In the past years, several examples of waveguides or gratings manufactured by laser radiation in various glasses or transparent polymers have been demonstrated. As these devices rely on a modification of the refractive index in the vicinity of the focal point of a writing laser, phase retrieval methods are well suited to determine the induced phase change inside the material experimentally. Thus, phase retrieval can assist parameter-studies and reveal information about the internal structure of the modifications which can hardly be obtained using other methods. Here, we demonstrate the application of a modified phase retrieval technique especially suited to determine the optical properties of phase objects such as femtosecond written gratings or waveguide arrays inscribed in transparent materials. Using the presented algorithm, we quantify the correlation between pulse energy and phase deviation as well as the structure width for femtosecond laser induced refractive index modifications in poly(methyl methacrylate) which, in the future, can be used to create tailored micro-optical structures for sensing applications.
© 2016 Optical Society of America
Fabrication of optical waveguides, gratings and other microstructures in dielectric materials using tightly focused femtosecond laser radiation has become a wide field of research over the past years. While early work was focused on waveguide formation in glass and crystalline materials , recent research includes the fabrication of more sophisticated devices such as diffraction gratings [2,3], three-dimensional splitter devices  and combined microfluidic and optical structures .
In recent years, direct writing of waveguides [6–8] and gratings  in polymer bulk material as an alternative to materials such as glass was under investigation. Writing in polymer material is especially interesting since it opens the possibility for mass-production of micro-optical devices in addition to already existing technologies for high-volume polymer processing such as hot-embossing  and lamination . Additional beneficial properties of polymer substrates such as low mass, high chemical resistivity, high flexibility and, lastly, low material costs expand the potential fields of applications for future plastic, femtosecond-laser processed optical devices. These include devices for mechanical and biomedical sensing applications and especially disposable lab-on-chip sensors.
For a precise adjustment of process parameters as a requirement for a reliable mass-production, however, it is essential that the underlying process yielding optical modifications in polymers is well understood. Femtosecond laser writing in polymers depends on mechanical (surface quality, writing speed), chemical (base material, additives) and optical (wavelength, pulse energy, repetition rate, pulse length, NA of focusing device) properties and parameters. Considering this large number of variables, parameter studies – which are required to fully grasp the mechanism of waveguide formation – become hardly feasible due to the vast amount of experimental runs. This is illustrated by the fact, that the refractive index modification induced in polymers can be quite different in shape from material alterations in glasses. For instance, instead of waveguiding inside the material modification, in femtosecond-processed polymer waveguides were observed around  and also below  the modified volume, which was attributed to material densification as result of thermal expansion at the focal point.
To assist future characterization of femtosecond-written structures, we here present an application of a phase-retrieval algorithm, originally proposed by Farn , based on iterative calculation of the phase α(n) of the diffraction image produced by a diffracting structure such as a waveguide array or a grating and the phase φ(k) of the diffracting structure itself. A technique using the standard Gerchberg-Saxton algorithm was shown by Berlich et al.  just recently as a proof-of-concept. The algorithm is based on the assumption of a phase-only diffracting structure and is especially suited for one-dimensional problems which increases computational efficiency. Its result, the height and width of the phase function φ(k) can assist in investigation of relative refractive index changes of the inscribed modifications as well as their spatial dimensions. We shortly introduce the algorithm as well as a method for determination of suitable starting values, apply the method to a set of femtosecond-written gratings fabricated with different pulse energies and verify the results using a set of microscope images as reference.
2. Experimental method
2.1. Specimen fabrication
We inscribed a periodic sequence of linear phase discontinuities in commercially available poly(methyl methacrylate) (PMMA) provided by microfluidic ChipShop GmbH (PMMA-03). The material modifications were created using a lab-made 1048 nm femtosecond laser system based on two Yb:KYW laser crystals in a chirped pulse oscillator  with an output repetition rate of 1 MHz. The pulse duration was compressed to 600 fs and the repetition rate was reduced to 100 kHz. This set of parameters already yielded good results in an earlier waveguide writing experiment performed with the same laser system . To obtain a tightly focused laser spot inside the bulk material (150 μm below the surface), we utilized an NA = 0.55 Newport 5722-H-B aspheric lens which was placed behind an automatic aperture which was open during writing of a modification and closed otherwise while the laser system was operated continuously. The PMMA substrate with a thickness of 1.5 mm was fixed on an Aerotech ABL 1000 positioning system and was moved with a feedrate of v = 60 mm/s during writing of a single linear modification in x-direction, see Fig. 1. After the full length of the substrate was reached, the aperture was closed to prevent further material modification and the specimen was repositioned at x = 0 mm and the next y-position to write another adjacent linear modification. The laser light was linearly polarized in writing direction. In this work, we fabricated linear gratings with a grating period of 10 μm. Six different specimen were manufactured using pulse energies ranging from 100 nJ to 200 nJ for investigation with the phase-retrieval method. For optical inspection and comparison, an additional specimen was fabricated with a pulse energy of 300 nJ.
2.2. Optical characterization setup
To deduce the phase modification created by the fabrication process as described in the preceding section, we recorded the far-field diffraction pattern generated by the grating structure using the setup depicted in Fig. 2. The device under test was positioned perpendicular to a HeNe laser beam of 4.5 mW with a diameter of 0.8 mm and a wavelength of 633 nm. The intensities of the maxima of the resulting diffraction image for the diffraction orders of n = −8 . . . + 8 were monitored using a Thorlabs laser powermeter setup consisting of a S120C photodiode attached to a PM200 handheld device which was located 400 mm away from the specimen. All diffraction maxima were measured 100 times and the average value was calculated. The standard deviation of the measurement was typically 0.1 % of the average value. The intensity values were used for later processing by the phase retrieval algorithm.
3. Phase retrieval algorithm and generation of start values
To calculate the phase change introduced to light propagating through the specimen as described previously, a method for phase retrieval is required. The algorithm utilized in this work is based on work of Farn  who proposed an iterative approach to design one-dimensional beam shaping devices such as 1-to-5 couplers with a predefined intensity distributions. A comprehensive explanation of the algorithm is given in , therefore, we restrict ourselves to a brief outline of the functioning principle and introduce our extension for this application. The phase retrieval algorithm is based on the assumption of a phase only grating which produces a diffraction pattern with a discrete number of diffraction orders. The complex electrical field which is obtained in the far field for each diffraction order n can be described asEquation 1 represents the discrete Fourier transform (DFT) of the phase-only grating function. The goal of the algorithm is to find a grating phase whose properties generate a diffraction pattern that matches the efficiency and shape of the measured diffraction pattern. To optimize both features simultaneously we utilized the merit function from . (1) into (2) leads to the conclusion that M will be maximized if φ(k) satisfies the following relation
The algorithm now calculates the phase α0(n) of c(n), starting with a random phase φ0(k). With α0(n) it then calculates a new phase φ1(k) applying equation (3). During an iterative loop, as displayed in Fig. 3(a), every c(n) generated by the current φj (k) is monitored and compared to the measured diffraction intensities. From the difference, a set of weights w(n) – one for each diffraction order – is generated which is used in a negative feedback loop to control the direction of convergence.
Since the random starting phase φ0(k) leads to a random starting point in the solution space of possible phase functions it is desirable to place the starting point of the algorithm near the supposed solution of the problem to achieve a fast convergence. From our manufacturing process the period of the grating formed by the linear discontinuities is well known and is set to Λ = 10 μm and, thus, we can infer that the resulting phase function φ(k) must exhibit a peak of unknown width and height and be zero elsewhere. To find suitable starting parameters based on this assumption, we constructed a starting phase function based on an inverted parabola with defined height and width. The actual values for both parameters were determined using a brute-force approach where width and height were varied. We calculated the resulting |c(n)|2 for each parameter pair using Eqn. 1 and chose the starting phase function for which the RMS error between calculated and measured diffraction image was minimal.
Figure 3(b) shows the constructed starting phase function and Fig. 3(c) the RMS error to a given measured intensity distribution as a function of width and height. The minimum value corresponding to the specific phase function φ0(k) which is then used to start the iterative algorithm is marked with a red circle in Fig 3(c). From there, the algorithm converges to the next minimum which is the desired solution for the given parameter set. The stability of the algorithm depends largely on the amplification of the feedback, i.e. the calculation of the weights. Here, we calculate the weights to wj (n) = wj−1(n) − 0.05 · (cj−1(n) − c * (n)), generating a slightly integrating feedback. To avoid instabilities we also allow only positive weights and normalize wj (n) after calculation, so that max(wj (n)) = 1.
To evaluate the robustness of both the brute-force and the iterative algorithm, we ran the method multiple times with a measured intensity distribution where normally distributed noise was added. The set of intensity distributions that was generated showed a standard deviation of 0.5 % for each diffraction maximum. While height and width of the resulting phase function for the brute-force approach did not show any remarkable differences to the results generated with the original data, the iterative algorithm adapted to the noisy data with a skewed phase function of almost the same width and height, depending on the skewness of the modified diffraction pattern caused by the added noise. In general, phase retrieval algorithms might converge to non-physical solutions due to a wrong selection of starting parameters such as the initial starting phase function. In our case the brute force approach generates an initial phase function which is similar to the real physical solution and, hence, no convergence issues were observed.
4. Results and discussion
To analyze the cross-section of the fabricated modifications by visual inspection, the specimens were cut perpendicular to the writing direction of the laser, as depicted in Fig. 4(a), and polished afterwards. The facets were evaluated using a conventional microscope. Corresponding microscope images of cross-sections for specimens fabricated at pulse energies of 100 nJ, 200 nJ and 300 nJ are exemplarily shown in Figs. 4(b)–4(d), respectively. For all three pulse energies, material modifications are observable along the writing track which are most likely caused by an increased laser intensity in the vicinity of the focal spot of the writing laser. While for pulse energies between 100 nJ and 200 nJ the diameter of the tracks increased with pulse energy, a smaller secondary modification can be observed directly below the first modification for a pulse energy of 300 nJ, as shown in Fig. 4(d). According to , these can be explained by a refocusing effect caused by interaction between the existing upper index modification and unabsorbed light.
The exact transition point from a single modification to dual modifications by refocusing is hard to determine with bright field microscopy alone. Alterations to the material at the secondary focus might already occur before they become apparent via optical inspection. We therefore restrict ourselves to a maximum pulse energy of 200 nJ where we are confident that only a single modification occurs. Experiments with more complex modification morphologies will be subject of a future study.
The evaluation of the gratings performed by recording the intensities in the diffraction orders showed an increasing diffraction efficiency with increasing pulse energy. This is consistent with observations of increased refractive index change with increased pulse energy . For the lowest pulse energy, the diffraction efficiency in the +1st diffraction order was η = 0.4 %, for the largest pulse energy η = 5.9 %, calculated as the measured energy in the +1st diffraction order over the sum of energies in all measured diffraction orders. To determine the phase change introduced by the material modifications from the diffraction orders, we utilized the algorithm presented in Section 3. Fig. 5 shows the phase function over a single grating period for manufactured gratings using pulse energies between 100 nJ and 200 nJ. Modifications by the femtosecond-laser beam caused an approximately Gaussian-shaped phase distribution with a width ranging from below 2 μm to over 3 μm depending on the pulse energy.
The maximum phase deviation increases with increasing pulse energy (see Fig. 6) up to the biggest evaluated pulse energy of 200 nJ where it amounts to 0.43 rad. Towards higher energies, the gradient seems to decrease. Since the microscope image presented in Fig. 4(d) shows the generation of a second modified region for higher pulse energies, here 300 nJ, we attribute this effect to a stagnation of induced energy into the first modification. Using the already observed [6, 8] ring-shape waveguiding around femtosecond written structures can hint to a possible explanation. The material in the center of the modification is assumed to be damaged in a way that waveguiding is prevented, as sketched in Fig. 7. For our gratings, this would translate into regions of increased scattering, indicated in white, increasing in size with rising pulse energy.
In our experiments, we also conducted a series of tests with increased pulse energies which showed a further increase in maximum phase deviation. However, once second-order modifications developed, the gratings diffraction images and phase-functions proved to be increasingly non-symmetric. This effect could be caused by the mutual influence between tracks due to their increasingly spatial dimensions when compared to the grating period itself.
We presented the application of an alternative robust and very efficient phase retrieval approach for the characterization of femtosecond-written diffracting structures in polymers such as PMMA. We developed methods for a start-value generation to reduce computation time. Gratings fabricated in PMMA at various pulse energies as the only varying parameter were tested and correlations between pulse energy and phase deviation as well as pulse energy and width of the modified volume are observed. For the near future, we plan to investigate the influence of other parameters such as feedrate, repetition rate or pulse length. Since our main goal is to produce a new type of optical sensor relying on femtosecond-written diffraction gratings in polymer substrates, we will also investigate ways to influence the diffraction behavior of the manufactured gratings by variation of the above mentioned parameters. Also, we aim to fully understand the formation of modifications in polymer by femtosecond-laser radiation and will therefor conduct additional experiments in different polymers using the method presented here.
We acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG) within the scope of the Collaborative Research Centre SFB/TRR 123/1 “Planar Optronic Systems”.
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