**4. Conclusions**

10 Will-be-set-by-IN-TECH

(a) (b) (c)

(a1) (a2) (a3)

(b1) (b2) (b3)

(c1) (c2) (c3)

(0) *<sup>s</sup>* , *<sup>I</sup>* (1) *<sup>s</sup>* , *<sup>I</sup>*

(0) *<sup>s</sup>* , *<sup>I</sup>* (1) *<sup>s</sup>* , *<sup>I</sup>* (2) *<sup>s</sup>* for

(2) *<sup>s</sup>* for

Fig. 5. Typical results of the computer simulations: (a) interference fringes, *Is*(*x*, *y*), for

sin *θ* = 1.4*λ*/3*d* > *λ*/3*d* and *ξ*<sup>0</sup> = 1.4/3*d* = 2.8*ξN*/3, (c) interference fringe, *Is*(*x*, *y*), for sin *θ* = 0.8*λ*/6*d* < *λ*/6*d* and *ξ*<sup>0</sup> = 0.8/6*d* = 0.8*ξN*/3, (a1)–(a3) corresponding sub-holograms

(2) *<sup>s</sup>* for sin *<sup>θ</sup>* <sup>=</sup> *<sup>λ</sup>*/3*d*, (b1)–(b3) corresponding sub-holograms *<sup>I</sup>*

sin *θ* = *λ*/3*d* and *ξ*<sup>0</sup> = 1/3*d* = 2*ξN*/3, (b) interference fringe, *Is*(*x*, *y*), for

sin *θ* = 1.4*λ*/3*d* > *λ*/3*d*, and (c1)–(c3) corresponding sub-holograms *I*

*I* (0) *<sup>s</sup>* , *<sup>I</sup>* (1) *<sup>s</sup>* , *<sup>I</sup>*

sin *θ* = 0.8*λ*/6*d* < *λ*/6*d*.

We have investigated theoretically tolerance of the incident angle of plane reference wave in space-division multiplexed single-shot phase-shifting digital holography presented by Sato *et al* Toge et al. (2008). It is found from our analysis that the rigorous alignment of the incident angle of the plane reference wave is very important requirement to reconstruct the correct complex amplitude of the object wave by direct application of the phase-shifting algorithm. It is also found that even if the incident angle of the plane reference wave departs from the condition given by Eq. (1), we have phase-shifted sub holograms from the single digital hologram by sparsely sampling. However, the direct application of the phase-shifting algorithm for these phase-shifted holograms results in wrong complex amplitude especially in phase distribution. We described the way to obtain correct phase distribution in this case from our theoretical analysis. That is to subtract the linear phase distribution introduced by misalignment of the reference plane wave. This solution was first derived and brought to light by our theoretical analysis within the authors knowledge, and widen field of the single-shot phase-shifting digital holography using off-axis plane wave illuminations.
