Abstract

Fiber optical light diffusers that enable interstitial light delivery have become a useful tool for various illumination tasks, such as in photodynamic therapy. However, existing methods based on light diffusing fiber tips are not applicable for spatially selective light delivery in more complex structures. Here, we employ femtosecond laser induced scattering centers without mechanical manipulation and removal of the outer coatings for generating customized emission patterns. Tailoring of the cumulative emission profile is achieved through controlling the step-width between modification spots. An in-depth evaluation shows that the side-emission pattern is the result of an interplay of several scattering mechanisms that interact with cladding and core modes.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

1. Introduction

Photodynamic therapy has become widely applied in treating superficial cancers, for example in skin, lung, esophagus, and bladder, with good clinical and cosmetic outcomes. This method of treatment employs light of a defined wavelength to activate a photosensitizing drug which causes localized cell death or tissue necrosis [1]. Beyond surface treatments, however, the method is difficult to apply to voluminous tumors because of the low light penetration depth in tissue due to scattering and absorption. Interstitial light delivery, which is a more efficient illumination method for such tissues, employs light scattering optical fiber tips which are placed directly inside the tumors [2]. Such tips are usually fabricated by fixing a mixture of epoxy resin and light-scattering particles (quartz, TiO$_2$) to the exposed core at the fiber end [3,4]. These diffusers provide for a flat emission profile over a length of several centimeters, but are mechanically stiff and have a large diameter; both factors may result in unwanted tissue damage. In addition, the maximum length of the light diffusing region is limited [3]. In recent years, several authors have hypothesized that effective light delivery, especially in organs or tumors of complex shape, could benefit from specifically tailored light emission profiles on the diffusive fiber tip [2,5,6]. For example, it was estimated that collateral damage on healthy tissue could be reduced in the range of ${15}{\%}$ to ${58}{\%}$ in terms of tissue volume when using adapted light sources for treating brain cancer tissue [5].

Current technology for tailored light diffusers employs long period fiber gratings, which couple light from guided core modes to cladding modes [3]. This light is eventually scattered out of the fiber by a layer of TiO$_2$ particles, which is applied as a coating to the modified region after the grating is written. This technique is able to generate ${250}\,{\mu \textrm {m}}$ thin and flexible light diffuser tips with a flat or customized longitudinal emission profile. Unfortunately, this mode of fabrication can neither generate point-like emission in the fiber nor emission profiles with large peak powers towards the fiber end [2].

To address the limitations of current approaches, here, we study tailored light diffusion from a commercial optical fiber which is modified through focused femtosecond laser irradiation. Contrary to continuous laser writing, the high energy density of this method enables the creation of local variations in the refractive index of any transparent fiber material due to nonlinear absorption processes [7]. The irradiation pulse generates a rapid increase in temperature followed by rapid cooling, which results in a refractive index change of the material. Repeated focused irradiation of a single spot by the laser pulses then create localized refractive index perturbations which scatter light guided in the fiber.

Using established technology, such scattering centers can be arranged in periodic or aperiodic longitudinal chains, thereby resulting in a light emitting fiber segment as shown in Fig. 1. The emission profile could be tailored by the number of pulses per spot, by the employed pulse energy and by the step-width between the scattering centers. Since the first two methods include non-linear processes in the creation of the scattering effect, which are difficult to control, we chose the latter to demonstrate the feasibility of our approach. We created two series of modifications under similar irradiation conditions: one with constant step-width, and another one with decreasing step-width. In this way, we make use of the usually undesired light scattering properties of femtosecond laser induced refractive index perturbations. According to this purpose, the writing process itself underlies less critical constraints in terms of writing accuracy and aberrations of the focal spot as compared to the established writing process of fiber gratings. Most noteworthy, this enables to perform the writing procedure in a non-contact process without removal of the fiber coating or other mechanical handling. On the one hand, we retain almost all benefits of light scattering by a long period fiber grating [3], in particular, the possibility to manufacture thin and long filaments which are able to illuminate large volumes of tissue (or other media [8,9]) with low laser power and very high spacial selectivity. On the other hand, the advantages of retaining an intact cladding (by placing the emission centers into the fiber core) also come with the challenge to include the light guiding ability of the cladding into the conception of efficient side-emission designs.

 

Fig. 1. Schematics and microscope image of light scattering on femtosecond laser modifications. a) Creation of scattering centers through focused femtosecond laser irradiation of the fiber core. b) Light scattering on the laser modifications in a two stage process: Light is scattered out of the core into free space and into the cladding, where it is again guided by total internal reflection or, eventually, scattered out into the environment. c) Microscope image of light scattering on laser modifications.

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The paper is organized as follows: Section 2 presents the three-level light flux exchange between core, cladding and surrounding in comparison to the common one-level light flux exchange considerations. In Section 3, we describe the experimental parameters as well as the measurement device for quantifying fiber transmission and emission. Section 4 examines the scattering centers and the measured emission and transmission profiles of modified fibers. Building on these analyses, in Section 4 we discuss singular as well as chains of scattering centers. Further, we test the theoretical analysis by applying it to a customized emission profile.

2. Theory

2.1 General

We initially analyze the interaction of scattering centers with guided light in an optical fiber. The proposed model is based on the Lambert-Beer law, which is expanded to describe the interaction of discrete scattering events with a core-cladding optical fiber structure. From the result we derive the fiber transmission and emission, which can be experimentally observed. The model to describe the fiber emission and transmission relies on two central assumptions: first, all cladding modes as well as all core modes exist in a steady state power distribution [10]. This is warranted by the constant coupling of modes due to light scattering. Second, the scattering centers are independent, i.e., they do not interact with each other. This means that a single total loss function can be applied to all modes in the cladding or in the core (viz., the Lambert-Beer law). This total loss in transmitted spectral flux $\phi$ [W/nm] can be decomposed into scattering $\sigma$ and attenuation $\alpha$, which may or may not vary over distance:

$$\frac{d\phi}{dz}=-(\alpha+\sigma)\phi \ \longleftrightarrow \ \phi(z) = \phi_0 \exp{\left\lbrace -\int_{0}^z\alpha(z')+\sigma(z') \: dz'\right\rbrace }\, .$$
If the integrand is constant, the result is a simple exponential decline in light flux with increasing distance $z$. From this the fiber emission can be deduced from conservation of energy: The absorbed light is transformed to heat and the scattered fraction will leave the fiber through its surface. This results in a simple expression for the emission of the fiber per unit length $\frac {d\phi _E}{dz}=\sigma \phi$, which is proportional to the fiber transmission.

Because a regular optical fiber consists of at least two layers with different refractive index, namely, core and cladding, this simple model (which is mostly applied to the transmission of the fiber core) is insufficient to describe the emission behavior of a real fiber. Therefore, we expand Eq. (1) based on similar assumptions, but include the general observation that light scattered from the fiber core partially ends up in the cladding where it is still able to propagate. The exchange of flux between the core $\phi _{1}$, the cladding $\phi _{2}$ and the surrounding $\phi _{3}$ is now described as the energy exchange between different states where the scattering is responsible for the coupling between the states. Light, which is scattered into free space, is transported away from the fiber so there is no reverse process of light being coupled into the fiber from the outside. This energy exchange is schematically shown in Fig. 2.

 

Fig. 2. Three-level representation of the energy exchange between the core, the cladding and free space: At first all light is contained in the core and gets scattered into free space (green) and the cladding (red). After some distance this leads to a mixed case where light is also contained in the cladding and scattered into free space (blue) and back into the core (yellow). Because the latter effect is very small, the core is eventually depleted and only the remaining light in the cladding is scattered.

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The scattering of cladding and core modes is caused by discrete refractive index perturbations (as will be generated by focused laser irradiation). The fiber is a low-loss silica fiber, so we approximate scattering to be the dominant loss mechanism and set all other contributions to zero ($\alpha \approx 0$). Under these conditions, we can make the transition from distance-dependent scattering $\sigma$ to constant scattering per modification $s$. This enables us to write the differential equations governing the energy exchange with the number of scattering centers $n$ as the independent variable $z \rightarrow n$:

$$\frac{d\phi_{1}}{dn} = -(s_{12} + s_{13}) \phi_{1} + s_{21} \phi_{2}\, , $$
$$ \frac{d\phi_{2}}{dn} = s_{12} \phi_{1} - (s_{23} + s_{21}) \phi_{2}\, , $$
$$ \frac{d\phi_{3}}{dn} = s_{13} \phi_{1} + s_{23} \phi_{2}\, . $$
Here, $s_{ab}$ represents the probability for the light to be scattered from state $a$ to $b$. To illustrate the process, the three general cases for energy exchange are shown in Fig. 2. At the start of the modified region the whole light flux is contained within the core; therefore the initial conditions are $\phi _{1}(0)=\phi _0$, $\phi _{2}(0)=0$ and $\phi _{3}(0)=0$. The scattering centers are regarded as identical, so all coefficients $s_{ab}$ are constant. In this experiment, the NA of the fiber core is very small, so the reverse process of scattering light from the cladding into the core is negligibly small ($s_{21}\approx 0$); this approximation decouples Eqs. (2) to (4).

With these conditions, the solutions to the ordinary differential equations can be found by standard methods (e.g. integrating factor). We are only interested in the solutions for $\phi _{1}$ and $\phi _{2}$ because Eq. (4) already describes the fiber emission at the respective scattering center:

$$ \phi_{1}(n) = \phi_0 \exp\lbrace{-(s_{12} + s_{13}) n}\rbrace\, ,$$
$$\phi_{2}(n) = \frac{\phi_{0} s_{12}}{s_{12} + s_{13} - s_{23}} \left[ \exp\lbrace{ -s_{23} n}\rbrace - \exp\lbrace{-(s_{12} + s_{13}) n}\rbrace \right]\, .$$

2.2 Observed quantities

The two quantities of this scattering process, which will be measured in the experiment, are the transmitted flux and the emitted flux of the fiber. Both quantities are normalized by the initial flux in the fiber $\phi _{0}$ to yield the transmission $T$ and the emission $E$. For the former, we just have to consider that the core as well as the cladding act as light guides, therefore the measured transmitted flux is the sum of flux of the two levels:

$$T(n) = \frac{\phi_1 + \phi_2}{\phi_0} = (1 - A_s)\exp\lbrace{-(s_{12} + s_{13}) n}\rbrace + A_s \exp\lbrace{-s_{23} n}\rbrace\, . \\$$
Here $A_s$ is substituted for
$$A_s = \frac{s_{12}}{s_{12} + s_{13} - s_{23}}\, .$$
The fiber emission is described by the normalized flux emitted by the fiber per unit length $E$. (In radiometric convention, this would be expressed as the radiant exitance $M = E/(2\pi R)$ of the fiber surface with the radius $R$.) With the general assumption that $n$ is some function of z [$n=n(z)$], the relation between the emission per scattering center and the emission per unit length is found via the chain rule.
$$ E(z) = \frac{1}{\phi_0} \frac{d\phi_{3}}{dz} = \frac{1}{\phi_0} \frac{d\phi_{3}}{dn} \frac{dn}{dz} \, ,$$
$$= \left[(s_{13} - s_{23} A_s) \exp\lbrace{-(s_{12} + s_{13}) n(z)}\rbrace + s_{23} A_s \exp\lbrace{-s_{23} n(z)}\rbrace \right] \frac{dn}{dz}\, . $$
This is the primary result of this analysis; it means that the functional form of the light scattering per scattering center is always a sum of two exponential functions, regardless of their respective distance to each other. So by placing the scattering centers at different distances, the emission profile can be locally compressed or stretched to yield a customized cumulative profile according to some specific design requirements.

A more sophisticated treatment would expand the consideration further to include the fiber cladding as an additional third light guiding layer and the convolution of the emission signal with the point spread function of the measurement device. Furthermore, it could be considered that light is never coupled out instantaneously but has to undergo several partial reflections which results in an additional exponential decay [11]. These considerations were omitted for clarity.

3. Experimental

3.1 Laser writing

A commercially available step-index optical fiber made from fused silica with a germanium doped core and a fluoroacrylate coating (Nufern 20/400 Precision Matched Passive LMA Double Clad Fibers; Core: NA = 0.065, diameter = ${20}\,{\mu \textrm {m}}$; Cladding: NA = 0.46, diameter = ${400}\,{\mu \textrm {m}}$) was coupled to a super-continuum light source (NKT SuperK) and clamped to a motorized xy-table. Due to the fiber coupling only core modes were excited, so the initial conditions are fulfilled. The femtosecond laser was focused through the transparent coating into the fiber core with an NA = 0.5 microscope objective. The laser, which was employed for writing, was a Ti:sapphire (Spectra Physics, Spitfire) regenerative laser amplifier system emitting pulses at $\lambda = {800}\,\textrm {nm}$ with a pulse length of 200 fs (FWHM). We used $4\,\mu \textrm {J}$ pulse energy and a ${1}\,{\textrm {kHz}}$ repetition rate. A mechanical shutter was set to 0.4 s, so the number of pulses per scattering center was approximately 400. The transparent fiber coating was not removed for writing.

The free fiber end was connected to an integrating sphere measurement system (see Fig. 3 in the following subsection) in order to record the transmission spectrum after each step of modification. After irradiation, the position of the focal spot was changed and the procedure was repeated until the desired length of the modified region was reached. In one of the examples demonstrated in the following, 1000 modifications with a distance of ${60}\,{\mu \textrm {m}}$ were written. After writing, the fiber was removed from the xy-table and clamped to the linear stage without changing the coupling to the light source, as shown in Fig. 3. We performed the side emission measurement with a longitudinal resolution of 0.2 mm and an aperture $\Delta z$ of 2 mm. All measurements were normalized to the baseline $\phi _{0}$, which is the transmitted spectral flux of the unmodified fiber. The limited range of the light source and the spectrometer constrained the spectrum to the wavelength interval from 600 nm to 1000 nm.

 

Fig. 3. a) Motorized linear stage: The integrating sphere is moved incrementally alongside the optical fiber; the spectrometer measures an emission spectrum for every position. b) Integrating sphere: Light emitted by the fiber segment $\Delta z$ - limited by the fiber guide - is homogeneously distributed on the sphere wall by multiple diffuse reflections. The irradiance on the detector port is proportional to the emitted flux.

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After the light measurements were finished, we performed optical examination of the refractive index modification with a microscope (ZEISS, JENAPOL interphako). For this, the fiber was immersed in oil ($n$ = 1.52) to yield a sharp picture by compensating the aberrations on the cylindrical fiber surface. Then the modifications were imaged with 50 times magnification. We additionally used phase contrast (interphako method) [12] to enhance the image quality, because the refractive index perturbations have a very low contrast with the surrounding glass.

3.2 Scanning stage

Quantitative light measurement is notoriously difficult due to many different kinds of instrument attenuation, which are hard to determine. Therefore, we chose a relative measurement approach and normalized all spectra to the transmitted flux of the unmodified fiber. Also, we made as little changes as possible to the measurement device in between the change from the transmission to the emission set-up. This is especially important for the light coupling and the measurement system because they can introduce unknown errors. Under these conditions, we assumed that the constant unknown instrument attenuations will cancel out with the baseline normalization, as shown in Eq. (12).

The measurement device was a fiber coupled spectrometer (Ocean Optics: Maya2000 Pro) with a custom-made integrating sphere for measurement head (Fig. 3). The integration sphere was machined from optical PTFA (Berghof Fluoroplastics) with a reflectivity of $\rho \approx 0.98$. It consists of two fiber guides and a baffle to protect the detector port from direct irradiation. An optical fiber connects the sphere to the spectrometer.

The sphere was operated in two modes: In transmission measurement mode one fiber guide was blocked with optical PTFA and the cleaved fiber end was put in the middle of the aperture. In this way, the flux emitted from the fiber end is scattered on the blocked fiber guide and captured by the sphere with almost no change in the setup. In emission measurement mode, the fiber was threaded through the sphere with the help of both hollow fiber guides, leaving only a small segment of the length $\Delta z$ exposed to the interior of the sphere. Because the fiber guides limit the size of the measured fiber segment, they also act as the aperture of the integrating sphere. The light flux radiated into the sphere by the fiber is distributed homogeneously by multiple diffuse reflections on the sphere walls. The irradiance $M$ [W/m$^2$] measured by the detector of the port area $A$ is proportional to the flux emitted inside of the sphere [13]:

$$M=\frac{ \rho \phi}{A(1-\rho(1-f))}\, .$$
Where $f$ is the ratio of open port area to total sphere area $A_{Port}/A_{Sphere}$ and $\rho$ is the coefficient of reflection of the wall material. When $f$ and $\rho$ are kept constant between measurement modes, the relation of a measured value to a baseline value will be the same as the relation between the baseline flux value and the measured flux value because the unknown attenuations cancel out. So the irradiance is proportional to the flux collected by the sphere:
$$\frac{M_i}{M_0}=\frac{\phi_i}{\phi_0}\, .$$

4. Results

4.1 Observation

The results of the focused femtosecond laser irradiation are irregular shaped refractive index distortions in the fiber core. Fig. 4 shows microscope images of the core and the modifications viewed perpendicular to the direction of irradiation (side view). In brightfield microscopy the written features show up as faint blurs with irregular but defined boundaries which cover the fiber core partially in a regularly spaced pattern. Phase contrast shows that the modifications consist of a mixture of regions of higher (darker) and lower (brighter) refractive index compared to the surrounding glass matrix. If the fiber is rotated by ninety degrees (top view), the modifications are revealed to be flat, as they cover the core only partially. Here the induced phase shift exceeds the instrument threshold with the consequence that the contrast is inverted. The approximate dimensions of the modifications are ${24}\,{\mu \textrm {m}}$ (length) by ${28}\,{\mu \textrm {m}}$ (height) by ${7}\,{\mu \textrm {m}}$ (width).

 

Fig. 4. Microscope images of the laser modifications in the fiber core. Top view is in the direction of laser irradiation and side view is orthogonal to it. The contrast of the brightfield images is low, so additional phase contrast images are provided. Here a higher refractive index shows up darker e.g. the fiber core is the dark band in the center of the pictures.

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If light is coupled into the fiber, it interacts with the localized modifications and is partially scattered out. This can be seen under the microscope (Fig. 1c), where the modifications appear as clear, bright spots. Without optical magnification, the spots merge into a cumulative continuous emission profile.

For the quantitative investigation of the scattering phenomenon, transmission and emission spectra were recorded. The extensive data set of one series of modifications is displayed in Fig. 5 as two color maps, one for transmission and one for emission. The independent variable is either the number of the modification in the first- or the position in the second case. To better visualize the decline in flux we averaged the data over three different wavelength intervals (which roughly represent the upper-, lower and the whole wavelength range). This is shown in the graphs on top of the corresponding heat maps.

 

Fig. 5. Wavelength-resolved transmission and emission plots for scattering centers with constant spacing. Transmission spectra are plotted as a function of the modification number and the emission spectra as a function of position starting from the initial maximum. The top plots show the average spectral flux in certain wavelength intervals. This shows a steady decline in transmitted power with consecutive modifications for in transmission and an overall decline but with local maximums and minimums in emission.

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The averaged transmission curves are declining monotonously with increasing number of scattering centers from their highest value at the start to the lowest value at the end of the examined section. The emission (which is zero outside of the modified area) jumps to its highest value at the first modification. Then it declines rapidly for 10 mm before it crosses over into a section where the overall rate of decline is less and several local minima and maxima appear.

The spectrally resolved measurement shows that the transmission as well as the emission is wavelength-dependent. Figure 6 displays both sets of spectra in corresponding stages. In transmission, the strongest decline is found after 1000 modifications at the shortest observed wavelength of 600 nm, where 55% of the initial flux is transmitted through the modified length. Going to higher wavelength, the dampening decreases until its lowest value at around 820 nm where more than 70% is transmitted. Then it declines again to 65% at the local minimum in the near infrared wavelength range at around 900 nm. The shape of the emission curves resembles the shape of the transmission when the order of spectra and the magnitude is inverted: it is approximately a downscaled mirror image. Its highest value of 3% at 600 nm is found at the very start of the modification. Then the emission declines with increasing wavelength to the minimum value of 1.75% at 820 nm and it increases again to 2% at 900 nm which is a local maximum. The magnitude of the emission spectra decreases from the start to the end of the modified length.

 

Fig. 6. Transmission spectra for increasing amounts of scattering centers (indicated by the labels) with their corresponding emission spectra, measured at the indicated positions.

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4.2 Model fit

For data processing, the analysis from Fig. 2 is applied to the measurements to yield the scattering spectrum of a single modification. According to Eq. (7) and Eq. (9) a double-exponential decay governs both the transmission and emission behavior. This translates to a fit of either Eq. (7) or Eq. (9) to each row in the heat-map of a transmission or emission plot, respectively. To improve the data quality for the fitting, we limited the data range from 0 to 600 modifications. Some selected fit results are shown in Fig. 7. We numerically integrated the data over the wavelength intervals provided in the legend as indicated in the figure captions. This has no consequence for the underlying analysis and was solely done for better visualization.

 

Fig. 7. Transmission and emission spectra integrated over three different wavelength ranges and plotted as a function of the number of scattering centers or the position with their respective fits according to Eq. (7) and Eq. (9)

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We obtained three scattering coefficients from every fit which represent the fraction of energy transfer between core, cladding and free space for a single wavelength. These separate values were then stacked and plotted over their respective wavelength to yield the average scattering spectrum per modification $s_{ab}(\lambda )$ as shown in Fig. 7 for $s_{23}$. The initial decay $s_{12} + s_{13}$ is too short to yield a meaningful spectrum, therefore we give the average value of ${0.03}\,{\textrm {per}}\ {\textrm {modification}}$ for transmission and emission. The fit results for the $s_{23}$ scattering spectrum for transmission and emission are plotted in Fig. 8 together with their respective standard error. This represents the scattering spectrum of one modification for cladding modes.

 

Fig. 8. Fit results of Eq. (7) and Eq. (9) to every wavelength of the emission and transmission data set. This yields scattering spectra $s_{23}$ of a modification in emission or transmission with its corresponding standard error (shaded area). The spectra show an increase in scattering for lower wavelength as well as a local maximum at 900 nm and the difference in magnitude $\Delta$. A scattering function $\propto \lambda ^{-4}$ was fitted to the transmission data in the range 600 nm to 840 nm with an $R^2=0.967$.

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In transmission the highest amount of 0.07% light scattering per modification is found for the lowest measured wavelength of 600 nm. The scattering power then declines with increasing wavelength until the minimum is reached at around 820 nm with 0.035%. Then the scattering increases again up to 0.05% at the local maximum which is located at around 900 nm. The emission spectra show a similar shape and value range as compared to the transmission data. The difference $\Delta$ of both spectra has the shape of a broad maximum centering around 800 nm. The relative fit error in transmission lies between 1% (740 nm) and 4% (600 nm) with an average value of 2%, which is too small to be visible in Fig. 8. The relative fit error in emission lies between 4% (740 nm) and 20% (600 nm) with an average value of 7%. Here, relative errors greater than 10% are only found for wavelength smaller than (650 nm).

5. Discussion

5.1 Light scattering centers

The combination of microscope imaging and dampening measurement allows us to obtain information on the inner structure of the laser modifications. We can see from the optical images that the modifications consist of refractive index fluctuations, which are contained in a flat but otherwise irregular shaped region. This form is different from a normal focal spot [14] which would be rotationally symmetric when viewed from the top. The straightforward explanation for the derivations is the refraction of the laser on the cylindrical shaped fiber surface. This introduces aberrations which could result in such a shape. Additionally, such aberration depends on the alignment of the fiber with respect to the laser. The outer shape of the modification will probably influence the symmetry of the angular scattering pattern and the total scattering power of the modification. While the former is compensated by the integrating sphere measurement device, which covers the whole solid angle, the latter is determined experimentally by the fitting procedure. Therefore, this asymmetry does not influence the outcome of this experiment and could even be used to customize the angular emission profile.

The emission and transmission data show that the average decay constant of the core flux $s_{12}+s_{13}$ is approximately 60 times larger than that of cladding flux. This has the effect that the fiber core is rapidly depleted and the second exponent (representing the decay of cladding flux) becomes dominant after about 100 modifications. Then it defines the larger part of the emission and transmission profiles. These different roles are caused by the different diameters of core and cladding. The cross-sectional area of the scattering center is a larger fraction of the core than of the cladding. Therefore, the scattering coefficient is smaller for the cladding flux.

The obtained spectra for $s_{23}$ provide us with additional insight into the scattering mechanisms of an average modification. Both show similar features but differ by a wavelength dependent amount $\Delta$. Because this additional attenuation only affects the light emission, we conclude that it is an effect of the plastic fiber coating, probably true absorption. The remaining distinct features we will investigate further are the local maximum around 900 nm and the increase of scattering when decreasing the wavelength from 800 nm to 600 nm. Both features are found in transmission and emission, so we infer that they are caused by elastic scattering.

The increase in scattering for decreasing wavelength shows the classical proportionality with $\lambda ^{-4}$ ($R^2=0.967$ over the wavelength range of 600 nm to 840 nm). The maximum around 900 nm indicates the existence of another scattering mechanism. It is probably also caused by fluctuations, but with a period comparable to that of the scattered light: Marcuse showed that the power loss of an optical waveguide with random wall perturbations is at its maximum if the correlation length of the distortions is approximately equal to the wavelength of the guided light [15]. The same idea can be applied to refractive index fluctuations in the fiber core.

These kinds of perturbations should be resolvable with a microscope and indeed, the modifications in Fig. 4 show inner structures with fluctuations in the same order of magnitude. We assume that these inner structures are a result of multiple rapid heating and cooling cycles caused by laser irradiation. Their emergence might be similar to the formation of nanogratings, which are caused by a feedback between the laser beam and the induced microstructure [16]. This interaction leads to a structural evolution from single spots for one laser shot to oriented elongated shapes with increasing number of pulses. A similar process, but with the addition of cylindrical aberrations, might be responsible for the observed fluctuations. To further investigate these scattering phenomena, an evaluation of the angular spectrum of the scattered radiation is necessary. This would yield a distribution of the mechanical frequencies.

5.2 Emission profile

In Section 2, we postulated that the scattering centers can be regarded as sufficiently independent of each other to allow for direct shaping of the emission profile by placing the scattering centers at well-controlled step-width.

According to our model, a constant spacing between the modifications should result in a double-exponential decay in transmission and emission, which is also consistent with a longitudinally homogeneous scattering function [17]. This is confirmed by the fit of Eq. (7) and Eq. (9) to both measurement series shown in Fig. 7. In transmission, the curve shape is in good agreement with the theoretical considerations. Regarding emission, Eq. (9) could also be fitted to the data set, except for several local maxima and minima, which cause the scattering spectrum of $s_{23}$ to be a lot noisier than in transmission. Still, both spectra span the same range and show roughly the same features. This indicates that both procedures record the same effects.

The local extrema we can observe in the emission profile are the result of laser writing errors which are caused by a loss of beam focus. This is due to strong aberrations which are brought about by fluctuations in coating thickness: the affected modifications are either smaller or fainter than the average scattering center, or they have not been written at all. A possible solution to this problem would be to use refractive index matching oil, which would compensate such outer surface irregularities.

Surprisingly, deviations from the ideal exponential curve shape are far stronger in emission than in transmission. This is a counterintuitive consequence of the property discussed in Section 2: contrary to the emission, the transmission is independent of scattering center spacing when there is no interaction. In transmission, only measurements after a successful modification were recorded. Therefore, it shows a double-exponential decay which is in agreement with the model. In emission, the whole length of the chain of modifications was recorded, so missing scattering centers show up as minima. Additionally, according to Eq. (4), the local emission of one modification is determined by the product of its scattering coefficient with the present flux in the core or in the cladding. This makes it even more susceptible to small variations in scattering power.

Besides these irregularities, the model is able to describe the emission and transmission behavior. Nevertheless, the consistency of the theoretical prediction with the measured values might be coincidental for the special case of constant modification spacing. As an additional test for the proposed approach, we generated an arbitrary emission profiles with aperiodic spacing,

$$d_{n+1} = 0.002 ({60}\,{\textrm{mm}}-d_n)\, .$$
Hence, the first distance is $d_0 =$ 0.120 mm and the last is $d_{1000} =$ 0.016 mm. The last interval is smaller than the length of the modification, so at the end of the modified region scattering centers start to overlap. Again, we created 1000 modifications and performed transmission and emission measurements as in the previous section. Results are provided in Fig. 9.

 

Fig. 9. a) Measured transmission with fits according to Eq. (7) at three selected wavelength ranges shows a second degree exponential decay. b) Comparison of the measured emission behavior and the calculated emission profile. The right-side maximum is caused by the decrease of the scattering center distance leading to an increase in emission per unit length.

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The measured emission profile in Fig. 9 is U-shaped. This shape is caused by the deformation of the two exponential functions by consecutive decrease of the modification spacing. The high emission values on the left side of the curve correspond to the initial decay of the two exponentials. The increasing line density of scattering centers towards the end of the profile causes the high emission on the right side. An overlap of both effects leads to partial compensation and therefore causes the almost flat emission profile in the central section.

Again, the transmission curve follows a double-exponential decay function, but with lower dampening due to a difference in focal spot position with respect to the fiber core. Qualitatively the overall behavior is in accordance with our hypothesis that the emission profile is shaped by scattering center positioning but the transmission is not.

We extracted the three different scattering spectra $s_{ab}$ from the transmission measurement by fitting Eq. (7) to the data set. The obtained values where then inserted into Eq. (9) to calculate the emission spectra. To apply this formula, the derivation of the function $n(z)$ was approximated in the following way:

$$\frac{dn}{dz} \approx \frac{\Delta n}{\Delta z}=\frac{1}{d_n}\, .$$
This gives the emission per unit length at the position of every modification. The last step was to plot this emission as a function of the position of each modification known from the iterative formula Eq. (13). This calculated emission profile is also shown in Fig. 9. Apart from the irregularities in emission caused by writing errors, the shape of the calculated distribution closely follows the measured one; this holds true for all calculated wavelengths.

Nevertheless, the model shows deviations at the start of the modified length. This is due to the additional effects mentioned in Section 2: the convolution of the measured data with the instruments point spread function and the fiber cladding as a possible third light guiding layer. Also, the small number of measurement points, where the $s_{12}+s_{13}$ exponent dominates the transmission profile, makes the fit of this exponent more prone to errors. Despite these shortcomings, the model was capable of predicting the trend of the emission behavior from the transmission data. Therefore, we conclude that the assumption, which leads to the development of the model, i.e., the independence of the scattering centers, can also be applied to modifications of varying distance. This means that arbitrary emission profiles can be designed just by deliberate spacing of similar scattering centers.

This property can also be used to increase the scattering power of the modified area. Figure 6 indicates the range of the transmission through the modified fiber segment after 1000 modifications from 50% to 70%. Depending on the application, this value should be close to zero or at least controllable. In the context of our model, the way to enhance the emission from the fiber and thereby decrease the transmission is to increase the number of scattering centers. This can be achieved either by increasing the length of the light emitting fiber segment or decreasing the distance between the scattering centers. If this is not desired, then another possibility is to make a second chain of scattering centers, which is slightly parallel displaced to the first one. In this way, the number of scattering centers is doubled, therefore just the scattering coefficients $s_{ab}$ increases but the longitudinal distance of scattering centers stays the same. All these methods results in a double exponential decay with lower transmission but with different longitudinal emission properties. These can again be predicted with the presented model.

However, one needs to acknowledge the possibility that in high light power application, the interaction of the guided light with the scattering centers could cause the destruction of the fiber due to a process called optical fuse [18]. Here, the interaction of the incident light with the backscattered light could lead to the creation of a new defect just in front of the first modification. Repetition of this process may cause a chain reaction in which the defect front moves backward in the optical fiber towards the light source. Driscoll et al. [18] gave an approximate threshold value for the radiant flux density $q$ in the order of 1MW/cm−2 for the creation of the fuse. Fortunately, radiant flux $Q$ required for photodynamic therapy lies between 1W and 5W [1], which would lead to an average radiant flux density $q=Q/(2\pi r)$ between 0.3MW/cm−2 and 1.5MW/cm−2 in the present fiber core with a radius $r$ of ${10}\,{\mu m}$. This puts the upper limit of the required flux on the lower end of the threshold value. If necessary, the problem can easily be avoided by taking a fiber with a slightly higher core radius. For example, a core radius of ${15}\,{\mu m}$ would lead to an average radiant flux density of 0.7MW/cm−2 while the radiant flux is 5MW.

6. Conclusions

In summary, we tested a new method for creating tailorable emission profiles for side-emitting optical fiber segments by focused femtosecond laser irradiation, generating chains of refractive index perturbations alongside the fiber core. The scattering spectrum of each individual such emission center is broadband and wavelength-dependent. Cumulation of all individual scattering centers leads to an overall emission profile which is determined by the employed step-width between the laser-written modifications. Thereby, the writing can be periodic or aperiodic.

Using a model based on simple considerations of Lambert-Beer’s law and a three-level transfer process, the emission behavior of longitudinal chains of scattering centers can be predicted with satisfying accuracy, for regular as well as for irregular spacing. As the scattering centers can be treated as acting independent of each other, they can be used as building blocks for creating a virtually arbitrary emission pattern such as required in a certain illumination task.

Funding

Thüringer Ministerium für Wirtschaft, Wissenschaft und Digitale Gesellschaft (TMWWDG) (41-5507-2016); Leibniz-Institut für Naturstoff-Forschung und Infektionsbiologie – Hans-Knöll-Institut (HKI) (SAS-2015-HKI-LWC); Deutsche Forschungsgemeinschaft (DFG) (GRK2101/1).

Acknowledgments

This work was supported by a Strategy and Innovation Grant from the Free State of Thuringia (41-5507-2016), the Leibniz ScienceCampus InfectoOptics (SAS-2015-HKI-LWC) and Deutsche Forschungsgemeinschaft (DFG) (GRK 2101/1). We thank Dietmar Güttler for assistance with the PC based automated measurement system and Thomas Kittel for assistance with the phase contrast microscopy.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. 53(9), R61–R109 (2008). [CrossRef]  

2. A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008). [CrossRef]  

3. L. Vesselov, W. Whittington, and L. Lilge, “Design and performance of thin cylindrical diffusers created in ge-doped multimode optical fibers,” Appl. Opt. 44(14), 2754–2758 (2005). [CrossRef]  

4. T. M. Baran and T. H. Foster, “Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy,” Med. Phys. 41(2), 022701 (2014). [CrossRef]  

5. A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018). [CrossRef]  

6. A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

7. K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006). [CrossRef]  

8. L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015). [CrossRef]  

9. L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019). [CrossRef]  

10. D. Marcuse, Principles of Optical Fiber Measurements (Academic Press, 1981), pp. 11–68.

11. A. W. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 2012), pp. 135–140.

12. G. Andrehs, H. Beyer. Theorie und Praxis der Interferenzmikroskopie (Akademische Verlagsgesellschaft Geest & Portig K.-G., 1974).

13. J. F. Clare, “Comparison of four analytic methods for the calculation of irradiance in integrating spheres,” J. Opt. Soc. Am. A 15(12), 3086–3096 (1998). [CrossRef]  

14. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006). [CrossRef]  

15. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972), pp. 376–387.

16. F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016). [CrossRef]  

17. Z. Pan and L. Wondraczek, “Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission,” Sci. Rep. 8(1), 9527 (2018). [CrossRef]  

18. T. Driscoll, J. Calo, and N. M. Lawandy, “Explaining the optical fuse,” Opt. Lett. 16(13), 1046–1048 (1991). [CrossRef]  

References

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  1. B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. 53(9), R61–R109 (2008).
    [Crossref]
  2. A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
    [Crossref]
  3. L. Vesselov, W. Whittington, and L. Lilge, “Design and performance of thin cylindrical diffusers created in ge-doped multimode optical fibers,” Appl. Opt. 44(14), 2754–2758 (2005).
    [Crossref]
  4. T. M. Baran and T. H. Foster, “Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy,” Med. Phys. 41(2), 022701 (2014).
    [Crossref]
  5. A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
    [Crossref]
  6. A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.
  7. K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
    [Crossref]
  8. L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
    [Crossref]
  9. L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
    [Crossref]
  10. D. Marcuse, Principles of Optical Fiber Measurements (Academic Press, 1981), pp. 11–68.
  11. A. W. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 2012), pp. 135–140.
  12. G. Andrehs, H. Beyer. Theorie und Praxis der Interferenzmikroskopie (Akademische Verlagsgesellschaft Geest & Portig K.-G., 1974).
  13. J. F. Clare, “Comparison of four analytic methods for the calculation of irradiance in integrating spheres,” J. Opt. Soc. Am. A 15(12), 3086–3096 (1998).
    [Crossref]
  14. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006).
    [Crossref]
  15. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972), pp. 376–387.
  16. F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
    [Crossref]
  17. Z. Pan and L. Wondraczek, “Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission,” Sci. Rep. 8(1), 9527 (2018).
    [Crossref]
  18. T. Driscoll, J. Calo, and N. M. Lawandy, “Explaining the optical fuse,” Opt. Lett. 16(13), 1046–1048 (1991).
    [Crossref]

2019 (1)

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

2018 (2)

A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
[Crossref]

Z. Pan and L. Wondraczek, “Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission,” Sci. Rep. 8(1), 9527 (2018).
[Crossref]

2016 (1)

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

2015 (1)

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

2014 (1)

T. M. Baran and T. H. Foster, “Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy,” Med. Phys. 41(2), 022701 (2014).
[Crossref]

2008 (2)

B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. 53(9), R61–R109 (2008).
[Crossref]

A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
[Crossref]

2006 (2)

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006).
[Crossref]

2005 (1)

1998 (1)

1991 (1)

Andrehs, G.

G. Andrehs, H. Beyer. Theorie und Praxis der Interferenzmikroskopie (Akademische Verlagsgesellschaft Geest & Portig K.-G., 1974).

Baran, T. M.

T. M. Baran and T. H. Foster, “Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy,” Med. Phys. 41(2), 022701 (2014).
[Crossref]

Beck, J.

A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

Beck, J. C.

A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
[Crossref]

Betz, V.

A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
[Crossref]

Brakhage, A. A.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Calo, J.

Clare, J. F.

Driscoll, T.

Foster, T. H.

T. M. Baran and T. H. Foster, “Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy,” Med. Phys. 41(2), 022701 (2014).
[Crossref]

Gottschaldt, M.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Itoh, K.

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

Köhler, A.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Küsel, K.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Lasser, T.

Lawandy, N. M.

Leitgeb, R. A.

Leutenegger, M.

Lilge, L.

A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
[Crossref]

A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
[Crossref]

L. Vesselov, W. Whittington, and L. Lilge, “Design and performance of thin cylindrical diffusers created in ge-doped multimode optical fibers,” Appl. Opt. 44(14), 2754–2758 (2005).
[Crossref]

A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

Love, J.

A. W. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 2012), pp. 135–140.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972), pp. 376–387.

D. Marcuse, Principles of Optical Fiber Measurements (Academic Press, 1981), pp. 11–68.

Méndez-Ramos, J.

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Müller, F. A.

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Nolte, S.

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

Okawa, J.

A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

Pan, Z.

Z. Pan and L. Wondraczek, “Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission,” Sci. Rep. 8(1), 9527 (2018).
[Crossref]

Patterson, M. S.

B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. 53(9), R61–R109 (2008).
[Crossref]

Plech, A.

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

Pohnert, G.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Rao, R.

Rendon, A.

A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
[Crossref]

A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

Richter, S.

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

Schacher, F. H.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Schaffer, C. B.

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

Schubert, U. S.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Snyder, A. W.

A. W. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 2012), pp. 135–140.

Tunnermann, A.

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

Tyystjärvi, E.

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Vesselov, L.

Watanabe, W.

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

Weersink, R.

A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

Whittington, W.

Wilson, B. C.

B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. 53(9), R61–R109 (2008).
[Crossref]

Wondraczek, L.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Z. Pan and L. Wondraczek, “Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission,” Sci. Rep. 8(1), 9527 (2018).
[Crossref]

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Yassine, A.-A.

A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
[Crossref]

Zhang, Q.

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Zimmermann, F.

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

Adv. Mater. (1)

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Adv. Sci. (1)

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Appl. Opt. (1)

J. Biophotonics (1)

A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
[Crossref]

J. Opt. Soc. Am. A (1)

Laser Photonics Rev. (1)

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

Med. Phys. (1)

T. M. Baran and T. H. Foster, “Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy,” Med. Phys. 41(2), 022701 (2014).
[Crossref]

MRS Bull. (1)

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (2)

B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. 53(9), R61–R109 (2008).
[Crossref]

A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
[Crossref]

Sci. Rep. (1)

Z. Pan and L. Wondraczek, “Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission,” Sci. Rep. 8(1), 9527 (2018).
[Crossref]

Other (5)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972), pp. 376–387.

D. Marcuse, Principles of Optical Fiber Measurements (Academic Press, 1981), pp. 11–68.

A. W. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 2012), pp. 135–140.

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Figures (9)

Fig. 1.
Fig. 1. Schematics and microscope image of light scattering on femtosecond laser modifications. a) Creation of scattering centers through focused femtosecond laser irradiation of the fiber core. b) Light scattering on the laser modifications in a two stage process: Light is scattered out of the core into free space and into the cladding, where it is again guided by total internal reflection or, eventually, scattered out into the environment. c) Microscope image of light scattering on laser modifications.
Fig. 2.
Fig. 2. Three-level representation of the energy exchange between the core, the cladding and free space: At first all light is contained in the core and gets scattered into free space (green) and the cladding (red). After some distance this leads to a mixed case where light is also contained in the cladding and scattered into free space (blue) and back into the core (yellow). Because the latter effect is very small, the core is eventually depleted and only the remaining light in the cladding is scattered.
Fig. 3.
Fig. 3. a) Motorized linear stage: The integrating sphere is moved incrementally alongside the optical fiber; the spectrometer measures an emission spectrum for every position. b) Integrating sphere: Light emitted by the fiber segment $\Delta z$ - limited by the fiber guide - is homogeneously distributed on the sphere wall by multiple diffuse reflections. The irradiance on the detector port is proportional to the emitted flux.
Fig. 4.
Fig. 4. Microscope images of the laser modifications in the fiber core. Top view is in the direction of laser irradiation and side view is orthogonal to it. The contrast of the brightfield images is low, so additional phase contrast images are provided. Here a higher refractive index shows up darker e.g. the fiber core is the dark band in the center of the pictures.
Fig. 5.
Fig. 5. Wavelength-resolved transmission and emission plots for scattering centers with constant spacing. Transmission spectra are plotted as a function of the modification number and the emission spectra as a function of position starting from the initial maximum. The top plots show the average spectral flux in certain wavelength intervals. This shows a steady decline in transmitted power with consecutive modifications for in transmission and an overall decline but with local maximums and minimums in emission.
Fig. 6.
Fig. 6. Transmission spectra for increasing amounts of scattering centers (indicated by the labels) with their corresponding emission spectra, measured at the indicated positions.
Fig. 7.
Fig. 7. Transmission and emission spectra integrated over three different wavelength ranges and plotted as a function of the number of scattering centers or the position with their respective fits according to Eq. (7) and Eq. (9)
Fig. 8.
Fig. 8. Fit results of Eq. (7) and Eq. (9) to every wavelength of the emission and transmission data set. This yields scattering spectra $s_{23}$ of a modification in emission or transmission with its corresponding standard error (shaded area). The spectra show an increase in scattering for lower wavelength as well as a local maximum at 900 nm and the difference in magnitude $\Delta$ . A scattering function $\propto \lambda ^{-4}$ was fitted to the transmission data in the range 600 nm to 840 nm with an $R^2=0.967$ .
Fig. 9.
Fig. 9. a) Measured transmission with fits according to Eq. (7) at three selected wavelength ranges shows a second degree exponential decay. b) Comparison of the measured emission behavior and the calculated emission profile. The right-side maximum is caused by the decrease of the scattering center distance leading to an increase in emission per unit length.

Equations (14)

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d ϕ d z = ( α + σ ) ϕ ϕ ( z ) = ϕ 0 exp { 0 z α ( z ) + σ ( z ) d z } .
d ϕ 1 d n = ( s 12 + s 13 ) ϕ 1 + s 21 ϕ 2 ,
d ϕ 2 d n = s 12 ϕ 1 ( s 23 + s 21 ) ϕ 2 ,
d ϕ 3 d n = s 13 ϕ 1 + s 23 ϕ 2 .
ϕ 1 ( n ) = ϕ 0 exp { ( s 12 + s 13 ) n } ,
ϕ 2 ( n ) = ϕ 0 s 12 s 12 + s 13 s 23 [ exp { s 23 n } exp { ( s 12 + s 13 ) n } ] .
T ( n ) = ϕ 1 + ϕ 2 ϕ 0 = ( 1 A s ) exp { ( s 12 + s 13 ) n } + A s exp { s 23 n } .
A s = s 12 s 12 + s 13 s 23 .
E ( z ) = 1 ϕ 0 d ϕ 3 d z = 1 ϕ 0 d ϕ 3 d n d n d z ,
= [ ( s 13 s 23 A s ) exp { ( s 12 + s 13 ) n ( z ) } + s 23 A s exp { s 23 n ( z ) } ] d n d z .
M = ρ ϕ A ( 1 ρ ( 1 f ) ) .
M i M 0 = ϕ i ϕ 0 .
d n + 1 = 0.002 ( 60 mm d n ) .
d n d z Δ n Δ z = 1 d n .

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