Effects on Polarization and Coherence

Apart from the spatial homogenization of radiation in a multimode optical fiber, the statistical mode coupling processes also yield a reduction of the degree of polarization [94] and a reduction of the radiation’s coherence [95]. Although a detailed discussion of these effects is beyond the scope of this work, the appearance of these effects will be briefly exemplified by measurements that have been conducted within this work.

A convenient description for the degree of polarization of an electro-magnetic wave is given by the Stokes parameters fragmenting the po-larization into its linear and circular popo-larization contributions [96, pp.

373-379]. In Fig. 5.12 the Stokes parameters for radiation emitted by 2 m multimode optical fiber with NA = 0.22 (M25L02/Thorlabs) are given in a spatially resolved manner. The measurements have been conducted

18Please note that this fiber length, marking invariance of spatial intensity distribu-tion for further elongadistribu-tion of the fiber, is not meant to be generalized; instead, it only holds for the specific conditions applied in this illustrative example. Any other launch conditions might yield a significantly different evolution of spatial intensity distribution over fiber lengths.

Figure 5.12: Spatially resolved Stokes parameters of radiation emitted from a 2 m long multimode fiber with NA = 0.22 (M25L02/Thorlabs).

(a) denotes the degree of linear polarization at 0^◦(S1), (b) the degree of linear polarization at 45^◦ (S2), (c) the degree of circular polariza-tion (S3) and (d) the overall degree of polarization (VStokes). The color coding denotes the amplitudes and orientations (-1 and 1, re-spectively) of the measured polarization contributions.

by coupling linearly polarized ultrashort laser pulses into the fiber and analyzing the emitted radiation field by a 4f imaging setup, a quarter-waveplate and a linear polarizer.¹⁹ The spatial information has been ob-tained by imaging the radiation field transmitted by the described optical setup onto a CMOS-camera, that has been calibrated pixel-wise regarding its polarization dependent current response prior to the measurements. In Fig. 5.12a the amount of linear polarization oriented at 0^◦or 90^◦is shown (S1), in (b) the amount of linear polarization oriented at 45^◦or 135^◦ (S2) and in (c) the amount of circular polarization (S3). The color-coding ranging from -1 (blue) to 1 (red) denotes the amplitude and orientation of the particular polarization contributions (e.g. blue indicates 90^◦ po-larization and red 0^◦ polarization in Fig. 5.12a or clockwise (blue) and counterclockwise (red) circular polarization in (c)). Fig. 5.12d gives the

19For details of the measurement procedure the reader is referred to e.g. [96, p.

374].

degree of polarization computed from VStokes=

pS₁²+S₂²+S₃²

I (5.25)

with overall intensity I (measured without polarizer or waveplate).

VStokes= 0 denotes unpolarized andVStokes= 1 polarized radiation.

Taking into account that the degree of polarization without multimode fiber (pure laser pulse) has been measured to be approximately 0.95, the reduction in VStokes to (roughly) 0.5 after only 2 m of multimode fiber highlights the strong impact of the multimode fiber on the radiation’s po-larization properties. Furthermore, it is noteworthy, that the popo-larization properties are spatially uniform across the fiber end facet.

Remaining variations in Fig. 5.12d are mainly due to speckle patterns that are intensity variations caused by statistical interference effects across a given surface. For highly coherent radiation these patterns can be quite obvious, however, for lower coherence lengths they eventually vanish.

Thus, the contrast of speckle patterns is an appropriate indicator for the coherence of radiation [97]. In multimode fibers the coherence of radiation propagating through the fiber is reduced due to the incoherent superpo-sition of the intensity distributions of all present modes [95]. Therefore, multimode fiber approaches are also exploited in projector applications [98].

Speckle reduction in multimode fibers is demonstrated by the results in Fig. 5.13b. The measurements have been conducted by coupling ultra-short laser pulses with 100 fs pulse duration and 800 nm center wavelength into a multimode fiber with NA = 0.39 (FT200EMT/Thorlabs) of various lengths. For an evaluation of the speckle patterns the output facet of the multimode fiber has been imaged onto a CMOS-camera with a 4f-setup and the speckle contrast has been computed according to [98]

C=

phP_i²i − hPii²

hPii , (5.26)

withP_ias intensity of the i-th camera pixel andhidenoting spatial aver-aging over all camera pixels.

(a) (b)

Figure 5.13: (a) Enlargement of the spatial intensity distribution at the end facet of the FT200EMT/Thorlabs multimode fiber af-ter 2 m (i) and 175 m (ii) fiber length. (b) Speckle contrast of a FT200EMT/Thorlabs fiber (NA = 0.39) at various lengths for ultra-short laser pulses of 100 fs pulse duration and 800 nm center wave-length. Please note, that the absolute values for speckle contrast are potentially biased by significant measurement uncertainties in this uncalibrated measurement.

The results demonstrate the general tendency of speckle contrast and, hence, coherence reduction while the radiation propagates through the multimode optical fiber. For compensating the ongoing variation of spa-tial intensity distribution with fiber length (see Fig. 5.11), the fiber output images have been normalized by their radially averaged intensity distribu-tion prior to computadistribu-tion of the speckle contrast. However, the absolute values of speckle contrast might still suffer from measurement uncertain-ties in this uncalibrated measurement so that rather the general tendency shown in Fig. 5.13b should be noted instead of the absolute values ofC.

In Fig. 5.13a two enlargements of sections of the fiber end facet are shown for 2 m and 175 m fiber lengths. In the top image (2 m fiber lengths) speckle features of approximately 5 to 10 µm size can be observed. For 175 m fiber length the speckles vanish and only fluctuations caused by noise remain.

Both effects, the randomization of polarization and reduction of co-herence, are welcomed side effects regarding the application of ultrashort laser pulses for characterization of current responses of solar cells. Be-sides temporal stretching and spatial homogenization of radiation inside a multimode optical fiber a further randomization of the distinct laser radiation properties in the polarization and coherence domain is greatly appreciated as this serves an improved imitation of the sun’s radiation characteristics.

5.5 Fiber Concept for Conditioning of