Near-field diffraction of irregular phase gratings with multiple phase-shifts

A phase-step in a phase mask is not copied into the substrate but is split into two half-amplitude phase-shifts in the near-field because of the presence of an additional interference fringe system of the two beams diffracted from the two grating sections separated by the phase-step. In the case of multiple phase-shifts, the split phase-shifts from two adjacent phasesteps can crossover in the propagation without interfere. This paper contributes to understanding the near-field diffraction of irregular phase gratings with multiple phase-shifts, and provides a theoretical base for designing multiple phase-shifted phase masks for high channel-count phaseonly sampled fiber Bragg gratings [1,2]. © 2005 Optical Society of America OCIS codes: (050.1960) Diffraction Theory; (050.5080) Phase mask; (220.3740) Lithography;


Introduction
The region of diffraction from a few wavelengths to hundreds of micrometers from the mask is of interest for microlithography.The side-writing of fiber Bragg gratings (FBGs) through a phase mask is a type of interference lithography.The scalar diffraction Talbot imaging and rigorous coupled-wave analysis have been used to analyze the near-field beyond the phase mask in order to ensure high performance of the FBGs [3][4][5][6][7][8].It is well known that the intensity distributions in the near-field are typically noisy with complex structures due to the fractal Talbot effect, and that any small contributions from the diffracted orders other than 1  ± orders can cause significant deviation from the ideal two-beam interference pattern [3].In this paper we are concerned with the phase-shifted phase masks.For periodic phase shifts, such as the stitching errors in the e-beam phase mask, the diffraction has been analyzed with Fourier optics [9].Diffraction intensity distribution of the phase mask with a single phase-shift was computed with integration of the Fresnel-Kirchoff equation.The result showed asymmetry error in the FBG spectrum.However, the near-field structure was not analyzed [5].
The phase masks containing a high number of phase-shifts along the grating length are irregular gratings and they are used for side-writing phase-shifted FBGs such as the phaseonly sampled multi-channel FBGs [11][12][13][14].Although the phase-shifts can be introduced to the FBGs by relative motions between fiber and interference fringes in the interferometric direct writing system or by over-dithering of the phase mask in the scanning writing approach [15].The phase-shifted phase mask approach remains advantageous over the others for the high accuracy of the feature locations provided by e-beam lithography.It was believed that the phase-shifts in the phase mask were replicated in the FBG [10].However, the pattern transfer from the mask to the substrate is only possible when they are in contact.When writing FBGs there is a minimum distance of ≈ 62.5 m μ from the phase mask to the fiber core for a fiber diameter of 125 m μ .The numerical finite difference in time domain (FDTD) calculation showed the split of the phase-shift into two half-magnitude phase shifts that are propagated at the angle of the ±1 diffracted orders.The split of the phase shifts caused large roll-off errors in the multi-channel spectrum that can be corrected by new designs of the phase-shifted phase masks [1,2].However, the physical reason of the phase-shift split was not understood in Refs.1 and 2.Moreover, the high-channel-count phase-only sampled FBGs contains a high number of phase-shifts in one sampling period (~1 mm for 100 GHz channel spacing) [1,2,14].Phase masks having multiple phase-shifts were not analyzed by the FDTD due to computer memory limitation.
In this paper we analyze an ideal near-field diffraction of the phase masks with multiple phase-shifts in order to get a physical understanding of the phase-shift split.Our model shows that the multiple phase-shifts are split individually.The split phase-shifts can crossover without changing their values.This paper provides a theoretical base for the new design of the multiple phase-shifted phase masks for phase-only sampled high channel-count FBGs [1,2].

Ideal diffraction near field of phase-shifted phase mask
The phase mask used for the side-writing of FBGs is in general a one-dimensional surface relief binary phase grating with a square-wave profile.The phase-shifts are phase-steps inserted to the phase mask.Although the numerical solutions for the phase-shifted phase masks are accurate and include all the diffracted orders, only physical understanding on the near-field structure permits new designs of the phase masks with corrections of the diffraction effects [1,2,14].The basic fringe structure of the ± 1 order interference in the near-field is disturbed by the zero and high diffracted orders, which give rise to noise.For the physical understanding of the near-field diffraction, we get rid of noise and consider only the 1 ± diffracted orders.While most works were focused on the near field intensity distribution, we investigate the fringe phase distribution.Similarly to the Fourier analysis, the phase of the FBG coupling coefficient would have more impact on the FBG spectrum than its amplitude.
First, we consider a phase mask with a single phase-shift.Inserting a phase-shift into the phase mask is equivalent to cutting the grating into two sections and then separating them with a phase-step of width δ .Note that the value of δ can be also negative [1,14].At a minimum distance from the phase mask of 62.5 m μ , the evanescent waves vanish.Only homogeneous waves should be considered.We use the scalar diffraction theory and we neglect the diffraction at the ends of the finite length gratings.When the gap width 2 / λ δ > , the interaction between the two gratings separated by δ can be neglected [16].This is natural in our model as we omit all the high diffracted orders.Now assume that a phase step 0 > δ is inserted in a groove, and separates the grating into two sections of lengths L 1 and L 2 .Without loss of generality, we assume L 1 and L 2 to be integer multiples of the grating period Λ .We define the phase reference points x 01 and x 02 that are at the same location within the grating period profile of the two gratings respectively.The values of x 01 and x 02 can change by any integer multiples of Λ .Thus the phase-step takes place from ) ( as shown in Fig. 1(a).
At normal incidence and directly behind the phase mask in the plane y=0, every point is a source emitting 1  ± diffracted orders resulting from interference of the Huygens wavelets.We write down the four diffracted beams: beam (  .Hence, we define the specific regions in the near field where the four beams are superposed respectively, as shown in Fig. 1(a).In addition to the ordinary interference fringe patterns formed by the superposition of beams (1) and (2) both diffracted from grating L 1 in region A, and that of beams (3) and (4) both diffracted from grating L 2 in region C, there is a The half-amplitude phase-shifts of the fringes can also be easily demonstrated by analytically adding the beams ( 1)-( 4) as follows: In region A, the fringe intensity of the superposition of beams ( 1) and ( 2 . In region B, the superposition of beams ( 2) and (3), diffracted by gratings L 1 and L 2 respectively, is In region C, the intensity distribution of the superposition of beams ( 3) and ( 4) is . If the fiber core was in contact with the phase mask, the width of region B would be zero at y=0 and the signal propagating along the FBG would experience a phase-shift ) 2 / /( 2 Λ πδ when passing from region A to C, as expected for a phase-shifted FBG.However, at a given distance y>0 from the phase mask, the interference of beams ( 2   We conclude that the physical cause for the split of the phase shift in the phase mask is the presence of the interference fringe pattern between the two beams diffracted from the two grating sections separated by the phase step δ .
In the case of a phase mask with multiple phase shifts, we assume that two adjacent phase shifts ) ( are separated by L 2 , where x 03 is the phase reference point in the grating section L 3 , as shown in Fig. 2(a).The beams (1)-( 4) are described in the same manner as previously.We define two new beams diffracted by the grating L 3 , as beam ( 5 three fringe systems.All have the same period of 500 nm.However, the boundaries between the middle fringe system and the right and left fringe systems are blurred.At distance y=15 m μ there is a set of 5 periods of 525 nm and 2 periods of 487.5 nm and 512.5 nm, in the left and right boundaries respectively as shown in Right-Bottom of Fig. 3.The variations of those 7 periods make a total phase shift of 125 nm, which is equal to half of the phase step of 250 nm in the phase mask.This is a clear indication that the phase shift of 250 nm in the phase mask is split into two half-magnitude phase-shifts of 125 nm each, which appear to propagate from the original phase step in the phase mask at the angle θ of ±1 diffracted orders.Their separation along the fiber core at distance y from the mask is equal to θ

Conclusion
We analyzed the near-field diffraction of irregular phase gratings which contain multiple phase-steps.The physical cause of the phase-shift split is the additional interference fringe system of the two first orders diffracted from the two grating sections separated by the phase step.The multiple phase-shifts in the phase mask are split individually and the split halfamplitude phase-shifts can crossover in the propagation without changing their values.Another popular irregular phase grating is chirped grating.In our model the grating sections have a uniform period and a minimum length (>10 periods), so our analysis is valid for the step-chirped gratings, where the grating period changes stepwise.However, our analysis can not be applied to continuous chirped gratings, where grating period varies continuously.

Fig. 1 .
Fig. 1.Near-field diffraction of a phase mask with one phase step δ ; (a) regions of superposition of the 4 diffracted beams; (b) a small central part of the calculated interference pattern of the 4 beams, illustrating the phase shift split.At y=0, the widths of the four beams are equal to their respective grating section lengths.Then, each beam propagates at the diffracted angle θ ± determined by the grating equation λ θ = Λ sin.Hence, we define the specific regions in the near field where the four beams are superposed respectively, as shown in Fig.1(a).In addition to the ordinary interference fringe patterns formed by the superposition of beams (1) and (2) both diffracted from grating L 1 in region A, and that of beams (3) and (4) both diffracted from grating L 2 in region C, there is a

9μm(
C) 2005 OSA 8 August 2005 / Vol. 13, No. 16 / OPTICS EXPRESS 6113 triangular region B, where beams (2) and (3) diffracted from the two gratings L 1 and L 2 overlap.The vertex of region B is at the location of the phase step.The boundaries of region B are at the diffraction angle θ ± with respect to the y-axis.There is an uncertainty of δ − Λ 2/ on the location of the vertex of the region B, because of the degeneracy of the diffracted beams in the middle segment of the enlarged groove.We calculated the superposition of beams (1)-(4) in the respective regions with a phase mask period m the near field intensity distribution is shown in Fig.1(b).A shift of a quarter of the fringe period between the fringes in regions B and A, and another 2 / π shift between the fringes in regions C and B can be seen clearly at the boundaries of region B.
) and (3) produces an additional fringe system of width θ boundaries between regions A, B and B, C, respectively.

Fig. 2 .
Fig. 2. Near-field diffraction of a phase mask with two phase shifts; (a): regions of superposition of the 6 diffracted beams; (b): part of the interference pattern calculated by superposing the beam 1-6, described in the text, in the respective regions.

Fig. 3 .
Fig. 3. Near-field intensity distribution of a phase mask with one phase-shift.;Left-Top: from 0 to 5 m μ computed by the FDTD; Left-Bottom from 5 to 15