Multimode fibers are conveniently used to couple low spectral resolving power \( ( \lambda /\delta \lambda \simeq 1000 ) \) spectrometers to light sources. However, as spectral resolution increases modal noise in fibers introduces systematic errors in fiber-fed spectrographs. Here we summarize the origin of this noise and implement a remedy in the form of a ball lens based fiber coupler for a high spectral resolving power spectrometer \( (\lambda /\delta \lambda \simeq 10,000 ) \).
Modal Noise in Fibers
Figure 1 shows the far-field diffraction pattern from a 50 \(\mu\)m multimode fiber. When this fiber is illuminated using a quasi-monochromatic 650-nm diode laser, the resultant speckle pattern due to interference between the various modes propagating in the fiber is evident. Since the optical path length and hence the resultant interference pattern is wavelength dependent, the transmission of the fiber also varies with wavelength. This wavelength dependent transmission is superimposed on any spectra measured using this fiber, causing systematic errors. Small perturbations to the fiber changes the optical path length, e.g., by bending the fiber, hence the speckle pattern is not static. Consequently, the transmission is variable and cannot easily be measured and removed.
One solution to modal noise is to agitate the fiber mechanically, which changes the speckle pattern. If the pattern can be refreshed many times during an exposure, then a form of averaging occurs and the effect is mitigated. The alternative is to use a fiber that supports more modes and therefore more speckles. This approach relies on spatial averaging rather than temporal averaging. The simplest way to increase the number of modes is to use a larger diameter fiber. However, as the spectral resolving power of a dispersive spectrometer is typically determined by the size of the entrance aperture (the fiber diameter) a simple replacement with a larger diameter fiber will degrade resolution.
Computing the Number of Modes & Speckle Noise in a Fiber
Originally, in our application a high-resolution spectrometer was fed using inexpensive telecom OM2 fiber that is used for 10 Gigabit Ethernet. This graded index, multimode fiber has a 50 \(\mu\)m-diameter core, \( 2a\), a 125 \(\mu \)m cladding, and a numerical aperture, \( NA = 0.2\). The normalized frequency or dimensionless \(V\) number,
\[ V = \frac{2\pi a }{\lambda}\; NA , \]determines the number of propagating modes in a fiber. For a circular cross section, graded-index fiber the number of supported modes is approximately \( M =V^2/4 \) for a fiber with power law profile that minimizes modal dispersion. At 650 nm an OM2 fiber (\(NA =0.2 \)) has \(V =110 \) and \(M = 580 \). This is a relatively small number of modes, as the signal-to-noise ratio is set by Poisson statistics and is approximately equal to \(M^{1/2} =24 \).
Alternatives to telecom fibers are larger 100, 200, or 400 \(\mu\)m diameter step index, \(NA =0.22\) fibers. The smallest diameter is preferred given that the spectral resolving power decreases inversely with fiber diameter. The number of modes supported by the preferred choice of 100 \( \mu\)m step index fiber is
\[ M = \left( \frac{2V}{\pi} \right)^2\]or \(M\) = 5050, which represents a substantial reduction in speckle noise.
Ball Lens Based Fiber Coupler
The simplest way to couple a larger diameter fiber to the spectrograph without degrading the spectral resolving power is to re-image the end of the fiber onto a entrance slit with width equal to the diameter of the original fiber. This approach loses light which spills over the edges of the slit, but because light up and down the slit can be imaged onto the sensor the light loss for a 100 \(\mu\)m fiber and 50 \(\mu\)m slit is only approximately 40%.
The adopted design to couple a fiber to a slit using ball lenses is shown in Figure 2.
Ball lenses are compact and inexpensive and although they suffer spherical aberration. Figure 3 shows the geometric spot size (20 \(\mu\)m diameter), which is sufficient to illuminate the spectrograph entrance slit without additional losses.
The fiber coupler is compact; the total track from fiber input to slit is about 13 mm. The ball lenses are glued into compact lens adaptors (Figure 4 & 5) that are 2 mm thick. The small size of the assembly means that no major changes are needed to the mechanical structure of the spectrometer and the 1:1 magnification permits reuse of the collimator optics.
An image of the 50 \(\mu\)m spectrograph entrance slit is shown in Fig. 6.
The final assembly of the fiber coupler, slit, and 100-mm focal length collimator is shown in Fig. 7. A cage system was adopted for the mechanical structure. Since the working distance from the ball lenses is only 1.1 mm, care has to be taken to allow the fiber tip and the slit to be close enough to the lens holder so that they can be in focus. Accurate element spacing was established using precision gauge blocks. The slit holder (Fig. 6) was glued into a 1-inch threaded lens tube and a retaining ring was used to allow this holder to be oriented so that the image of the slit is vertical at 590 nm. The slit is held in a stage with adjustable X/Y position so that the image of the fiber is accurately centered on the slit.
Results
Figure 8 shows a high resolution spectrum of the Sun obtained with a 100 \(\mu\)m fiber and the new fiber coupler. In this very high signal-to-noise data, multiple weak atomic absorptions lines are visible as well as molecular bands due to oxygen and water vapor in the Earth's atmosphere.
© Eikonal Optics