## Abstract

It is well known that the Ronchi test has two equivalent interpretations, Physical, as an interferometer, or geometrical, as if the fringes were just shadows from the fringes on the ruling. The second interpretation is nearly always used in practice because it is simpler. However, the disadvantage is that the irradiance profile of the fringes cannot be calculated with this theory. Here, the interferometric interpretation of the test will be used to obtain the irradiance profile and the sharpness of the fringes.

© 1990 Optical Society of America

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### Equations (30)

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(1)
$${B}_{n}=\frac{{(-1)}^{n}}{n\pi}\text{sin}(n\pi K).$$
(2)
$$f(x)=\sum _{n=-\infty}^{\infty}{B}_{n}\hspace{0.17em}\text{exp}\left[i\left(\frac{2\pi nx}{d}\right)\right].$$
(3)
$$G(x,y)=\sum _{n=-\infty}^{\infty}{B}_{n}\hspace{0.17em}\text{exp}[i\varphi (x+nS,y)],$$
(4)
$$\varphi (x+nS,y)=\frac{2\pi}{\mathrm{\lambda}}W(x+nS,y),$$
(5)
$$S=\frac{\mathrm{\lambda}(r-1)}{d},$$
(6)
$$\varphi (x+ns,y)={\varphi}_{0}+{\varphi}_{\text{odd}}+{\varphi}_{\text{even}}.$$
(7)
$$G(x,y)=\text{exp}(i{\varphi}_{0})\sum _{n=-\infty}^{\infty}{{B}_{n}}^{i({\varphi}_{\text{even}}+{\varphi}_{\text{odd}})},$$
(8)
$$=\text{exp}(i{\varphi}_{0})\left[{B}_{0}+2\sum _{n=1}^{\infty}{B}_{n}(\text{cos}{\varphi}_{\text{odd}})\hspace{0.17em}\text{exp}(-i{\varphi}_{\text{even}})\right].$$
(9)
$$I(x,y)=G(x,y){G}^{*}(x,y).$$
(10)
$$\alpha =\frac{\mathrm{\lambda}ln}{rd},$$
(11)
$$\text{OPD}=\frac{{\mathrm{\lambda}}^{2}l(r-l){n}^{2}}{2r{d}^{2}}.$$
(12)
$${\varphi}_{\text{even}}=\frac{2\pi}{\mathrm{\lambda}}\text{OPD}=\frac{\pi \mathrm{\lambda}l(r-l){n}^{2}}{r{d}^{2}},$$
(13)
$${\varphi}_{\text{odd}}=\frac{2\pi}{\mathrm{\lambda}}\alpha x=\frac{2\pi lnx}{rd}.$$
(14)
$$W(x+nS,y)=\sum _{i=0}^{k}\sum _{j=0}^{i}{A}_{ij}{[x+nS]}^{j}{y}^{i-j},$$
(15)
$$W(x+nS,y)=\sum _{i=0}^{k}\sum _{J=0}^{i}\sum _{r=0}^{J}{A}_{ij}\left[\begin{array}{c}j\\ r\end{array}\right]{x}^{J-r}{y}^{i-j}{(nS)}^{r},$$
(16)
$$\left[\begin{array}{c}j\\ r\end{array}\right]=\frac{j!}{r!(j-r)!}.$$
(17)
$$W(x+nS,y)=D{[x+nS)}^{2}+{y}^{2}]+E{[{(x+nS)}^{2}+{y}^{2}]}^{2}.$$
(18)
$$\begin{array}{l}W(x+nS,y)=D{n}^{2}{S}^{2}+D{y}^{2}+S{y}^{4}+2E{y}^{2}{n}^{2}{S}^{2}+E{n}^{4}{S}^{4}+\hspace{0.17em}[2DnS+4E{y}^{2}ns+4E{n}^{3}{S}^{3}]x+[D+2E{y}^{2}+6E{n}^{2}{S}^{2}]{x}^{2}+\hspace{0.17em}4EnS{x}^{3}+E{x}^{4}.\end{array}$$
(19)
$$W(x+nS,y)={W}_{0}+{W}_{\text{even}}+{W}_{\text{odd}},$$
(20)
$$\begin{array}{l}{W}_{0}=D{y}^{2}+S{y}^{4}+[D+2E{y}^{2}]{x}^{2}+E{x}^{4},\\ {W}_{\text{even}}=D{n}^{2}{S}^{2}+2E{y}^{2}{n}^{2}{S}^{2}+E{n}^{4}{S}^{4}+6E{n}^{2}{S}^{2}{x}^{2},\\ {W}_{\text{odd}}=[2DnS+4E{y}^{2}ns+4E{n}^{3}{S}^{3}]x+4EnS{x}^{3}.\end{array}$$
(21)
$$\begin{array}{l}{W}_{\text{even}}=D{n}^{2}{S}^{2}+E{n}^{4}{S}^{4}+6E{n}^{2}{S}^{2}{x}^{2},\\ {W}_{\text{odd}}=[2DnS+4E{n}^{3}{S}^{3}]x.\end{array}$$
(22)
$$\begin{array}{l}{\varphi}_{\text{even}}=2\pi {n}^{2}{S}^{2}[D+E{n}^{2}{S}^{2}]/\mathrm{\lambda},\\ {\varphi}_{\text{odd}}=4\pi xnS[D+2E{n}^{2}{S}^{2}]/\mathrm{\lambda}.\end{array}$$
(23)
$$D=\frac{l}{2r(r-l)},$$
(24)
$$E=\frac{1}{4{r}^{3}}.$$
(25)
$${\varphi}_{\text{even}}=\frac{\pi \mathrm{\lambda}l(r-l){n}^{2}}{r{d}^{2}}\left[1+\frac{{n}^{2}{\mathrm{\lambda}}^{2}{(r-l)}^{3}}{2{r}^{2}{d}^{2}l}\right],$$
(26)
$${\varphi}_{\text{odd}}=\frac{2\pi lnx}{rd}\left[1+\frac{{n}^{2}{\mathrm{\lambda}}^{2}{(r-l)}^{3}}{{r}^{2}{d}^{2}l}\right].$$
(27)
$${\varphi}_{\text{even}}=\frac{\pi \mathrm{\lambda}l(r-l){n}^{2}}{r{d}^{2}}\left[1+\frac{{n}^{2}{\mathrm{\lambda}}^{2}r}{2{d}^{2}l}\right],$$
(28)
$${\varphi}_{\text{odd}}=\frac{2\pi lnx}{rd}\left[1+\frac{{n}^{2}{\mathrm{\lambda}}^{2}r}{{d}^{2}l}\right].$$
(29)
$$\frac{\mathrm{\lambda}l(r-l)\pi}{r{d}^{2}}=2M\pi .$$
(30)
$$\frac{2\mathrm{\lambda}l(r-l)}{r{d}^{2}}=\pm 1.$$