## Abstract

It is shown that, when the scalar field associated with the propagation of a distorted wave function has nulls in its intensity pattern, the phase function that goes with that scalar field has branch points at the location of these nulls and that there are unavoidable 2π discontinuities across the associated branch cuts in the phase function. An analytic proof of this supposition is provided. Sample computer-wave optics propagation results are presented that manifest such unavoidable discontinuities. Among other things, the numerical results are organized in a way that demonstrates that for those cases the branch points are unavoidable. It is found in the sample numerical results that the branch cuts can be positioned so that the 2π discontinuities are located along lines of minimum intensity. This location tends to minimize the physical significance or importance of the discontinuities, a significant consideration for deformable-mirror adaptive optics, for which there is an unavoidable correction error in the vicinity of the branch cut. An algorithm is briefly described that allows the branch cuts to be located automatically and a phase function to be calculated that has discontinuities equal only to 2π discontinuities that are located at the branch cuts.

© 1992 Optical Society of America

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### Equations (42)

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(1)
$$\mathrm{\varphi}(\mathbf{r})=\text{Arg}\{U(\mathbf{r})\}+2\mathrm{\pi}n(\mathbf{r}),$$
(2)
$$n(\mathbf{r})=\dots ,-3,-2,-1,0,1,2,3\dots ,$$
(3)
$$U(\mathbf{r})=\hspace{0.17em}\mid U(\mathbf{r})\mid \text{exp}[i\mathrm{\varphi}(\mathbf{r})],$$
(4)
$$\mathbf{r}\equiv (x,y)$$
(5)
$$\begin{array}{l}U(x,y)=u(x,y)+iv(x,y)\\ =({u}_{0}+i{v}_{0})+({u}_{x}+i{v}_{x})(x-{x}_{0})+({u}_{y}+i{v}_{y})(y-{y}_{0})+\cdots ,\end{array}$$
(6)
$$\mathrm{\varphi}(x,y)=2\mathrm{\pi}n(x,y)+\mathrm{\phi}(x,y),$$
(7)
$$-\mathrm{\pi}<\mathrm{\phi}(x,y)\le \mathrm{\pi},$$
(8)
$$\text{cos}[\mathrm{\phi}(x,y)]=u(x,y)/{\{{[u(x,y)]}^{2}+{[v(x,y)]}^{2}\}}^{1/2},$$
(9)
$$\text{sin}[\mathrm{\phi}(x,y)]=v(x,y)/{\{{[u(x,y)]}^{2}+{[v(x,y)]}^{2}\}}^{1/2},$$
(10)
$$u(x,y)={u}_{0}+{u}_{x}(x-{x}_{0})+{u}_{y}(y-{y}_{0})+\cdots ,$$
(11)
$$v(x,y)={v}_{0}+{v}_{x}(x-{x}_{0})+{v}_{y}(y-{y}_{0})+\cdots .$$
(12)
$$x={x}_{0}+\mathrm{\rho}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta},$$
(13)
$$y={y}_{0}+\mathrm{\rho}\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\theta}.$$
(14)
$$u(x,y)={u}_{0}+{u}_{x}\mathrm{\rho}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta}+{u}_{y}\mathrm{\rho}\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\theta}+\cdots ,$$
(15)
$$v(x,y)={v}_{0}+{v}_{x}\mathrm{\rho}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta}+{v}_{y}\mathrm{\rho}\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\theta}+\cdots .$$
(16)
$${\mathrm{\mu}}_{u}={[{({u}_{x}\mathrm{\rho})}^{2}+{({u}_{y}\mathrm{\rho})}^{2}]}^{1/2},$$
(17)
$${\mathrm{\mu}}_{v}={[{({v}_{x}\mathrm{\rho})}^{2}+{({v}_{y}\mathrm{\rho})}^{2}]}^{1/2},$$
(18)
$${\mathrm{\theta}}_{u}={\text{arctan}}_{2}\hspace{0.17em}({u}_{x},{u}_{y}),$$
(19)
$${\mathrm{\theta}}_{v}={\text{arctan}}_{2}\hspace{0.17em}({v}_{x},{v}_{y}),$$
(20)
$$u(x,y)={u}_{0}+({\mathrm{\mu}}_{u}\hspace{0.17em}\text{cos}\hspace{0.17em}{\mathrm{\theta}}_{u})\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta}+({\mathrm{\mu}}_{u}\hspace{0.17em}\text{sin}\hspace{0.17em}{\mathrm{\theta}}_{u})\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\theta}+\cdots \cong {u}_{0}+{\mathrm{\mu}}_{u}\hspace{0.17em}\text{cos}(\mathrm{\theta}-{\mathrm{\theta}}_{u}),$$
(21)
$$v(x,y)={v}_{0}+({\mathrm{\mu}}_{v}\hspace{0.17em}\text{cos}\hspace{0.17em}{\mathrm{\theta}}_{v})\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta}+({\mathrm{\mu}}_{v}\hspace{0.17em}\text{sin}\hspace{0.17em}{\mathrm{\theta}}_{v})\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\theta}+\cdots \cong {v}_{0}+{\mathrm{\mu}}_{v}\hspace{0.17em}\text{cos}(\mathrm{\theta}-{\mathrm{\theta}}_{v}).$$
(22)
$$A={{\mathrm{\mu}}_{u}}^{2}\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}{\mathrm{\theta}}_{u}+{{\mathrm{\mu}}_{v}}^{2}\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}{\mathrm{\theta}}_{v},$$
(23)
$$B=2{{\mathrm{\mu}}_{u}}^{2}\hspace{0.17em}\text{cos}\hspace{0.17em}{\mathrm{\theta}}_{u}\hspace{0.17em}\text{sin}\hspace{0.17em}{\mathrm{\theta}}_{u}+2{{\mathrm{\mu}}_{v}}^{2}\hspace{0.17em}\text{cos}\hspace{0.17em}{\mathrm{\theta}}_{v}\hspace{0.17em}\text{sin}\hspace{0.17em}{\mathrm{\theta}}_{v},$$
(24)
$$C={{\mathrm{\mu}}_{u}}^{2}\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}{\mathrm{\theta}}_{u}+{{\mathrm{\mu}}_{v}}^{2}\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}{\mathrm{\theta}}_{v},$$
(25)
$${R}^{2}={(u-{u}_{0})}^{2}+{(v-{v}_{0})}^{2}=A\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}\mathrm{\theta}+B\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta}\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\theta}+C\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}\mathrm{\theta},$$
(26)
$${\left(\frac{\partial {R}^{2}}{\partial \mathrm{\theta}}\right)}_{\mathrm{\theta}=\mathrm{\vartheta}}=0.$$
(27)
$$\mathrm{\vartheta}=1/2\hspace{0.17em}{\text{tan}}^{-1}(B/A-C).$$
(28)
$${a}^{2}=A\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}\mathrm{\vartheta}+B\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\vartheta}\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\vartheta}+C\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}\mathrm{\vartheta},$$
(29)
$$\begin{array}{l}{b}^{2}=A\hspace{0.17em}{\text{cos}}^{2}(\mathrm{\vartheta}+1/2\mathrm{\pi})+B\hspace{0.17em}\text{cos}(\mathrm{\vartheta}+1/2\mathrm{\pi})\times \text{sin}(\mathrm{\vartheta}+1/2\mathrm{\pi})+C\hspace{0.17em}{\text{sin}}^{2}(\mathrm{\vartheta}+1/2\mathrm{\pi})\\ =A\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}\mathrm{\vartheta}-B\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\vartheta}\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\vartheta}+C\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}\mathrm{\vartheta}.\end{array}$$
(30)
$$\mathrm{\Theta}={\text{tan}}^{-1}\left[\frac{{\mathrm{\mu}}_{v}\hspace{0.17em}\text{cos}(\mathrm{\vartheta}-{\mathrm{\theta}}_{v})}{{\mathrm{\mu}}_{u}\hspace{0.17em}\text{cos}(\mathrm{\vartheta}-{\mathrm{\theta}}_{u})}\right].$$
(31)
$$u={u}_{0}+{u}^{\prime}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\Theta}-{v}^{\prime}\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\Theta},$$
(32)
$$v={v}_{0}+{u}^{\prime}\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\Theta}+{v}^{\prime}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\Theta},$$
(33)
$${u}^{\prime}=a\hspace{0.17em}\text{cos}(\mathrm{\theta}-\mathrm{\vartheta}),$$
(34)
$${v}^{\prime}=b\hspace{0.17em}\text{sin}(\mathrm{\theta}-\mathrm{\vartheta}).$$
(35)
$$I(x,y)=\xbd\mid u(x,y)+iv(x,y){\mid}^{2}=\xbd\{{[u(x,y)]}^{2}+{[v(x,y)]}^{2}\},$$
(36)
$${\oint}_{({x}_{0},{y}_{0})}\text{d}\mathrm{\theta}{\mathrm{\varphi}}_{\mathrm{\theta}}(\mathrm{\rho}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta},\mathrm{\rho}\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\theta})=\{\begin{array}{ll}\pm 2\mathrm{\pi},\hfill & \text{if}\hspace{0.17em}{u}_{0}=0,{v}_{0}=0\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}.$$
(37)
$$\oint \text{d}\mathrm{\theta}{\mathrm{\varphi}}_{\mathrm{\theta}}(\mathrm{\rho}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta},\mathrm{\rho}\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\theta})=\{\begin{array}{ll}\pm 2\mathrm{\pi},\hfill & \text{if}\hspace{0.17em}I({x}_{0},{y}_{0})=0\hfill \\ 0\hfill & \text{if}\hspace{0.17em}I({x}_{0},{y}_{0})>0\hfill \end{array}.$$
(38)
$$\u3008{[\mathrm{\varphi}(\mathbf{r})-\mathrm{\varphi}({\mathbf{r}}^{\prime})]}^{2}\u3009=6.88{(\mid \mathbf{r}-{\mathbf{r}}^{\prime}\mid /{r}_{0})}^{5/3}.$$
(39)
$${\mathrm{\Delta}}_{x}\mathrm{\phi}(x,y)=\mathrm{\phi}(x+\mathrm{\Delta},y)-\mathrm{\phi}(x,y)+2\mathrm{\pi}{n}_{x}(x,y),$$
(40)
$${\mathrm{\Delta}}_{y}\mathrm{\phi}(x,y)=\mathrm{\phi}(x,y+\mathrm{\Delta})-\mathrm{\phi}(x,y)+2\mathrm{\pi}{n}_{y}(x,y),$$
(41)
$$\text{exp}[i\mathrm{\varphi}(x,y)]=U(x,y)/\mid U(x,y)\mid ,$$
(42)
$$R=\mathrm{\alpha}{{r}_{0}}^{2}/\mathrm{\lambda}.$$