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We present a compact design for an integrated interconnect based on a hybrid imaging setup combining microchannel and conventional imaging. Within this setup the conventional imaging is performed by an aluminum-coated spherical lens. The aberrations introduced by this spherical mirror to the channels of the interconnect can be compensated by channel-wise adapted microlenses located at the in- and output interfaces. These microlenses are used for collimating or refocusing the beams, respectively. Within this paper we present the design of the microlens array with individually shaped lenses referred to as chirped mircolens array (cMLA) based on numerical optimization and the use of fitting functions. Further on we focus on the fabrication of the chirped microlens arrays by laser lithography and first experimental results of coupling efficiencies of singlemode as well as multimode fibers for the realized prototypes.
©2006 Optical Society of America
1. Introduction
Optical interconnects for telecommunication applications are utilized to establish optical connections between input and output devices for many channels in parallel. Both in- and output devices are usually arranged in arrays and may e.g. consist of singlemode fibers as input sources and detectors as outputs, respectively. Here, the small I/O components are not densely packed but are placed on a regular grid leading to gaps between the electro-optical elements which are black or simply not required for achieving the optical functionality [1].
Free-space optical interconnects perform an imaging of the input array onto the output array. In order to achieve low loss and low crosstalk connection for all channels the quality of the imaging must be acceptably good over the whole array. Usually the pitches of the in- and output arrays are equal, therefore a 1:1 imaging is requested. A conventional imaging setup possesses constant resolution over the complete image field if aberrations can be neglected. Since high resolution is required for the imaging of the small optical I/O devices, most of the space-bandwidth product is wasted in case of dilute arrays and rather complex, bulky and expensive imaging optics is required. A more suitable approach is the so called hybrid imaging being a combination of microchannel and conventional imaging (Fig. 1) [2, 3, 4]. Here, a first microlens array MLA1 channel-wise collimates the light emanating from the input sources. In a second step a conventional 4F-setup consisting of lenses L1 and L2 images the first array MLA1 onto the second MLA2. The latter refocuses the light channel-wise to the output devices. Since the microlenses provide high resolution locally at the areas of interest, the hybrid imaging scheme distributes the space-bandwidth product as needed.
This approach can be applied for the design of planar-integrated free-space optics. Here, the optical beam path is folded into a monolithically integrated rugged substrate [5].
In a 4F-hybrid imaging setup (Fig. 1) the collimated beams of the different channels hit the first lens L1 of the conventional imaging system at different heights. In case of poorly corrected systems, like single spherical elements, the optical bundles will suffer from different levels of spherical aberration introduced by the macroscopic lens system. Besides using more complex well-corrected optics for the conventional stage of the hybrid imaging system the aberrations can easily be compensated channel-wise by implementing individually adapted mircolenses. Consequently, the microlens arrays are no longer static arrangements of equally shaped and equidistantly distributed lenses but have an appearance referred to as chirped microlens array (cMLA). Each single lens of the array can be described by parameters like e.g. focal length or decenter. In case of an easy modeling of the imaging system the parameter values of each cell can be obtained by analytical functions [6, 7] or otherwise by numerical optimization techniques. The functions depend strongly on the geometry of the system and the position of the cell within the array indicated by its position index (i,j). The use of an array with individually adapted cells in a hybrid imaging setup was already shown in Ref. [8]. However, this was done using a different hybrid imaging configuration (2F/2F instead of a 4F setup). Furthermore it was based on diffractive optical elements leading to low transmission efficiencies and was designed using analytical functions.
Our proposal is based on refractive microlenses and a single off-the-shelf spherical mirror for the conventional part of the hybrid imaging system. This provides the opportunity for building very cost effective free-space interconnects at optimum optical performance and high transmission efficiencies.
In section 2 we discuss the optical system design especially of the cMLA. We introduce an easy to use and time effective approach for calculating the parameters of the cMLA based on numerical optimization of a small number of significant cells and interpolation for calculating the entire array. Section 3 is dedicated to the fabrication of the lens arrays by laser lithography. In section 4 we present measurements obtained on a first demonstrator.
2. Optical system design
The basic design concept is a conventional hybrid imaging system in a 4F configuration (Fig. 1) which is folded into a single glass substrate. For folding the beam path the lenses L1 and L2 need to replaced by curved mirrors. Additionally, a folding mirror placed in the plane of the common focal points is required in the center of the system. A further simplification results by substituting the two curved mirrors by one reflection-coated standard off-the-shelf spherical lens. In this case half of the object plane can be used as the whole system becomes symmetric to the optical axis of the spherical mirror (Fig. 2). Due to the symmetry of the system both in-and output microlens arrays can be expected to be identical except for their orientation.
A further consequence of the symmetry of the system is that the parameters of the microlenses of the array are a function of the radial distance r of the microlens centers to the optical axis of the system. Consequently after the analysis of the fitted functions the parameters for all cells can easily be obtained as they only depend on the radial distance r.
The design was performed for singlemode fibers SMF 28 with a mode field diameter of 10.5μm as in-and output devices using numerical optimization functions provided by the commercial design software ZEMAX. Here, several different configurations for different radial distances r reaching from 1mm to 4mm distance to the optical axis were analyzed.
Within the software the degree of optimization of the optical system is described by a quality function often referred to as merit function. The optimization algorithms try to find the best suited set-up by minimizing the merit function. In a first design step the optimization was performed on geometrical and Gaussian beam propagation considerations for achieving maximum coupling between the in- and output fibers. From the geometrical optics point of view constraints on the telecentricity of the system (chief rays in parallel to the optical axis of the system) as well as on the lateral position of the chief ray -which has to coincide with the fiber location- were implemented. These constraints are necessary for optimum coupling since a lateral displacement d and angular displacementϕ of the incoming optical bundle are in direct connection with coupling losses (Eq. 1 and 2) [9].
Fig. 1. Setup of the hybrid imaging system. F - focal length of the Fourier lens, f - focal length of the microlens.
Herein de and ϕe are normalization values given by Eq. 3 and 4:
with τa being the coupling efficiency without tilt or offset (which can be approximated as unity), ω 01,2 the waist radius of the incoming Gaussian mode and mode field diameter of the single-mode fiber, k the wavenumber = 2π/λ and ω 1,2 the field radius of the incoming Gaussian mode and that of the singlemode fiber at the alignment plane which can be approximated as the waist radii ω 01,2.
Since the system is symmetric to the optical axis, the focal points of the different channels have to coincide on the plane folding mirror. When using a regular microlens array (rMLA) this requirement can not be fulfilled as a consequence of the spherical aberration of the mirror (Fig. 3a) and therefore the symmetry of the system is broken. As a consequence a tilted and off-set bundle at the end face of the fiber leads to a decreased coupling efficiency. An effective way to compensate this effect of spherical aberration caused by the spherical mirror is to tilt the bundle with respect to the optical axis of the system. This can be accomplished by means of a decenter of the microlens with respect to the optical axis of the fiber (Fig. 3b). Since the amount of spherical aberration depends on the ray height on the spherical mirror (Fig. 3c), the tilt angle and therefore the decenter has to be adapted channel-wise leading to a cMLA configuration.
Fig. 3. a) Mismatch of the beam positions at the plane folding mirror and the output fibers caused by the spherical aberrations introduced by the spherical mirror. b) Aberration compensation using a cMLA. c) Optical path difference vs. ray height at spherical mirror (λ=1.55μm).
Another important aspect of the design is to make sure that the waist diameter at the output fiber equals the mode field diameter of the fiber by defining Gaussian propagation constraints. Further on the beam waist needs to be located axially at the fiber’s end face in order to achieve maximum coupling efficiency ηa (Eq. 5) [10].
Herein F(x,y) and W(x,y) denote the electric field distribution of the mode guided by the singlemode fiber and the incoming wave field to be coupled, respectively. The asterisk stands for the mathematical operation of the conjugated complex value. Thus it is possible to calculate the total coupling efficiency ηtotal by multiplying the results for ηa, ηd and ηϕ:
Again, due to the symmetry of the system the beam waist of every single channel after hitting the spherical mirror the first time has to be located at the plane folding mirror. Therefore, due to the different propagation distances the focal lengths of the microlenses have to be adapted for each channel.
The calculation of the Gaussian propagation parameters like waist radius and waist location is based on Fresnel diffraction in a completely analytical manner. Using these operations in the optimization routines leads to a reasonably good starting system for further final optimization with the advantage of short calculation times.
For final optimization the propagation of the wave field trough the system was modeled using the physical optics propagation (POP) tool within ZEMAX. Here, the wave field is described as a set of complex amplitude values at discrete sampling points [11]. The propagation through free space is modeled by the angular spectrum of the plane wave method [12, 13] or employing Fresnel diffraction. Based upon the Fresnel number F of the beam the software automatically decides which propagation method to use for the propagation between the surfaces:
with A being the radius of a single microlens, λ the wavelength and Z its focal length. For the interaction of the beam with optical surfaces the wave field is transformed into a set of rays. This set of rays is traced through the optical boundary and subsequently transformed back into a wave field. The software therefore takes into account the refractive properties of the optical elements and is not limited by scalar diffraction theory (with its approximation of infinite thin elements).
Finally the coupling integral (Eq. 5) is calculated based on the propagated wave field and the mode field of the output fiber to optimize the system for maximum coupling efficiency. For the numerical optimization standard fibers for telecommunication (singlemode fibers SMF 28) were used as in- and output devices at a wavelength of 1.55μm. The optimization was carried out for seven equidistantly spaced channels at radial locations r of 1mm up to 4mm distance to the optical axis. Since laser lithography was intended for the fabrication of the microlenses, they were not limited to spherical shapes. Therefore the lens shape was allowed to be anamorphotic (different radii of curvature in two perpendicular directions) and to have non zero conic constants, again possibly different in two perpendicular directions. Together with the decenter of the lenses with respect to the optical axis of the fiber five parameters are needed for a complete description of a single microlens. The optimization results for the seven equidistantly spaced channels are shown in Fig. 4 marked with diamonds.
Fig. 4. Optimized mathematical functions (dashed line) for the five parameters needed for the description of a single lens of the chirped MLA. Diamonds mark the optimized results for the seven calculated channels.
For calculating the five lens parameters for cells with radial distances r different to the seven optimized channels, the parameters as a function of the radial distance r have to be known by means of mathematical functions. These can easily be obtained using fitted spline functions (Fig. 4, dashed lines). Especially in case of arrays with a huge number of channels this is a time efficient approach if analytical modeling is not possible. However, the use of interpolation functions is always involved with the risk of discrepancies between the really best suited and the calculated values and it is only possible if the context legitimates its acceptance. The application of the fitted functions will e.g. lead to wrong parameters in case of an unsteady functional context or when the sampling is insufficient. Consequently a verification of the results is considered to be indispensable. However, this verification process is usually much less time consuming compared to the optimization of the whole microlens array.
For calculating the parameters of all cells, first the radial distance r of the center of the cell to the symmetry axis has to be calculated. Fig. 5 shows a schematic top view of the system. Using regular arrays of constant pitch p in x- and y-direction the radial distance r of the cell with index (x,y) is given by:
Fig. 5. Schematic top view of the system and parameters needed for the calculation of the radial distance ri,j of each cell of the MLA.
where r 0 is the distance from the first row of the array to the symmetry center in y-direction, Nx and Ny the number of channels in x- and y-direction, respectively. The terms (Nx - 1)/2 and (Ny - 1)/2 arise from the declaration of the cell with index (0,0) located in the lower left corner of the array.
Furthermore each lens of the array has to be rotated such that its plane containing the y-radius of curvature coincides with the tangential plane of the overall imaging system for each channel. From simple geometrical considerations the rotation angle can be calculated by:
For our prototype we chose an array pitch of 125μm, which fits to commercially available fiber arrays with 30 channels in x- and 22 channels in y-direction. The distance r 0 of the first row to the symmetry center was set to 1mm. In Fig. 6 the angle of rotation Θ and the radial distance r for all cells of the array are plotted.
After the calculation of the radial distance ri, j for each cell, the parameters of the lenses can be calculated using the fitted functions. From simple geometrical considerations the decenter of the cells can be split into portions in x-and y-direction (dxi,j and dyi,j) according to Eq. 10 and 11:
with Θi,j being the angle of rotation of the cell with index (i,j), di,j the overall decenter. The required lens parameters for all cells are plotted in Fig. 7.
As already mentioned a verification of the values calculated using fitting functions is strictly recommended. Therefore a macro for importing the lens parameters to ZEMAX and calculating the coupling efficiency for each single channel was used. Fig. 8 shows a plot of the theoretical coupling efficiencies, which are close to 100% for all channels.
The major advantage of using fitting functions is the opportunity to calculate the parameters of cMLA with a huge number of channels quickly whenever a sufficiently complete analytical description of the system is not possible. In our case the optimization of the seven channels used as grid values took about 20 minutes. The calculation of the complete array using the fitting functions is done in seconds and the channel-wise verification of the complete array took about 10 minutes on a standard PC. When optimizing each channel by the numerical optimizer it would take about 32 hours. This time issue clearly becomes even more relevant when dealing with larger arrays.
3. Generation of the cMLA by laser lithography
Due to the asymmetry of the arrays single lenses caused by the decenter as well as the anamorphotic and aspherical shape, well-established fabrication techniques for the generation of microlenses like reflow of photo resist cannot be applied [14]. An evolving technology for the fabrication of asymmetrical lens profiles is the direct writing of the structure by laser lithography [15]. Here, a tightly focused beam is scanned over a photo sensitive layer while its intensity is modulated. According to the locally deposited dose different height levels of the structure result after a subsequent development step. This technology is especially suited for the generation of diffractive elements. However, several groups have tried to enhance this method for the generation of refractive microoptical components with larger sag heights [16]. For our system structure heights of 26.5μm in maximum are necessary. The first obstacle one has to face results from the limited number of intensity levels of the laser lithography system. Our machine (modified Heidelberg Instruments DWL 400) is able to address 64 intensities and therefore corresponding height levels. Considering the structure height of 26.5μm, a step height of 0.27μm results which can be interpreted as the rms value of the profile deviation. According to the Maréchal criterion the system will inherently not be able to obtain diffraction limited performance even without the appearance of any other fabrication errors. A refinement of the minimal step height can be accomplished using a writing scheme based on the decomposition of the structure into a coarse and a fine structure (Fig. 9). For a refinement of a factor of M the coarse structure of the entire array has to be written (M-1) times. The resulting profile due to the exposure of the coarse structure would lead to something looking like an Aztecan pyramid (Fig. 9 bluish colored areas). In a final exposure step the fine structure is written into the resist (Fig. 9 yellowish colored areas) leading to the desired profile with smaller deviations.
Fig. 6. Angle of rotation Θ and radial distance r for all cells of the MLA. The position of the z-axis marks the cell with index (0,0).
Fig. 7. Decenter, focal length and conic constant for all lenses of the chirped MLA. The position of the z-axis marks the cell with index (0,0).
Fig. 8. Calculated coupling efficiency for all channels of the system. The position of the z-axis marks the cell with index (0,0). Plotted scale reaches from 0.92 to 1.
Fig. 9. Plotting scheme for writing the lenses. Blue areas are to be exposed when writing the coarse structure, the yellowish colored areas correspond to the structures when writing the fine structure.
The total dose deposited in the resists for creating the deepest structures has to be adapted to the sensitivity of the resist and the subsequent development process. Fig. 10 shows a grey-scale drawing of the coarse (Fig. 10a) and the fine structure (Fig. 10b) for a section of the array.
Fig. 10. Grey-scale drawing of the splitting scheme for a) the coarse structures and b) the fine structures for a detail of the array. c) detail picture of the fabricated micro lenses.
For controlling the development process after the exposure the measurement of the achieved profile depth has to be carried out. Due to the chirp of the lenses a direct measurement of the lenses is somehow unhandy. A blazed grating with a depth of 15μm and a period of 62.5μm (half of lens pitch) as a simple test structure which can be accessed and measured easily with a profilometer was added (Fig. 5).
4. Experimental Verification
For the experimental setup the cMLA structures were UV-replicated in Ormocer material on Schott BOROFLOAT 33 wafers using the laser written structures as masters [17]. Before replicating the arrays onto the wafers the plane aluminum folding mirrors were deposited through a simple rectangular mask resulting in reflecting stripes of 1.3mm width. The master structures were aligned to the mirror stripe on the wafer to be centered between the in- and output arrays. As the spots of every single channel have a diameter of about 120μm and hit the mirror on the same location (see Fig. 3b) precise alignment is not necessary.
The alignment of the entire interconnect was done actively. Due to the lack of 2D-fiber arrays with acceptable pitch tolerance we used a single line silicon v-groove fiber array with 500μm pitch accommodating the in- and output fiber. Within the test setup it was only possible to take measurements along the symmetry axis of the cMLA in y-direction (dashed line in Fig. 5) as all other channels do not meet the exact pitch of the fiber array due to tilted channels in relation to the symmetry axis as e.g. the input in the upper left micro lens in the left hand side cMLA1 is imaged to the lower right microlens in the right hand side cMLA2. However, due to the fixed arrangement of the in- and output fiber pointing errors caused by insufficient lens profiles cannot be compensated by lateral adjustment unlike using single fibers. Since the location of the in- and output devices in a real setup will be fixed as well as this kind of test setup is more realistic besides being much more critical. For adjusting the components to each other the AL-coated spherical mirror, the wafer containing the cMLA and the plane mirror and the fiber array were mechanically connected to manipulation drives providing all required degrees of freedom (see Fig. 11). By iterative optimization of all translational and rotational degrees of freedom the optimal alignment for high coupling efficiencies was adjusted.
The fiber array is assembled with SMF 28 singlemode fibers. The losses for coupling within the system along the center line in y-direction at the wavelength λ= 1.55μm were 15 to 17dB, including about 2dB Fresnel losses at air glass interfaces and losses at the three AL mirrors within the system.
Besides losses caused by insufficient alignment another source are fabrication tolerances. In Fig. 12 a plot of the designed and measured height profile for two lenses of the array is given (lenses with index (14,21) and (14,22), which corresponds to the microlenses located on the center line in x-direction and outermost in the y-direction of the left hand side cMLA, see Fig. 5). The consequences of the large peak-to-valley deviation of almost 4μm are twofold. On the one hand a pointing error of the input beam results. The result is a lateral displacement of the field at the corresponding focusing microlens in cMLA2 and consequently a lateral and angular displacement of the focus with respect to the output fiber. On the other hand the waist of the Gaussian beam does not coincide axially with the end face of the output fiber.
The deviation of the fabricated to the ideal profile is even more obvious when looking at the grating test structure (Fig. 13). The development of the resist was carried out until the desired height of the teeth meet the nominal value of 15μm. The slope angle of the grating is steeper than intended and further on the angle of the slope is getting smaller at the bottom of the grating, nearly meeting the intended slope angle. The reason is an overexposure of the upper region of the photo resist due to the diffraction of the writing beam. During writing the beam is auto-focused on top of the photo resist and diffracts while propagating. The Rayleigh length of about 7μm of the writing beam is smaller than the resist height of about 30μm. Therefore the radiation dose implanted in the resist differs from the upper to the lower region. In consequence a nonlinear context between the light intensity and the profile height results which has not been taken into account. In that case the result is a variance of the radius of curvatures of the microlenses and thus a change of the focal length and the deflection of the beam.
The parameters of the fitted profile were implemented into the ZEMAX design file. The resulting coupling efficiency for this re-simulated channel is almost zero. However, when doing the assembly the axial distance of the wafer containing the cMLA with respect to the fiber and the spherical mirror are two degrees of freedom and can be used as compensators. The simulation of the influence of the axial distance on the coupling efficiency is given in Fig. 14.
With this setup a theoretical maximum efficiency of 1.25dB is possible in case of an axial displacement of the lens of about 200μm. The profiles have an rms-deviation of about 180nm, which corresponds to λ/10 and therefore being almost diffraction limited. However, even if the losses are much larger than the theoretical values, without using a cMLA no light would have coupled between the fibers due to the large aberrations of the spherical mirror.
Due to the small mode field diameters of singlemode fibers the coupling is very sensitive to axial displacements and tilts. A less critical set-up comprises singlemode fibers as input and multimode fibers or detectors as output devices. In our system we used a single SMF 28 as input and a multimode fiber with a core diameter of 100μm and NA of 0.3 as output device each connected to translation stages. We measured insertion losses from 1 to 3dB (Fresnel and reflection losses of 2dB subtracted) for certain channel pairs distributed over the entire cMLA. Here, the peak-to-valley deviations of the lenses are less critical, since the lateral mismatch and the defocus are widely tolerated by the large core diameter of the multimode fiber.
5. Conclusions
We demonstrated a compact design for an integrated interconnect based on a hybrid imaging setup using a refractive cMLA for channel-wise aberration correction. The measured efficiency for coupling singlemode fibers SMF 28 was in the range of 13 to 15dB, mainly caused by large peak-to-valley deviations of the laser written microlenses fabrication tolerances. These are due to the diffraction effects of the laser beam when writing the structures into the resist leading to a higher exposure in the areas next to the surface. In consequence this leads to an overdevelopment of the upper profiles of the lens and a deviation of the designed microlens shape. The improvement of the writing regime in order to minimize the peak-to-valley deviations is under current exploration. However, a system using a cMLA even with degraded microlenses provides better coupling compared to a system using a rMLA where the optical performance of the channels is always a trade-off. Using a multimode fiber as output device coupling losses of 1 to 3dB can be obtained, clearly stating the suitability in principal and the superior performance of the cMLA compared to rMLA.
Acknowledgements
We gratefully acknowledge the contributions of Peter Dannberg from the Fraunhofer Institute of Applied Optics and Precision Engineering (IOF), Jena, for the realization of the UV-replication of the mircolens arrays. Further on we like to thank Olga Kreis from the Center for Micro- and Nano Technologies of the Technical University Ilmenau (ZMN TU-Ilmenau) for the generation of the aluminum mirror structures.
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