1. Introduction

Diffractive optical elements (DOEs) [1, 2] provide powerful opportunities over spatial and spectral beam shaping that typically rely on wavelength-scale structuring of surface relief, opacity, or reflectance. Such DOEs find widespread application in science and industry that include spectral analysis [3], laser beam shaping [4], coherent laser combining [5], spatial filtered microscopy [6], optical imaging [7], data storage [8], holographic sorting of particles [9] and information encoding [10]. On the other hand, volume DOEs offer further utility in manipulating the phase and amplitude of the beam as it propagates and subsequently diffracts over a much longer grating depth [2, 8, 11]. Early approaches here considered stacking of multiple surface relief gratings [12–14] that promised benefits of improved efficiency. However, such high contrast dielectric-air interfaces are challenging to assemble with sub-wavelength phase precision over multiple layers and the delicate surface structures are susceptible to damage in contrast with DOEs or gratings formed inside bulk material.

Volume DOEs are also found having low contrast in the refractive index structure that can be generated in photosensitive materials, for example, by interference of laser light [15, 16] to inscribe uniform volume gratings. Alternatively, 3D direct writing with focused femtosecond laser light [17] has offered more flexibility in creating highly functional DOEs such as volume Fresnel lens [18], periodicity-doubled phase gratings [19] and gray-level information encoded holograms [10]. Generally, such low contrast phase gratings have been shown to offer very high 1st order diffraction efficiency of up to 90% [20], but only for the well-known Littrow configuration [21]. Otherwise, for broader DOE applications, low-contrast phase gratings are generally limited to low efficiency as diffracted light will dephase and destructively interfere when accumulated over a long grating length.

A promising direction for enhancing the efficiency of low-contrast volume gratings relies on breaking apart the phase structure into multiple layered phase elements, an effect first observed in models by Nordin et al. [13]. Ng et al. took advantage of the flexibility of femtosecond laser writing and showed in both experiments and modeling that strong 1D diffraction can be built up coherently when multiple phase grating layers were assembled on Talbot self-imaging planes [22]. In this way, low contrast volume gratings can now be extended to highly efficient high-resolution applications, by-passing the restrictive Littrow configuration.

In the present work we investigate and manipulate both amplitudes and phases of a large number (nine) of diffraction beams produced by high density and high resolution 2D-diffracting volume gratings. The femtosecond laser writing technique was applied in fused silica glass to optimally pattern and order the stacking of laser formed phase layers on Talbot planes that enhance and balance the diffraction efficiencies amongst several first order beams, developed for the first time in a 2D diffracting volume phase mask. Laser writing is further applied to tune the phase offset between orthogonal grating layers and thereby control the symmetry of 3D diffraction interference such that body centered tetragonal (BCT) and woodpile tetragonal (wTTR) structural symmetry could be recorded in photoresist. The results demonstrate the utility of low index contrast phase masks in managing both the phase and amplitude of the diffraction beams in new directions of advanced diffractive optics that are broadly promising in high-resolution 3D holographic lithography.

2. Volume grating lithography

The multi-layer architecture of a volume grating is depicted schematically in Fig. 1. Two sets of linear phase gratings have been positioned orthogonally and illuminated with a laser to generate a 3D interference pattern in the proximity of the DOE as shown by the woodpile structure in the inset image. In order to build up strong first order diffraction, each X- or Y-grating set followed our prior design [22], with track layers periodically formed on an integer number of Talbot distances, $c_g$, in the fused silica. When the offset, d, between the two orthogonal grating sets are also an integer or half integer number of Talbot distances, the combined interference pattern will follow BCT symmetry. In between these positions, offsets of $d=(n+1/4)c_g$ or $(n+3/4)c_g$ will transform the 3D interference pattern to wTTR symmetry as seen inset in Fig. 1. Hence, tuning of the grating displacement d will cycle the 3D structure between BCT and wTTR symmetry on each quarter Talbot distance, $c_r/4$.

 

Fig. 1 Schematic of a multi-layer volume phase grating written with a femtosecond laser to provide triple- layered sets of X-oriented and Y-oriented tracks on $\text{Λ}$ lateral period and offset vertically on Talbot planes at integer multiples of $c_g$ distance, to enhance the diffraction efficiency. A collimated laser beam normally incident into the volume grating creates a 3D interference pattern below the mask with symmetry controlled by the spatial displacement, d, between the two grating sets (i.e. between layers 1 and 3). For the case of $d=c_g/4$, a woodpile TTR symmetry is expected as shown in the inset, having X and Y periods of $\text{Λ}$, vertical period of $c_r$, and a vertical offset of $s=c_r/4$ in the relative positions of the X- and Y-oriented logs.

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A key challenge in forming a practical 2D-diffracting grating structure is in optimizing and balancing the X- and Y-oriented diffraction efficiencies amongst the multiple diffracted beams that overlap and interfere to form into a 3D periodic interference pattern. These interference patterns can be readily calculated with knowledge of the efficiency and relative phases of each diffracted beam [23, 24]. Practical recording of the 3D interference structure is typically made in photoresist which in turn can be compared with the isointensity surfaces [23] as calculated from the interference pattern.

A volume grating phase mask design (Fig. 1) is considered for the orthogonal combination of identical linear phase grating stacks, where the X- and Y-oriented grating periods of ${\text{Λ}}_{\text{x}}$ and ${\text{Λ}}_{\text{y}}$, respectively, are equal (${\text{Λ}}_{\text{x}}$ = ${\text{Λ}}_{\text{y}}$ = $\text{Λ}$) and illuminated with coherent light of wavelength $\text{λ}$. Following [22], the optical coherent stitching condition requires an assembly of the grating layers on an integer multiple of Talbot distances given generally for a medium, i, of index, n_i, by:

One typically desires a small lateral grating period together with a large diffraction angle such that Eq. (1) will give a small longitudinal period matched with the lateral period (i.e. $c_i$ = $\text{Λ}$ for ${\theta }_i$ = 90°) and thus provide an isotropic 3D interference patterns. However, this evanescent limit at the ${\theta }_i$ = 90° diffraction angle is not practical for a beam splitting phase grating and values of $c_i$ = 1.2$\text{Λ}$ to 1.44 $\text{Λ}$ have otherwise been found desirable in supporting a complete 3D PBG structure [25] for high index materials [26].

3. Volume grating fabrication

Volume phase gratings were fabricated in fused silica glass (Corning 7980, $n_g$ = 1.46, 1 mm thickness) by forming multiple laser modification track layers with a focused scanning ultrafast laser beam. A frequency-doubled Yb-fiber amplified femtosecond laser (IMRA µJewel D-400-VR) with output of 522 nm wavelength, ~300 fs duration, M² = 1.4 beam quality and 1 MHz repetition rate was focused with an oil-immersed objective lens (Zeiss, 100 × , N.A. = 1.25) to about 450 nm beam waist and 610 nm depth of focus. A non-uniform profile of refractive index change was induced within and surrounding the laser focal volume, while the strongest change was observed inside the focal volume. To remove any possible directional dependence of nanograting formation [27], the modification tracks were formed with circular polarization. All exposures were made below the critical power for self-focusing.

The possibility to generate a high-resolution single-layer phase grating that could meet the desired $c_g$/$\text{Λ}$ = 1.4 ratio for a complete PBG was attempted with our highest resolution focusing objective (N.A. = 1.25). For the illuminating wavelengths of 514 and 532 nm, this $c_g$/$\text{Λ}$ = 1.4 ratio requires the formation of $\text{Λ}$ = 544.0 nm and $\text{Λ}$ = 562.4 nm grating periods, respectively. Single layer arrays of laser tracks were formed on varying periods from 0.6 to 4 µm [22, 28] and studied for grating contrast and first order efficiency for the following range of laser exposure conditions: scanning speeds from 0.1 to 100 mm/s, laser power from 5 to 100 mW. The 1st order efficiencies were found to peak at 3.4% for 532 nm wavelength probing of a 1.5 µm grating period, formed with 1 mm/s and 35 mW exposure. This value fell off to 3.2% for $\text{Λ}$ = 2 µm, 1.2% for $\text{Λ}$ = 4 µm period and 2.6% for $\text{Λ}$ = 1 µm. Shorter periods of 0.6 µm provided only 0.35% efficiency, possibly owing to a washing out of the refractive index contrast as adjacent laser tracks become strongly overlapped. Hence, grating periods of $\text{Λ}$ = 1 to 1.5 µm were selected for further evaluation for their coherent stacking. For $\text{λ}$ = 532 nm wavelength exposure of a 1 µm period grating, a small diffraction angle of ${\theta }_g$ = 21.4° and large axial period of $c_g$ = 5.3 µm impose a large structural anisotropy of $c_g$/$\text{Λ}$ = 5.3. While this ratio will not support templating of a complete 3D PBG structure, one can explore the merits for developing high resolution volume gratings in glass materials.

The principle of assembling 1D multiple grating layers on Talbot periodicity was previously demonstrated [22] for the case of 1 µm grating period, using optimal laser exposure conditions of 1 mm/s scan speed and 35 mW power. Coherent stacking of grating layers on Talbot planes was designed for 532 nm probing wavelength, applied on a double period spacing of 2$c_g$ = 10.6 µm to accommodate the large axial modification zone of ~7 µm that would have otherwise coalesced the tracks if $c_g$ = 5.3 µm spacing was used. A progressive improvement in the 1st order efficiency from 2.6% for 1 layer to 6.5% for 2 layers, and 17% for 8 layers was reported in [22], thus demonstrating the principle of coherent stacking of phase gratings on Talbot planes for enhanced diffraction efficiency. Note that weak index contrast here undermines any formation of a stopband on this Talbot alignment.

Towards the present objective, the optimal combination of X- and Y- oriented stacks of 1 µm period gratings did not follow a straightforward assembly of stacked phase gratings as was found for the single orientation grating stacking in [22]. The balancing of diffraction efficiencies on X- and Y orientations were found to depend on the order of writing and stacking of the individual layers. For example, by following the schematic in Fig. 1 with two isolated orthogonal groups of triple grating layers, asymmetric 1st order efficiencies of 8.4% and 5.5% were first found for the Y- and X- oriented grating sets, respectively, when probed with 532 nm light. This asymmetry exceeded a statistical variation of ± 1% (absolute) noise in the first single order diffraction efficiencies. Such asymmetry is attributed to blazing effects in the arrangement of the grating lines as discussed further in Section 4.2. The laser writing exposure was 1 mm/s scan speed and 35 mW power. Here, the grating lines were written with a femtosecond laser incident and focused from the bottom of the sample of Fig. 1, in layers formed in order of the deepest (top layer in Fig. 1) to the shallowest (bottom layer in Fig. 1) grating planes. The stacking order of these crossed grating layers (X-X-X-Y-Y-Y) is represented as sample D in Fig. 2(a).This same design is seen to follow the top to bottom ordering in Fig. 1, where the sample had been inverted to position the grating volume closest to the photoresist for the laser holography step. Stronger diffraction response was observed from the Y-orientated gratings, which were written after the X-oriented grating formation.

 

Fig. 2 (a) Various configurations of X- and Y- oriented grating tracks assembled in order (left to right) with laser writing from the deepest to the shallowest layers in the glass, and ‘-‘ indicating a vertical spacing of 2$c_g$ between layers. (The layering for sample D appears inverted in Fig. 1) (b) The average of the 1st-order diffraction efficiencies measured for each grating type.

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In one approach to better balance the orthogonal grating efficiencies for the configuration of Sample D (Fig. 2(a)), the writing power for the Y-oriented grating lines was reduced from 35 mW to 20 mW while holding the X-grating writing power at the same level (35 mW). In this way, 1st order efficiencies of 7.0% and 5.5% were obtained for the Y- and X- oriented grating sets, respectively, which appear in Fig. 2(b) as Sample D.

The effect on grating layer writing order is further demonstrated by the highly asymmetric 2-fold differences in the 1st order diffraction efficiencies found in Sample A, B, and C in Fig. 2(b). Following Fig. 2(a), the three sets of X and Y tracks were assembled either in alternating order with X-oriented gratings written first (Sample A) and second (Sample B) or with the X and Y gratings written overlapped in single layers (Sample C), with X-gratings writing first. The laser writing exposure was 1 mm/s scan speed and 35 mW power for each set of X- and Y- grating lines. One can see in Fig. 2(b) that the last written grating layer was strongly dominating in the diffraction efficiency. The laser writing may be partially erasing the structuring modification in previously written nearby layers, and thus diminishing their index contrast. Diffraction models presented in Section 4.2 also predict an approximate 1.3-fold larger 1st order diffraction from the second grating set relative to the first grating set, when the light has first passed through the first grating set. This factor falls short of the ~2-fold difference measured from samples A, B, and C in Fig. 2(b), showing that the laser writing order remains a considerable unbalancing factor in optimizing the design of the multi-layer volume grating.

Further optimization of the volume grating to improve the diffraction efficiency and better balance the X- and Y-oriented grating efficiency was explored by unbalancing the number of X- and Y-oriented grating layers as well as by offsetting the laser powers used in writing the X- and Y- oriented tracks. For example, the 4 + 4 layer grating assembly of sample E (Fig. 2(a)) applied 35 mW and 20 mW power to write the X- and Y-tracks, respectively, and provide a better balanced 9.4% and 7.5% efficiency for each of the 1st order diffraction beams generated by the respective Y- and X-oriented track gratings. Further improvement to 9.4% and 8.1% efficiency was found in Sample F from the Y- and X-oriented gratings, respectively, when an additional layer of X-oriented tracks were written relative to Sample E.

The results show that both X- and Y-grating tracks can be combined into a single 2D diffracting volume phase grating to harness the principle of coherent stacking on Talbot planes. However, stressing and erasing effects associated with the writing order of grating planes on the presently small vertical stacking distances (~10 µm) necessitated a compensation method to balance an asymmetric diffraction efficiency between X- and Y-oriented gratings. Such compensation was demonstrated in designing X- and Y-oriented gratings with different layer numbers and different laser exposures, presenting 1st order efficiencies of 8.8% average with ± 0.7% absolute error in Sample F.

4. Volume grating lithography results

The multiple-layer volume grating was further developed to demonstrate the formation of photonic crystal structure in photoresist, and particularly to show the transition from BCT to wTTR symmetry by systematically varying the d offset between X- and Y-oriented gratings (Fig. 1). A longer Λ = 1.5 µm grating period was targeted here in an attempt to improve on both the efficiency and the balancing of the four 1st order diffracted beams that were otherwise challenging in the Λ = 1.0 µm case of Section 3. This larger period further permitted us to separate grating layers on a single Talbot spacing (c_g) with the opportunity to improve the assembly accuracy.

4.1 Multi-layer volume grating morphology

Laser exposure of 35 mW power and 1 mm/s scan speed was applied to form two sets of orthogonal linear volume gratings, each consisting of three layers in the arrangement of sample D in Fig. 2(a). For normal incidence illumination with 514 nm wavelength (Argon-ion laser, Sabre MotoFRED), Eq. (1) provides a diffraction angle of ${\theta }_g$ = 13.6° and a Talbot period of $c_g$ = 12.6 µm inside the glass. This large $c_g$ value exceeded the laser-modification track height (7 µm), permitting the gratings layers to be offset on adjacent Talbot planes (12.6 µm) for the present samples.

Figure 3 shows optical images of a representative volume grating fabricated in fused silica. A low-resolution image of the 6-layer grating is shown formed over an 0.5 by 5 mm² area in Fig. 3(a), and followed with optical microscope images (Nikon, Eclipse LV100) recorded in back-lighting with a low resolution lens (10 × objective) in Fig. 3(b) and front-illumination with an 100 × lens in Fig. 3(c) to 3(e). Laser modification tracks are noted in both of the X-and Y- oriented grating layers in Fig. 3(b), which become resolved into isolated tracks by imaging to different depths in the images of Fig. 3(c) to 3(e). In Fig. 3(c), Y-oriented grating lines are viewed from layer 1 as identified in Fig. 1, while X-oriented gratings lines are imaged in Fig. 3(e) from layer 3 in Fig. 1. Intermediate focusing between layers 1 and 3 captured both of these X and Y oriented grating lines in Fig. 3(d). The line-to-line spacing matched with the design period of Λ = 1.5 µm.

 

Fig. 3 Optical images of a volume phase grating at different magnifications recorded by (a) camera and microscope under (b) back-lighting and (c)-(e) front-lighting, where.(c), (d), and (e) resolve Y-oriented, X-Y oriented, and X-oriented laser tracks, respectively, corresponding to the imaging positions identified by layers 1, 2 and 3, respectively, as labeled in Fig. 1.

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4.2 Phase-tuned volume gratings

Each volume grating as defined in Section 4.1 was probed with various polarization states of collimated 514 nm laser light at normal incidence to measure the diffraction efficiencies of all prominent orders generated up to 2nd order. For orthogonal grating sets offset by approximately d = $(5/4)c_g$, average diffraction efficiencies of 13%, 3.7% and 1.0% were measured with circular polarized light for the respective 0, 1st and 2nd diffraction orders as plotted (green bars) in Fig. 4(a).After accounting for Fresnel losses (8%), a significant scattering loss of 47% was inferred from these measurements to originate with non-uniform structural modifications generated within the grating volume during laser scanning. The origin of optical scattering arises from non-uniform refractive index structure that has formed on sub-wavelength scale together with nanoporous structure [27]. These factors are known to manifest in fused silica waveguide losses of 0.2 dB/cm [29], but which is too small to explain the present 55% scattering loss. This high scattering loss may therefore be attributed to randomness of the center positions, randomness in symmetry, and high density writing proximity effects that require further investigation. The total 55% loss was attributed proportionally to each of the orders by the dark hatched zone in Fig. 4(a). The average efficiencies were obtained from measurements of individual diffraction orders, also plotted in Fig. 4(a) as red circles for 1st order diffraction by the X-gratings (0, ± 1), black squares for 1st order diffraction by the Y-gratings ( ± 1, 0) and diamonds for 2nd order diffraction by both X- and Y-gratings ( ± 1, ± 1). Similar measurements were made for other polarization states and for other grating offset values.

 

Fig. 4 (a) Comparison of FDTD calculated (red bar) and measured (green bar) diffraction efficiencies from a volume phase mask of 3 + 3 grating layers (Sample D in Fig. 2(a)). The hatched red bar represents the measured loss. The measured diffraction efficiencies of individual orders are shown in 1st order for Y-oriented (black square) and X-oriented (red circle) gratings, and in 2nd order (black diamond). The average measured diffraction efficiencies (green bar) for the 0th, the four 1st and the four 2nd orders of 13%, 3.7% and 1.0%, respectively, were applied to predict the interference patterns, yielding (b) the isointensity surfaces shown in b1 and b2 for only the 0th and 1st order beams and in b3 and b4 with all orders present. The isointensity surfaces in b5 and b6 were calculated using the experimental efficiencies measured on all nine individual orders. The simulations used offsets of d = 0 (b1, b3, b5) and $c_g/4$ (in b2, b4, b6) between the orthogonal linear gratings.

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A theoretical assessment of the present grating efficiency to examine the polarization and offset (d) sensitivity was explored by extending the theoretical modeling of Ng et al. [22] from 1D grating stacks to the present 2D gratings stack of 3 + 3 layers. A full-vector 3D electromagnetic wave theory based on finite difference time domain (FDTD) modeling (Lumerical FDTD solutions) was applied to the crossed and multi-stacked grating layers. Each grating layer was modeled with $\text{Λ}$ = 1.5 µm period, 50% duty cycle, average refractive index contrast of $\Delta n$≈0.015, and grating thickness of ~7 µm depth that best matched the average diffraction efficiencies in our present observations. A peak index contrast of $\Delta n$≈0.022 was measured in single laser tracks by refracted near-field (RNF) refractometry as reported previously [30]. The FDTD simulation for circularly polarized light yielded average diffraction efficiencies of 30% for the 0th order, 12% for the 1st orders and 4.0% for the 2nd orders which are plotted in Fig. 4(a) to represent the lossless case (solid plus hatched red bar). These values were scaled proportionately to diffraction efficiencies (solid red bar data) of 13.5%, 5.4% and 1.8% for the 0th, 1st and 2nd diffraction orders, respectively, to account for the experimentally observed 55% loss (hatched red bar data). In experiment, averaged efficiencies of 13%, 3.7% and 1.0% (green bar) were observed for respective 0th, 1st and 2nd orders. These theoretical values agree within a factor of 2 of the experimental values. The simulations followed from our prior work in Ref [22], where RNF profiles of single tracks were evaluated and then simplified into an array of uniform rectangles to define a single layer grating structure. These were stacked on Talbot planes to enable the present simulation and matching with the experimental data after tuning the rectangle length and correcting for the scattering losses. In this approach, the modeling was forgiving on variations of the index contrast and rectangle length. Hence, a more precise determination of the refractive index profile is warranted to improve the precision of the simulation as the laser modification has formed both positive and negative zones of nonuniform refractive index contrast that also can contribute constructively to the grating efficiency.

Theoretical modeling for the cases of linear polarization parallel with the X- and Y-grating lines, with illumination of X-grating layers followed by Y-grating layers, predicted only small differences in the first order efficiencies. Relative to circular polarization, the first order efficiencies increased 3.9% (i.e. + 0.47% absolute) for X-gratings and decreased 3.2% (i.e. −0.32% absolute) for the Y-gratings when illuminated by linear polarization along X-oriented grating lines. The magnitude of change was similar but reversed in sign when illuminated by linear polarization along Y-oriented grating lines. Further, the theoretical 1st order diffraction efficiencies were found to modulate with the offset distance d by up to ± 3.9% (i.e. ± 0.54%) relative to the mean value. Hence, laser polarization yielded a minor dependence on the diffraction efficiencies. In contrast, the absolute diffraction measurements were limited by ± 4% relative variability in reproducibility while the uniformity of diffraction efficiency over the area of a single volume grating had a moderate statistical deviation of ± 6.8%. These relative measurement errors therefore precluded any definitive confirmation of the polarization or offset (d) dependence on the diffraction efficiencies of the present gratings.

The theoretical models predicted an asymmetric diffraction efficiency from the X- and Y- oriented gratings, for example, yielding 10.5% and 13.6% 1st order diffraction efficiency (i.e. 4.7% and 6.1% after accounting for the 55% loss) from respective X- and Y-oriented gratings, when illuminated with circularly polarized light. Experimentally, average efficiencies of 2.3% (3.3% and 1.2% from Fig. 4(a)) and 5.1% (2.7% and 7.6% from Fig. 4(a)) were measured from the respective X- and Y-oriented grating sets, which is a much stronger three-fold asymmetry than predicted. Figure 4a also reveals that a large asymmetry arises between the + 1st and –1st orders as well as the + 2nd and −2nd orders. The ratio of + 1st to −1st order efficiencies are 2.77 and 2.83 from the X- or the Y-oriented gratings, respectively. Since only symmetric efficiencies are expected theoretically, this asymmetry may be attributed to blazing effects from imperfectly aligned grating layers and writing-laser-polarization-dependent formation of submicron gratings in the laser tracks [27]. This blazing-effect asymmetry is similar in scale with the asymmetry found between orthogonal gratings, and requires a different approach for compensation than presented in Section 3. For example, one may consider the application of small lateral shifts between adjacent layers within the same (X or Y) grating set to induce a countering blazing effect, but this approach was not tested here. Further, development of a higher resolution-imaging tool (< 0.5 µm) [22] that can directly resolve the asymmetry within the refractive index profile would be helpful in understanding and improving the diffraction asymmetries observed here.

The present 3 + 3 grating stack, having weak efficiency (Fig. 4(a)) of 1st order light (average 3.7%) against the 0th order (13.5%) together with additional interference terms generated by the presence of 2nd order light (1.0%), is examined to ensure sufficient interference can be generated in photoresist to form a stable 3D periodic photonic crystal template. These prospects are examined for the two cases of 0 and π/2 phase offsets that are introduced to the diffracted beams projected into the x-z and y-z planes of Fig. 1 by spatial offsets of d = $c_g$ and d = $(5/4)c_g$, respectively, between the X- and Y-grating sets. The multiple beam interference method [23, 24] was used here to calculate the 3D optical intensity pattern produced by the interference of the diffraction beams with knowledge of their amplitudes from grating efficiencies and phases from the X- and Y-grating offset, d. Isointensity values were set to ensure formation of a well cross-linked 3D bicontinuous structure [23].

The resulting 3D isointensity calculations for the 0 and π/2 phase offsets cases present the respective BCT and wTTR symmetry structures seen in the upper (example 1, 3, 5) and lower (example 2, 4, 6) rows, respectively, of Fig. 4(b). These isointensity surfaces were first calculated using the average experimental values of the diffraction efficiency found in Fig. 4(a) for the interference of 0th (13%) and 4 symmetric 1st order (3.7% each) beams (examples 1, 2). The influence of the 2nd order then followed (examples 3, 4) for interference combining the 0th (13%), 4 symmetric 1st order (3.7% each), and 4 symmetric 2nd order (1.0% each) beams. Finally, the effect of asymmetric diffraction was examined (examples 5,6) for interference between the 0th (13%), 4 asymmetric 1st order (2.7%, 3.3%, 7.6% and 1.2%), and 4 asymmetric 2nd order (1.0%, 2.2%, 0.6% and 0.3%) beams.

In each of the isointensity structures in Fig. 4(b), one notes the expected lateral and axial periodicities of 1.5 µm and 13.8 µm have been produced according to Eq. (1). As a result, the BCT and wTTR structures have a large 9:1 anisotropy. A comparison of examples 3 and 4 with the respective examples 1 and 2 predicts an insignificant change in the overall structural shape and symmetry due to the presence of the weak additional second order light in examples 3 and 4. Slightly more rounded elliptical structures are seen for both the BCT (example 3) and woodpile (example 4) cases in comparison with their respective counterparts (examples 1 and 2) but there is no evidence of periodic ripple structures forming on the surfaces as expected from the additional interference with the 2nd order wave vector on a ~100 nm computational grid resolution. Hence, the presently weak 2nd order diffraction beams are not seen to impart a significant distortion of BCT and woodpile symmetry 3D photonic crystal structure. The isointensity structure for the cases of highly asymmetry diffraction orders in examples 5 and 6 have also reproduced the basic motive and symmetry as expected for the respective BCT (example 1) and woodpile (example 2) cases. The most significant distortion is only seen by a tilting of the vertical posts in example 5 by an angle of ~3°. Therefore, one may expect that diffraction beams having both low efficiencies and high asymmetries can interfere with sufficient contrast and minimal distortion to enable formation of 3D photonic crystal templates.

4.3 Holographic interference lithography

Various volume grating phase masks were irradiated by a collimated 514 nm wavelength laser beam having circular polarization to expose negative-tone photoresist (SU-8 model 2050 from MicroChem modified with 0.3 wt.% HNu-470 photoiniator from Spectra) held in proximity to the phase mask according to the methods first demonstrated in references [31] and [32]. The photoresist was spun over 1-mm thick glass to ~40 µm thickness and pre-baked on a hotplate (Fisher Scientific, model) at 65°C for 3 minutes. After pre-baking, the photoresist layer was made to contact the surface of the phase mask to capture the 3D periodic light intensity pattern generated by the volume phase mask. The exposed photoresist was post-baked at 95°C for 10 minutes, developed in SU-8 developer (MicroChem) for 6~10 minutes, and dried with nitrogen gas flow. The 3D structures were coated with gold or carbon and evaluated by SEM.

For the 1.5 µm grating period, the 3 + 3 phase mask design of Fig. 4(a) generated a first order diffraction angle of ${\theta }_r$ = 12.4° and an axial periodicity of $c_r$ = 13.8 µm (Eq. (1)) inside the photoresist ($n_r$ = 1.6). The grating offset of d = 15.4 µm was expected to produce wTTR-like symmetry, which we examine in the SEM image of the photoresist as presented in Fig. 5 for a cleaved sample. The structural form has become distorted due to weak linkages provided by the high aspect ratio in the periodicities and the ~50% filling fraction. Nevertheless, the lateral period matches with the 1.5 µm grating period while the vertical period of 8.8 µm is also recognizable in the cross-sectional facet, matching with the theoretical value ($c_r$ = 13.8 µm) after accounting for a ~36% shrinkage that is often observed [33–36] during photoresist development. Cleaving of the weakly linked 3D structure has produced a distorted Gamma-X facet as shown in Fig. 5, however, tilted by ~6° to follow a weaker cleaving plane that can be compared with the calculated profile (inset image) as taken from the isointensity profile given in example 6 of Fig. 4(b) for 6° tilt. Although the structure has become distorted during cleaving, upright and inverted Y-branch links may still be identified as labeled by the dashed circles. One finds a loose correspondence of the Y-splitting links in both of the calculated (inset) and experimental observed SEM images, providing evidence of the wTTR symmetry expected from this volume grating. The cleavage plane presented in Fig. 5 therefore demonstrates experimentally that the weak and unbalanced diffraction orders as found in Fig. 4(a) are highly forgiving in generating 3D interference patterns that can be recorded inside photoresist.

 

Fig. 5 Tilted cross-sectional SEM image of photoresist developed after exposure through a 3 + 3 layered volume grating with d≈5$c_g$/4 offset. Y-branch links marked by the dashed circles conform approximately with the simulated isointensity image (inset). However, definitive formation of the expected 3D woodpile structure was not evident due to lattice distortion by cleaving of the fragile, high-aspect ratio (theoretical $c_r/\Lambda$ = 9.2) structure.

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A similar set of 3 + 3 volume grating masks were laser fabricated with the exposure conditions of Section 4.1, following the design of sample D in Fig. 2, and laid out in the 10 × 1 array as shown in Fig. 6(a). The offset distance (d) between the X- and Y-oriented gratings was finely offset in adjacent gratings by the increment $\text{Δ}d$ = 0.7 µm to provide a full offset range 9$\text{Δ}d$ = 6.3 µm from d = 12.6 µm in grating #1 to d = 15.8 µm in grating #10 as labeled in Fig. 6(a). This range corresponds to phase offsets in the diffraction patterns generated by the X- and Y-gratings ranging from 360° to 540° from grating #1 to grating #10, respectively. This ensures that pairs of gratings have been fabricated with nearly ideal phase offsets that reduce to 0° (grating #1) or 180° (grating #10) for generating BCT structure and to 90° (grating #5 or #6) for generating woodpile structure.

Photoresist samples were laser exposed from each grating sample according to the above procedure and examined by SEM, yielding the 3D photonic crystal templates in Fig. 4(b) and 4(c) for gratings #5 and #9, respectively. The images reveal that a strongly linked and open 3D structure has been formed at least within one axial period ($c_r$ = 13.8 µm) from the top surface. Without cleaving, the structure is more stable than presented in Fig. 5. Figure 6(b) shows Y-oriented connecting tracks (vertical direction in image) to dominate at the top surface while X-oriented connecting tracks (horizontal) are shifted deeper to the interior. These features follow the expected woodpile structure of wTTR symmetry as shown by the isointensity surface calculation for the 90° phase offset case as seen in the inset image of Fig. 6(b). In Fig. 6(c), a symmetric square grid array of posts is seen on the top surface while X-oriented and Y-oriented tracks appear with equal prominence to connect symmetrically in a single plane deeper inside the sample. This structure closely follows the isointensity surface shown in the inset image for the case of BCT symmetry structure calculated with 180° phase offset. The best examples of wTTR (Fig. 6(b)) and BCT (Fig. 6(c)) symmetry structures were found in gratings #5 and #9, respectively, at phase shifts of 360° + 80° and 360° + 160°, respectively, that slightly under reach the expected relative values of 90°and 180°, respectively. This discrepancy arises from an uncertain value for the refractive index of the glass ($n_g$ = 1.46) that has likely increased with the laser modification, creating positive refractive index changes up to $\text{Δ}n$ = 0.025 [22].

 

Fig. 6 Symmetry tuning of 3D photonic crystal structure in photoresist templates formed with (a) a 10-set array of gratings. The X- (horizontal) and Y-grating (vertical) displacements across this set were varied to provide phase offsets of $\text{Δ}\varphi$ = 360° + 0° to 360° + 180°, predicting systematic change in symmetry of interference patterns from BCT (grating 1) to wTTR (Grating 5 and 6) to BCT (grating 10). Woodpile symmetry is verified (b) in the SEM images of photoresist exposed by grating 5 in contrast with the BCT symmetry seen in (c). Simulated isointensity surfaces expected in the respective images are shown in the inset images. For comparison, a (d) 3D structure with similar BCT symmetry was also produced with a 3 + 3 grating where the X- and Y-grating lines crossed in the same layer (d = 0).

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Photoresist was also exposed from the 3 + 3 grating design of Sample C in Fig. 2, where the X- and Y-gratings were overlaid in the same layer to provide a $\text{Δ}\varphi$ = 0 phase offset. The resulting 3D structure shown in Fig. 6(d) reveals the symmetric square-arrayed posts that are characteristic of the expected BCT symmetry as previously demonstrated for the case of $\text{Δ}\varphi$ = 520° in Fig. 6(c). Hence, the systematic phase shifts studied here by the X-and Y-grating offsets provides means for controlling the 3D photonic crystal symmetry that cycles on the expected $c_g/2$ periodicity of half-Talbot planes inside the glass volume grating.

5. Discussion and conclusion

Laser direct writing was applied to form weakly contrasting volume gratings inside bulk glass, taking advantage of grating layers positioned on Talbot planes to coherently build up a strong diffraction efficiency outside the restricting Littrow configuration [22]. This approach was successfully extended here to combine X- and Y-oriented volume gratings, providing phase-locked 2D diffracting beams that enabled 3D holographic interference lithography inside photoresist. However, the combination of X- and Y-gratings presented challenges (Section 3) in fully harnessing a strong efficiency on all diffraction orders as well as in posing blazing effects that created asymmetric diffraction orders. Improved balancing of these asymmetric diffraction orders were demonstrated by varying the laser power, the writing order and the number of layers used for each of the X- and Y-oriented grating layers. These effects were further examined in isointensity surface models (Section 4.2) that showed a forgiving response to formation of 3D bicontinuous structures despite the moderately weak and asymmetric diffraction from the volume grating. Six-layer volume gratings were designed and fabricated, and applied to exposure of 40 µm thick photoresist to test the quality and symmetry-control in generating 3D photonic crystal templates (Section 4.3). Bicontinuous 3D photonic structure was thus formed with volume phase masks offering low (~4%) and asymmetric (3:1 intensity ratio) 1st order diffraction efficiencies.

Further, 3 + 3 layered volume gratings were designed and fabricated (Section 4.3) with the X- and Y-oriented grating displacement finely varied to tune the net phase offset over $0 to \pi /2$ range. In this way, holographic interference lithography with these phase masks demonstrated a controllable transition of the 3D structure from BCT to wTTR symmetry. Femtosecond laser writing is therefore seen to open new opportunities for generating novel designs of volume gratings. However, further improvement in the optical scattering loss and areal uniformity will be necessary together with tighter compensation for the index overwriting and blazing effects before highly flexible designs and high performance volume gratings become widely available.

In conclusion, we have demonstrated the design and fabrication of multi-layer 2D diffracting volume gratings in fused silica with a femtosecond laser. The principle of coherent stitching of grating layers on Talbot planes has been extended to X- and Y-oriented grating structures, where the opportunity of phase control between the orthogonal diffracted beams was demonstrated by the formation of 3D photonic crystal structure tunable from body centered tetragonal (BCT) to woodpile-like tetragonal (wTTR) symmetry. However, the generation of volume phase masks was found restricted by optical scattering, index overwriting, and blazing effects, as the resulting profile of refractive index modulation was not a simple linear superposition of the laser exposure steps. These effects could be compensated in part by modifying the grating design and laser exposures, thus opening a new platform for flexible design of 3D structure volume gratings. The development of such volume grating phase masks may open a broader range of applications to harness such low index contrast devices in new directions of advanced diffractive optics, high-resolution 3D holographic lithography, optical imaging, data storage and holographic small particle sorting.