1. 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 [3–8]. 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 phase-only sampled multi-channel FBGs [11–14]. 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 μm from the phase mask to the fiber core for a fiber diameter of 125 μm. 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].

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 μm, 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 δ > λ/2 , 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 δ > 0 is inserted in a groove, and separates the grating into two sections of lengths L₁ and L₂. Without loss of generality, we assume L₁ and L₂ 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 x₀₁ and x₀₂ can change by any integer multiples of Λ. Thus the phase-step takes place from (x ₀₁ + L ₁) to x ₀₂ with δ = (x ₀₂ -(x ₀₁ + L ₁))_{mod Λ} 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 (1) exp[j2π((x-x ₀₁)/Λ ycosθ/ λ)] and beam (2) exp[j2π((x-x ₀₁)/Λ ycosθ/ λ)] are diffracted from the grating L₁, beam (3) exp[j2π((x-x ₀₂)/Λ ycosθ/ λ)] and beam (4) exp[j2π(-(x-x ₀₂)/Λ ycosθ/ λ)] are diffracted from the grating L₂. In the enlarged groove of width Λ/2 + δ , the wavelets emitted from the segment from (x ₀₂ - δ) to x ₀₂ should be a part of the diffracted beams of grating L₂. The wavelets emitted from the segment from (x ₀₁ + L ₁ - Λ/2) to (x ₀₁ + L ₁ - Λ/2 + δ) should be that of grating L₁. In the middle segment of width Λ/2 - δ of the enlarged groove, the ±1 diffracted orders are degenerate. They can belong either to grating L₁ or to grating L₂.

 

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.

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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 sin θ = λ. 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 L₁ in region A, and that of beams (3) and (4) both diffracted from grating L₂ in region C, there is a triangular region B, where beams (2) and (3) diffracted from the two gratings L₁ and L₂ 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 Λ/2 - δ 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 Λ = 1μm and the wavelength λ = 250nm with a phase-step δ = 250nm in the middle of the phase mask at x = 10μm , corresponding to a phase-shift of 2πδ/(Λ/2) = π in the FBG of period Λ/2 . A part of 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 π/2 shift between the fringes in regions C and B can be seen clearly at the boundaries of region B.

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) is 1 + cos(2π(x - x ₀₁)/(Λ/2)). In region B, the superposition of beams (2) and (3), diffracted by gratings L₁ and L₂ respectively, is

where δ = (x ₀₂ - x ₀₁)_{mod Λ} and the intensity distribution is $1+\cos \left(\frac{2\pi \left(x-x_{01}-\frac{\delta }{2}\right)}{\left(\frac{\Lambda }{2}\right)}\right)$ In region C, the intensity distribution of the superposition of beams (3) and (4) is 1 + cos(2π(x - x ₀₁ - δ)/(Λ/2)). 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 2πδ/(Λ/2) 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) and (3) produces an additional fringe system of width ∆x = 2ytan θ in region B. The two half-amplitude phase shifts πδ/(Λ/2) occur at the boundaries between regions A, B and B, C, respectively.

 

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.

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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 δ = x ₀₂ -(x ₀₁ + L ₁) and γ = x ₀₃ -(x ₀₁ + L ₁ + δ + L ₂) = x ₀₃ -(x ₀₂ + L ₂) 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):exp[j2π:((x - x ₀₃)/Λ Λ ycosθ/λ,)] and beam (6): exp[j2π(-(x-x ₀₃)/Λ + ycosθ/λ)]. When the separation between δ and γ is large such that L ₂/2tan θ > y, the signal propagating to the fiber core at distance y will “see” the split of phase-shift δ in regions A, B and C, and then the split of phase-shifts γ in the regions C, D and E. The two phase-shifts are split independently. When the adjacent phase-steps δ and γare close such that L ₂ /2tanθ < y, the two fringe systems in regions B and D can overlap in region F , where beam (2) diffracted from grating L₁ and beam (5) diffracted from grating L₃ are superposed, as shown in Fig. 2(a). One can readily calculate the intensity distribution in region F by the interference between beams (2) and (5) as

At a given y > L ₂/2tanθ the signal in the fiber core “sees” the fringes in the regions, A, B, F, D and E as: in region A, 1 + cos(2π(x - x ₀₁)/(Λ/2)); region B $1+\cos \left(\frac{2\pi \left(x-x_{01}-\frac{\delta }{2}\right)}{\left(\frac{\Lambda }{2}\right)}\right)$; region F, $1+\cos \left(\frac{2\pi \left(x-x_{01}-\frac{\gamma +\delta }{2}\right)}{\left(\frac{\Lambda }{2}\right)}\right)$; region D, $1+\cos \left(\frac{2\pi \left(x-x_{01}-\left(\frac{\gamma }{2}+\delta \right)\right)}{\left(\frac{\Lambda }{2}\right)}\right)$; region E, 1 + cos(2π(x - x ₀₁ -(γ + δ))/(Λ/2)). A phase shift δ/2 occurs when crossing region A to B, and an additional phase shift γ/2 occurs when crossing region B to F, the third phase shift δ/2 occurs when crossing region F to D and the last phase shift γ/2 will be added when crossing region D to E. Each half-amplitude phase shift appears as being “propagated” from their respective original phase steps in the phase mask at the angle of first order diffraction to the y-axis. Thus, the multiple phase-shifts in the phase mask are split and individually and independently. All the split phase shifts occur at the boundaries between regions of different interference patterns. The split phase-shifts can then “propagate” and crossover during the beam propagation without interfere and change of their values. This model has been used without proving it in Ref [2] for designing the diffraction compensation phase mask for high channel count phase-only sampled FBGs with a high number of phase-shifts introduced in each sampling period.

3. Numerical solution with FDTD

We use two-dimensional FDTD to compute the near-field diffraction of the phase-shifted phase mask. The phase mask consists of 16 periods of 1 μm . A phase shift of 250 nm is located at x=8 μm . The structure was modeled by a fine meshing made of λ/20 square cells. A Gaussian pulse excitation was launched on the phase mask at normal incidence. The field amplitude and phase distributions were computed by a recursive temporal Fourier transform of the instantaneous electrical field intensity at an arbitrary chosen distance y=5μm from the phase mask. Then, we continued the free space propagation computation using the Fourier optics spatial filter for another 10 μm as shown in Fig. 3.

The numerical results are similar to the ideal diffraction field shown in Fig. 1. There are 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 μm 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 ∆x = 2y tan θ .

 

Fig. 3. Near-field intensity distribution of a phase mask with one phase-shift.; Left-Top: from 0 to 5 μm computed by the FDTD; Left-Bottom from 5 to 15 μm computed by Fourier free space propagation filter; Right-Top: Plot of the amplitude in arbitrary scale of the near field at y=15 μm ; Right-Bottom: Plot of the fringe periods at y=15 μm .

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4. 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 half-amplitude 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.