the input end of the top fiber. ${\tilde{E}}_{11}$ and ${\tilde{E}}_{12}$ represent the 1st and 2nd HOMs respectively in the first interferometer arm. Similarly, FM ${\tilde{E}}_{20}$ and HOMs ${\tilde{E}}_{21}$ and ${\tilde{E}}_{22}$ are excited in the lower fiber. After propagating through the fiber, the coherent light in various modes are separated in arrival time at the detector due to the difference in effective mode indexes. If the arrival time difference between modes becomes larger than the coherence length of the white light, each mode can be individually picked up through coherence interferometry by scanning the relative time delay between the two arms. Assuming, after propagating through the fiber, $| {\tilde{E}}_{10} | ≫ | {\tilde{E}}_{11} |, | {\tilde{E}}_{12} |$ and $| {\tilde{E}}_{20} | ≫ | {\tilde{E}}_{21} |, | {\tilde{E}}_{22} |$ , then the main coherent beating is generated between the FM and FM, and the FM and a HOM. As shown in the inset in Fig. 4, the interference can be found between ${\tilde{E}}_{12}$ and ${\tilde{E}}_{20}$ , ${\tilde{E}}_{11}$ and ${\tilde{E}}_{20}$ , ${\tilde{E}}_{10}$ and ${\tilde{E}}_{21}$ , ${\tilde{E}}_{10}$ and ${\tilde{E}}_{22}$ , which are symmetrically positioned on both sides of the main coherent peak arising from the FM and FM beating.

The HOM information including the mode intensity and phase distribution can be extracted from the coherent interferometry. Since the modes in the fiber are excited by white light, they all have very short coherence length. In order to present the limited coherence length in the electric field of the modes, we let:

{\tilde{E}}_{10} = E_{10} (x, y) \times e^{i (φ_{10} (\overset{⇀}{r}) - \bar{ω} t)} \times L (t - t_{1})

in which

E_{10} (x, y)

is the electric field distribution of the FM in x-y plane,

φ_{10} (\overset{⇀}{r})

is the phase,

\bar{ω}

is the average frequency of the white light,

t_{1}

is the temporal position of the FM from the top fiber in Fig. 4. The coherence length position function

L (t - t_{1}) = {[\exp (- 4 {(\frac{(t - t_{1})}{I_{C}})}^{2})]}^{\frac{1}{2}}

, where

I_{C} = \frac{C}{π \times Δ ν}

is the coherence length and

Δ ν

is the bandwidth of the white light, C is the speed of light. The expression:

\exp [- 4 {(\frac{(t - t_{1})}{I_{C}})}^{2}]

is the envelope of the autocorrelation of white light with a Gaussian spectral shape. In the same way, we let:

{\tilde{E}}_{11} = E_{11} (x, y) \times e^{i (φ_{11} (\overset{⇀}{r}) - \bar{ω} t)} \times L (t - (t_{1} - Δ t_{11}))

where

Δ t_{11}

is the group delay between

{\tilde{E}}_{10}

and

{\tilde{E}}_{11}

,

{\tilde{E}}_{12} = E_{12} (x, y) \times e^{i (φ_{12} (\overset{⇀}{r}) - \bar{ω} t)} \times L (t - (t_{1} - Δ t_{12}))

where

Δ t_{12}

is the group delay between

{\tilde{E}}_{10}

and

{\tilde{E}}_{12}

. The light coming out of the fiber 2 goes through a scanning stage to add an additional variable time delay

Δ T

in it. So the expression for

{\tilde{E}}_{20}

,

{\tilde{E}}_{21}

and

{\tilde{E}}_{22}

is the following:

{\tilde{E}}_{20} = E_{20} (x, y) \times e^{i (φ_{20} (\overset{⇀}{r}) - \bar{ω} t)} \times L ((t + Δ T) - t_{2})

where

t_{2}

is the temporal position of the fundamental mode carrying the short coherence length from the lower fiber Fig. 4,

{\tilde{E}}_{21} = E_{21} (x, y) \times e^{i (φ_{21} (\overset{⇀}{r}) - \bar{ω} t)} \times L ((t + Δ T) - (t_{2} - Δ t_{21}))

where

Δ t_{21}

is the group delay between

{\tilde{E}}_{20}

and

{\tilde{E}}_{21}

,

{\tilde{E}}_{22} = E_{22} (x, y) \times e^{i (φ_{22} (\overset{⇀}{r}) - \bar{ω} t)} \times L ((t + Δ T) - (t_{2} - Δ t_{22}))

where

Δ t_{22}

is the group delay between

{\tilde{E}}_{20}

and

{\tilde{E}}_{22}

. Let

\tilde{E}

represent the sum electric field of all the modes from both fibers, thus:

\tilde{E} = {\tilde{E}}_{10} + {\tilde{E}}_{11} + {\tilde{E}}_{12} + {\tilde{E}}_{20} + {\tilde{E}}_{21} + {\tilde{E}}_{22}

. In order to simplify the expression, some assumptions are made under the conditions that the white light is equally split into two beams and these two fibers are identical. Then we have:

E_{10} (x, y) = E_{20} (x, y) = E_{0}

,

E_{11} (x, y) = E_{21} (x, y) = E_{1}

,

E_{12} (x, y) = E_{22} (x, y) = E_{2}

,

Δ t_{11} = Δ t_{21} = Δ t_{1}

,

Δ t_{12} = Δ t_{22} = Δ t_{2}

. Then we can get:

\begin{matrix} \tilde{E} = E_{0} \times e^{i (φ_{10} (\overset{⇀}{r}) - \bar{ω} t)} \times L (t - t_{1}) + E_{1} \times e^{i (φ_{11} (\overset{⇀}{r}) - \bar{ω} t)} \times L (t - (t_{1} - Δ t_{1})) \\ + E_{2} \times e^{i (φ_{12} (\overset{⇀}{r}) - \bar{ω} t)} \times L (t - (t_{1} - Δ t_{2})) + E_{0} \times e^{i (φ_{20} (\overset{⇀}{r}) - \bar{ω} t)} \times L ((t + Δ T) - t_{2}) \\ + E_{1} \times e^{i (φ_{21} (\overset{⇀}{r}) - \bar{ω} t)} \times L ((t + Δ T) - (t_{2} - Δ t_{1})) + E_{2} \times e^{i (φ_{22} (\overset{⇀}{r}) - \bar{ω} t)} \times L ((t + Δ T) - (t_{2} - Δ t_{2})) \end{matrix}

The intensity of the light at (x,y) is given by:

I (x, y) = 〈 \tilde{E} (x, y) {\tilde{E}}^{*} (x, y) 〉

where the angle brackets indicate the time averaging done by the detector, so the high frequency oscillation term containing

\bar{ω} * t

is washed out. Also because of the conditions: E₀>>E₁, E₂, the terms containing E₁², E₂², E₁*E₂ are negligible. The intensity distribution can be simplified as:

\begin{matrix} I (x, y) = 2 E_{0}^{2} \\ + 2 E_{0}^{2} \times \cos (φ_{10} - φ_{20}) \times L (t - t_{1}) \times L ((t + Δ T) - t_{2}) |_{Δ T = t_{2} - t_{1}} \\ + 2 E_{1} E_{0} \times \cos (φ_{21} - φ_{10}) \times L (t - t_{1}) \times L ((t + Δ T) - (t_{2} - Δ t_{1}) |_{Δ T = (t_{2} - t_{1}) - Δ t_{1}} \\ + 2 E_{2} E_{0} \times \cos (φ_{22} - φ_{10}) \times L (t - t_{1}) \times L ((t + Δ T) - (t_{2} - Δ t_{2}) |_{Δ T = (t_{2} - t_{1}) - Δ t_{2}} \\ + 2 E_{1} E_{0} \times \cos (φ_{20} - φ_{11}) \times L (t - (t_{1} - Δ t_{1})) \times L ((t + Δ T) - t_{2}) |_{Δ T = (t_{2} - t_{1}) + Δ t_{1}} \\ + 2 E_{2} E_{0} \times \cos (φ_{20} - φ_{12}) \times L (t - (t_{1} - Δ t_{2})) \times L ((t + Δ T) - t_{2}) |_{Δ T = (t_{2} - t_{1}) + Δ t_{2}} \end{matrix}

On the right side of the equation, the first term is a DC component, the second term is from the beating between the two FMs at a delay ∆T = t₂-t₁, which represents the position where the two optical paths in the interferometer are balanced. The rest of the terms represent the beating between the HOMs in one fiber with the FM in the other fiber at a delay ∆T, which are symmetrically distributed on both sides of the position which represents the balanced optical paths. The ratio of the electric field between HOMs and FM can be derived from the ratio of the beating:

\frac{E_{i} E_{0}}{E_{0}^{2}}

, where i is either 1 or 2. This information can be used to reconstruct the mode pattern. The mode content ratio (MPI) between HOMs and FM can be calculated as:

M P I = 10 \log_{10} [\frac{\iint d x d y {(\frac{E_{i} (x, y)}{E_{0} (x, y)} \times E_{0} (x, y))}^{2}}{\iint d x d y E_{0}^{2} (x, y)}]

The experimental setup for the matched white-light interferometry is shown in Fig. 5.

Fig. 5 Schematic diagram of the matched white-light interferometry.

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A Bentham white light source WLS100 with quartz halogen lamp emitting from 350nm to 2500nm was spectrally filtered with a 780nm long pass filter, split into two beams and coupled into two identical straight fibers which were placed in an aluminum V-groove to keep them straight. From the simulation in the fiber design, the second-order mode in this fiber has a loss of ~62dB/m and a relative group delay of ~56fs/m to the FM. The length of the fiber used in the test was 1m. The beam out of one fiber was sent to a motorized scanning stage with 1µm accuracy and 100 mm travel. The beam from the other fiber was sent to a piezo-controlled scanning stage with 5nm accuracy and 20µm travel. After the two arms were roughly balanced by the motorized stage, the piezo stage was used to perform the fine interferometry scanning with a 50nm incremental step up to the 20µm maximum travel. Effectively, the path length was scanned with a 0.1µm incremental step up to 40µm maximum travel due to the double passed configuration. Meanwhile, a high speed silicon camera was used to record the two-beam interference pattern after each incremental step during the scan, which does not respond to the wavelengths larger than 1.1µm. In order to reduce the random noise on the camera, each 4 adjacent pixels in the original image are added together to reconstruct one pixel in the final image. The intensity oscillation on one pixel versus the delay time ∆T is the function $I (x, y)$ mentioned above.

Figure 6 shows the intensity oscillation throughout the scanning process on one pixel of the image, which is located at the lobe center of the reconstructed mode image corresponding to peak B. There are two horizontal axes in the figure. The lower one is the temporal delay in femtoseconds and the top one is the corresponding path length delay in micrometers. Five peaks are found in the plot. The major one marked as A comes from the coherent beating between the two FMs with the maximum amplitude of $E_{0}^{2}$ . Peaks B, D arises from beating between a FM and a second-order mode, with roughly the same maximum amplitude of $E_{1} E_{0}$ . The measured intermodal group delays between peaks A, B and A, D are 50fs and 59fs respectively, with an average of ~55fs which is very close to the calculated value of 56fs from the fiber design. The group delays between peaks A, C and A, E are measured to be 77.7fs and 78.9fs respectively. Figure 7 shows the reconstructed mode patterns and phases corresponding to the oscillation peaks from A to E found in Fig. 6. The mode pattern for peak A is a Gaussian-like beam with a flat phase pattern, which indicates it is the LP01 mode. The mode patterns for peaks B and C show two lobes with a π phase shift between them, which indicates they are LP11 modes. These two LP11 modes have different orientations and there is a 28fs time delay between them. The MPI values for these two LP11 modes are −29dB and −37dB for peaks B and C respectively. The modes for peaks D and E are similar to the ones for peaks B and C. Their MPI values are −33dB and −37dB respectively. The measured lower HOMs losses than the simulation could be due to some coherent reflection from the fiber outer boundary [15]. The linewidths of the envelopes of the coherent beating between FM and HOMs corresponding to peaks B, C, D and E are not broadened, as shown in Fig. 6, which indicates that there is insignificant mode coupling during the propagation in the fiber. The minimum intensity detection capability for this interferometry was tested by measuring the MPI value of the flat DC region without any coherent beating, which is determined to be −43dB.

Fig. 6 Intensity oscillation during the interferometry scanning at the pixel at the lobe center of the reconstructed image B.

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Fig. 7 Reconstructed mode patterns and phases corresponding to the peaks from A to E found in Fig. 6 with calculated MPI value for HOMs.

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The mode quality guided in this Re-LCF fiber was also tested qualitatively. The light from a LED source emitting from 890nm to 1090nm was collimated into the 1m-long straight Re-LCF. The mode pattern of the output beam was recorded while the launching beam at the input end was moved across the fiber core with the intention to excite HOMs. The mode pattern did not change at all while the launching beam was moved. The only thing that changed is the mode intensity which is due to the change in launching efficiency. M² of the mode was also measured when the LED light was carefully coupled into the fiber. The result is shown in Fig. 8, giving M² = 1.01.

Fig. 8 M² measurement of the output beam from the Re-LCF fiber in x, y directions.

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3. Conclusions

We have demonstrated, for the first time, the use of matched white-light interferometry (MWI) to perform mode characterization in an optical fiber with 20fs temporal resolution and −43dB MPI sensitivity. This quantitative mode characterization technique is especially useful for optical fibers with large cores. The fiber demonstrated here is a straight resonantly-enhanced LCF with 100µm core diameter. The group delay between the FM and second-order mode in this fiber was measured as ~55fs/m, very close to the 56fs/m expected from the simulation. The second-order mode was measured to be 29dB below the FM in this straight double-clad fiber. This result shows that significant HOM loss is possible in straight double-clad fibers, contrary to that observed in rod-type fibers [19]. The single-mode quality was also qualitatively tested by scanning the launching beam at the input end without finding any excited HOMs. The measured M² is 1.01. All the test results strongly show the robust single-mode operation in this Re-LCF, which makes it potentially very useful for high average and peak power fiber lasers.

Acknowledgments

This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under contract/grant number W911NF-12-1-0332 through a Joint Technology Office MRI program.

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Quantitative mode quality characterization of fibers with extremely large mode areas by matched white-light interferometry

Abstract

1. Introduction

2. Experiments

3. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Equations (10)

Optics Express