Abstract

A modified Ronchi test with an extended source is presented. By the use of a matched pair of source grating and Ronchi grating, a pure shearogram between two shifted wave fronts is obtained, and the extra interference with other disturbing grating orders is largely suppressed by the use of a specific grating layout.

© 1999 Optical Society of America

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References

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  1. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–451 (1964).
    [CrossRef]
  2. G. Toraldo di Francia, “Geometrical and interferential aspects of the Ronchi test,” in Optical Image Evaluation, Natl. Bur. Stand. (U.S.) Circ. No. 256 (U.S. GPO, Washington, D.C., 1954).
  3. D. Malacara, “Geometrical Ronchi test of aspherical mirrors,” Appl. Opt. 4, 1371–1374 (1965).
    [CrossRef]
  4. R. Barakat, “General diffraction theory of optical aberration tests from the point of view of spatial filtering,” J. Opt. Soc. Am. 59, 1432–1439 (1969).
    [CrossRef]
  5. A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 321–365.
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  7. J. Schwider, “Single sideband Ronchi test,” Appl. Opt. 20, 2635–2642 (1981).
    [CrossRef] [PubMed]
  8. G. Leibbrandt, G. Harbers, P. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt. 35, 6151–6161 (1996).
    [CrossRef] [PubMed]
  9. K. Creath, “Wyko systems for optical metrology,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 637–652.
  10. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).
  11. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  12. W. Groebner, N. Hofreiter, Integraltafel, Volume II: Bestimmte Integrale, 5th ed. (Springer, Vienna, 1973.

1996 (1)

1981 (1)

1969 (1)

1965 (1)

1964 (1)

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).

Barakat, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Cornejo-Rodriguez, A.

A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 321–365.

Creath, K.

K. Creath, “Wyko systems for optical metrology,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 637–652.

Groebner, W.

W. Groebner, N. Hofreiter, Integraltafel, Volume II: Bestimmte Integrale, 5th ed. (Springer, Vienna, 1973.

Harbers, G.

Hofreiter, N.

W. Groebner, N. Hofreiter, Integraltafel, Volume II: Bestimmte Integrale, 5th ed. (Springer, Vienna, 1973.

Kunst, P.

Leibbrandt, G.

Malacara, D.

Ronchi, V.

Schwider, J.

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Toraldo di Francia, G.

G. Toraldo di Francia, “Geometrical and interferential aspects of the Ronchi test,” in Optical Image Evaluation, Natl. Bur. Stand. (U.S.) Circ. No. 256 (U.S. GPO, Washington, D.C., 1954).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

Other (7)

A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 321–365.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

K. Creath, “Wyko systems for optical metrology,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 637–652.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

W. Groebner, N. Hofreiter, Integraltafel, Volume II: Bestimmte Integrale, 5th ed. (Springer, Vienna, 1973.

G. Toraldo di Francia, “Geometrical and interferential aspects of the Ronchi test,” in Optical Image Evaluation, Natl. Bur. Stand. (U.S.) Circ. No. 256 (U.S. GPO, Washington, D.C., 1954).

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Figures (7)

Fig. 1
Fig. 1

Ronchi test. An extended source covered by a grating structure GS is focused by the optics O to be measured onto the corresponding Ronchi grating GR. An auxiliary lens L is inserted to image the exit pupil EP of the optics onto the detection screen D. In the ray optics approximation, the local fringe deformation is given by the transverse aberration on GR of a ray originating from the corresponding point in EP.

Fig. 2
Fig. 2

Far-field pattern projected onto the detection screen D [coordinates (X, Y) in the case of a point source]. The pattern is the superposition of the diffraction orders generated by the Ronchi grating GR. The spacing between the centers of the diffraction orders is determined by the wavelength λ of the light, the grating period p, and the distance R between the grating plane and D.

Fig. 3
Fig. 3

Impression of the interference pattern produced by a single coherent source point Q on the detection screen D. In the plane of the Ronchi grating, a lens (not shown) takes care of the imaging of the exit pupil of the optics under test onto D.

Fig. 4
Fig. 4

Schematic layout of the Ronchi test showing the definition of the coordinates used in the analysis. The Ronchi grating is followed by a lens L that images the exit pupil of the optical system onto the detection screen D. The axial distances from the source to the entrance pupil, from the exit pupil to the Ronchi grating, and from the Ronchi grating to the detection screen are R0, R, and R, respectively. To simplify the analysis, we fix the pupil magnification of the optical system to +1.

Fig. 5
Fig. 5

View along the vertical direction of the periodic coherence distribution in the exit pupil EP of the optics (horizontal cross section with Y=y=Y=0). The interference in a general point P is determined by, e.g., the odd orders from the Ronchi grating GR of the points Q0 and its coherent satellites Qi. Further contributions in P originate from the even orders of the point P0 and its coherent satellites Pi. The period p of the Ronchi grating is twice the period pS of the projected source grating. Only the diffracted waves toward the point P, conjugate with the point P0 on the wavefront, have been shown. The auxiliary lens L has been omitted.

Fig. 6
Fig. 6

Ronchi test carried out with a cosine-type grating. The state of interference on the detection screen depends on the varying duty cycle that is encountered by the incoherent source points Q when they are imaged onto the cosine grating (points Q).

Fig. 7
Fig. 7

Binary grating with quasi-cosine transmissive behavior. A section of a grating with five periods has been shown. Integrated along vertical lines, the transmission varies in a cosine manner. The period of the grating in the x direction is p, and the extent in the y direction is q. The digitized appearance of the grating profile is an artifact that is due to the limited number of pixels used to represent the smooth profile. In a practical application, the ratio p/q will be much smaller than the value of 1/5 presented in the figure.

Equations (71)

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Δxt(X, Y)=mx px,Δyt(X, Y)=mypy,
A(X, Y; x0, y0)=[I(x0, y0)]1/2 exp2πiλR0(x0X+y0Y)×f(X, Y; x0, y0),
B(x, y; x0, y0)
=[I(x0, y0)]1/2 exp2πiλXx0R0-xR
+Yy0R0-yR f(X, Y)dXdY,
A(X, Y; x0, y0)
=t(x-Δx, y)B(x, y)
×exp-2πiλR(xX+yY)dxdy.
A(X, Y; x0, y0)=[I(x0, y0)]1/2
×f(X, Y)exp2πiλR0(Xx0+Yy0)t(x-Δx, y)
×exp-2πiλxXR+XR
+yYR+YRdxdydXdY.
FT{t(x-Δx, y)}=m exp2πimΔxp×δX-mλRpτmYR,
A(X, Y; x0, y0)=m[I(x0, y0)]1/2f(X, Y)
×exp2πiλR0(Xx0+Yy0)δXR+XR-mλp
×τmYR+YRexp2πimΔxpdXdY
=m[I(x0, y0)]1/2 exp2πimpΔx-RR0x0
×f-RRX-mλRp, Yτm-YR-YR×exp-2πiλR0RRx0X-Yy0dY.
Itot(X, Y)=mmI(x0, y0)T(x0, y0)×exp2πim-mpΔx-RR0x0×dY1dY2 f-RRX-mλRp, Y1×f*-RRX-mλRp, Y2×τm-YR-Y1Rτm*-YR-Y2R×exp2πiλR0(Y1-Y2)y0dx0dy0.
I(x0, y0)=1,T(x0, y0)=Tx(x0);
Itot(X, Y)
=mmTx(x0)exp2πim-mpΔx-RR0x0f-RRX-mλRp, ηf*-RRX-mλRp, ητm-YR-ηRτm*-YR-ηRdηdx0.
p=nm-mR0Rp,
Itot(X, Y)=n=-An exp2πinpΔx0+R0RΔx×mf-RRX-mλRp, η×f*-RRX+nλR0p-mλRp, η×τm-YR-ηRτm-n[(R0/R)(p/p)]*-YR-ηRdη.
Im,m(X, Y)=A(m-m)[(R/R0)(p/p)]×exp2πi(m-m)pRR0Δx0+Δx×f-RRX-mλRp, η×f*-RRX-mλRp, η×τm-YR-ηRτm*-YR-ηRdη,
Cm,m=τm(η)τm*(η)dη.
Δc=λRpS,
At(X, Y)=1(2K+1)pq-KpKp-q/2q/2t(x, y)
×exp2πiλR(Xx+Yy)dxdy,
t0(x, y)=1,-q41+cos2πxpy<q41+cos2πxp,-p2xp20,elsewhere.
At(X, Y)=1(2K+1)pqk=-K+Kexp2πikXpλR×-p/2p/2-q/2q/2t0(x, y)×exp2πiλR(Xx+Yy)dxdy;
At(X, Y)=12mJmπYq2λRπYq2λR×sinπ2YqλR+m×sincπm+XpλR×sincπ(2K+1)XpλRsincπXpλR.
sinc[(2K+1)Xp/λR]/sinc(Xp/λR)
At(0, 0)=1/2,
At(±λR/p, 0)=1/4,
At(±mλR/p, 0)=0,|m|2.
Bm(v)=12Jm(πv/2)πv/2sin[(π/2)(v+m)],
Cm,n=-+Bm(v)Bn(v)dv,
Cm,n=0m+nodd,m, n=1, 2, ,-2π21(n+m)2-11(n-m)2-1m+neven,m, n=1, 2, ,
C0,1=C1,0=18,
C0,n=Cn,0=0,n=3, 5, ,
C0,0=12-2π2,
C0,n=Cn,0=-2π21n2-12,n=2, 4, .
cm,n=2 AnCm,nA0C0,0,n=(m-n) RR0pp,
rm,n=Cm,nC1,-1,
At(X, Y)=1(2K+1)pqk=-K+K×exp2πikXpλR-p/2p/2-q/2q/2t0(x, y)×exp2πiλR(Xx+Yy)dxdy,
At(X, Y)=1(2K+1)pqk=-K+Kexp2πikXpλR×-p/2p/2exp2πiXxλRdx λR2πiY×expπiYq2λR1+cos2πxp-exp-πiYq2λR1+cos2πxp.
exp(iacos θ)=m=-+imJm(a)exp(imθ),
At(X, Y)=12(2K+1)k=-K+Kexp2πikXpλR×m JmπYq2λRπYq2λRsinπ2YqλR+m×sincπm+XpλR.
Cm,n=14Im,n,m, n=0, 1, ,
Im,n=2-+Jm(x)Jn(x)sin(12πm+x)sin(12πn+x)x2dx
=-+Jm(x)Jn(x)×cos[12π(m-n)]-cos[12π(m+n)+2x]x2dx.
Im,n=(-1)p-+Jm(x)Jn(x) sin 2xx2dxm+n=2p+1(-1)p-+Jm(x)Jn(x) (-1)n-cos 2xx2dxm+n=2p
forp=0, 1, .
ψm(ω)-+ exp(-iωx)Jm(x)dx=2(-i)mTm(ω)(1-ω2)1/2B(ω),
ϕn(ω)-+ exp(-iωx) Jn(x)xdx=2in(-i)n(1-ω2)1/2Un-1(ω)B(ω),
B(ω)=1,|ω|<10,|ω|1.
Im,n=0m+n=2p+1, m, n=1, 2,  ,(-1)(m-n)/2-+Jm(x)Jn(x)/x2 dx,=-8π1(n+m)2-11(n-m)2-1,m+n=2p,m, n=1, 2, .
sin 2x2x=14-22 exp(-iωx)dω.
I0,n=-+J0(x)Jn(x) sin 2xx2dx=12-2+2-+J0(x) Jn(x)xexp(-iωx)dxdω=14π-2+2-+ψ0(ω1)ϕn(ω-ω1)dω1dω=14π-+ψ0(ω1)dω1-+ϕn(ω)dω,
Jm(x)=12π-+ exp(iωx)ψm(ω)dω,
1xJn(x)=12π-+ exp(iωx)ϕn(ω)dω,
I0,2p+1=12πδp,p=0, 1, . .
-+ exp(-iωx) Jn(x)x2dx
=inn[1-ω2]1/21n+1Un(ω)-1n-1Un-2(ω)×B(ω)
I0,n=(-1)n/2-+J0(x)Jn(x)/x2dx=-8π1n2-12
sin xx2=14-22(2-|ω|)exp(-iωx)dω
exp(-iωx)J02(x)dx
=12π-+ψ0(ω1)ψ0(ω-ω1)dω1=2π-+ B(ω1)B(ω-ω1)dω1(1-ω12)1/2[1-(ω-ω1)2]1/2,
I0,0=2-J02(x)sin xx2dx=1π-22(2-|ω|)×- B(ω1)B(ω-ω1)dω1(1-ω12)1/2[1-(ω-ω1)2]1/2dω.
I0,0=2π-8/π.

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