Abstract

We formulate the problem of designing gradient-index optical coatings as the task of solving a system of operator equations. We use iterative numerical procedures known from the theory of inverse problems to solve it with respect to the coating refractive index profile and thickness. The mathematical derivations necessary for the application of the procedures are presented, and different numerical methods (Landweber, Newton, and Gauss–Newton methods, Tikhonov minimization with surrogate functionals) are implemented. Procedures for the transformation of the gradient coating designs into quasi-gradient ones (i.e., multilayer stacks of homogeneous layers with different refractive indices) are also developed. The design algorithms work with physically available coating materials that could be produced with the modern coating technologies.

© 2012 Optical Society of America

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2011 (4)

2010 (1)

2009 (2)

C. Polenzky, C. Rickers, and M. Vergöhl, “Properties of cosputtered SiO2-Ta2O5-mixtures,” Thin Solid Films 517, 3126–3129 (2009).
[CrossRef]

M. Scherer, “Magnetron sputter-deposition on atom layer scale,” Vak. Forsch. Prax. 21, 24–30 (2009).
[CrossRef]

2008 (10)

R. Ramlau, “Regularization properties of Tikhonov regularization with sparsity constraints,” Electron. Trans. Numer. Anal. 30, 54–74 (2008).

V. Janicki, J. Sancho-Parramon, and H. Zorc, “Refractive index profile modelling of dielectric inhomogeneous coatings using effective medium theories,” Thin Solid Films 516, 3368–3373 (2008).
[CrossRef]

D. Ristau, H. Ehlers, S. Schlichting, and M. Lappschies, “State of the art in deterministic production of optical thin films,” Proc. SPIE 7101, 71010C1–71010C14 (2008).
[CrossRef]

C.-J. Tang, C.-C. Jaing, K.-S. Lee, and C.-C. Lee, “Residual stress in Ta2O5-SiO2 composite thin-film rugate filters prepared by radio frequency ion-beam sputtering,” Appl. Opt. 47, C167–C171 (2008).
[CrossRef]

P. G. Verly, “Hybrid approach for rugate filter design,” Appl. Opt. 47, C172–C178 (2008).
[CrossRef]

X. Cheng, B. Fan, J. A. Dobrowolski, Li Wang, and Z. Wang, “Gradient-index optical filter synthesis with controllable and predictable refractive index profiles,” Opt. Express 16, 2315–2321 (2008).
[CrossRef]

J. Weber, H. Bartzsch, and P. Frach, “Sputter deposition of silicon oxynitride gradient and multilayer coatings,” Appl. Opt. 47, C288–C292 (2008).
[CrossRef]

S. Larouche and L. Martinu, “Optical filters with prescribed optical thickness and refined refractive indices,” Appl. Opt. 47, 4140–4146 (2008).
[CrossRef]

S. Larouche and L. Martinu, “Step method: a new synthesis method for the design of optical filters with intermediate refractive indices,” Appl. Opt. 47, 4321–4330 (2008).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Application of constrained optimization to the design of quasi-rugate optical coatings,” Appl. Opt. 47, 5103–5109 (2008).
[CrossRef]

2007 (6)

2006 (8)

D. Ristau, H. Ehlers, T. Gross, and M. Lappschies, “Optical broadband monitoring of conventional and ion processes,” Appl. Opt. 45, 1495–1501 (2006).
[CrossRef]

M. Lappschies, B. Görtz, and D. Ristau, “Application of optical broadband monitoring to quasi-rugate filters by ion-beam sputtering,” Appl. Opt. 45, 1502–1506 (2006).
[CrossRef]

U. B. Schallenberg, “Antireflection design concepts with equivalent layers,” Appl. Opt. 45, 1507–1514 (2006).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, M. A. Kokarev, N. Kaiser, O. Stenzel, S. Wilbrandt, and D. Gäbler, “New optimization algorithm for the synthesis of rugate optical coatings,” Appl. Opt. 45, 1515–1524 (2006).
[CrossRef]

A. G. Imenes and D. R. McKenzie, “Flat-topped broadband rugate filters,” Appl. Opt. 45, 7841–7850 (2006).
[CrossRef]

V. Janicki, D. Gäbler, S. Wilbrandt, R. Leitel, O. Stenzel, N. Kaiser, M. Lappschies, B. Görtz, D. Ristau, C. Rickers, and M. Vergöhl, “Deposition and spectral performance of an inhomogeneous broadband wide-angular antireflective coating,” Appl. Opt. 45, 7851–7857 (2006).
[CrossRef]

R. Ramlau and G. Teschke, “A Tikhonov-based projection iteration for non-linear ill-posed problems with sparsity constraints,” Numer. Math. 104, 177–203 (2006).
[CrossRef]

R. Leitel, O. Stenzel, S. Wilbrandt, D. Gäbler, V. Janicki, and N. Kaiser, “Optical and non-optical characterization of Nb2O5-SiO2 compositional graded-index layers and rugate structures,” Thin Solid Films 497, 135–141 (2006).
[CrossRef]

2004 (2)

2003 (1)

R. Ramlau, “TIGRA, An iterative algorithm for regularizing nonlinear ill-posed problems,” Inverse Probl. 19, 433–465 (2003).
[CrossRef]

2002 (1)

2001 (4)

2000 (2)

1999 (2)

J. Ciosek, J. A. Dobrowolski, G. A. Clarke, and G. Laframboise, “Design and manufacture of all-dielectric nonpolarizing beam splitters,” Appl. Opt. 38, 1244–1250 (1999).
[CrossRef]

D. Rats, D. Poitras, J. M. Soro, and L. Martinu, “Mechanical properties of plasma-deposited silicon-based inhomogeneous optical coatings,” Surf. Coat. Technol. 111, 220–228 (1999).
[CrossRef]

1998 (3)

S. I. Park and Y. J. Lee, “Design of multilayer antireflection coatings,” J. Korean Phys. Soc. 32, 676–680 (1998).

D. L. Donoho and M. Johnstone, “Minimax estimation via wavelet shrinkage,” Ann. Stat. 26, 879–921 (1998).
[CrossRef]

P. G. Verly, “Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films,” Appl. Opt. 37, 7327–7333 (1998).
[CrossRef]

1997 (2)

1996 (4)

1995 (4)

P. G. Verly, “Fourier transform technique with frequency filtering for optical thin-film design,” Appl. Opt. 34, 688–694 (1995).
[CrossRef]

S. Martin, J. Rivory, and M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. 34, 2247–2254 (1995).
[CrossRef]

D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet shrinkage: asymptopia?” J. R. Stat. Soc. Ser. B 57, 301–369 (1995).

A. Chambolle, “Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations,” SIAM J. Appl. Math. 55, 827–863 (1995).
[CrossRef]

1994 (3)

D. A. Tonova, “Inverse profiling by ellipsometry: a Newton-Kantorovitch algorithm,” Opt. Commun. 105, 104–112 (1994).
[CrossRef]

D. L. Donoho, and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[CrossRef]

A. V. Tikhonravov, B. T. Sullivan, and M. V. Borisova, “Discrete-Fourier-transform approach to inhomogeneous layer synthesis,” Appl. Opt. 33, 5142–5150 (1994).
[CrossRef]

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1992 (3)

1990 (2)

1989 (5)

1987 (1)

1984 (1)

1983 (1)

1981 (1)

1980 (1)

1978 (1)

1977 (1)

1967 (1)

1966 (1)

1965 (1)

Agarwal, G. S.

Amotchkina, T. V.

Anders, H.

Apolonski, A.

Arnon, O.

Bartzsch, H.

Baumeister, P.

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).

Bertram, R.

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).

Boivin, G.

Borisova, M. V.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Bovard, B. G.

Bulkin, P. V.

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Figures (6)

Fig. 1.
Fig. 1.

Geometry of the design problem of gradient-index coatings.

Fig. 2.
Fig. 2.

Gradient designs with starting physical thickness as indicated. Starting thickness (a) 200, (b) 350, (c) 400, (d) 500, (e) 600, (f) 800, and (g) 1000 nm.

Fig. 3.
Fig. 3.

Dependance of the average reflection Rm of the designs on their optical thickness nd.

Fig. 4.
Fig. 4.

First row: two-material; other rows: three-material designs and a five-material obtained from the gradient design with starting physical thickness as indicated. (a) Two-material; starting thickness 200 nm. (b) Two-material; starting thickness 500 nm. (c) Three-material; starting thickness 350 nm. (d) Three-material; starting thickness 400 nm. (e) Three-material; starting thickness 600 nm. (f) Three-material; starting thickness 800 nm. (g) Five-material; starting thickness 1000 nm.

Fig. 5.
Fig. 5.

Calculated reflection of the different gradient designs shown in Fig. 2 (solid lines) and their multilayer transformations shown in Fig. 4 (dashed lines). The reflections in (a) are practically identical. Starting thickness (a) 200, (b) 500, (c) 350, (d) 400, (e) 600, (f) 800, and (g) 1000 nm.

Fig. 6.
Fig. 6.

(a) Refractive-index profile at wavelength of 700 nm of the gradient coating design of a nonpolarizing edge filter. (b) Equivalent three-material refractive-index profile at wavelength of 700 nm of the gradient coating from (a). (c) Transmission of the multilayer coating design from (b); continuous line, Tp; dashed line, Ts.

Equations (92)

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dUdz(z)=ikV(z)d,
dVdz(z)=ik(ϵ(z)ϵ1sin2ϕ)U(z)d,
dUdz(z)=ikϵ(z)ϵ1sin2ϕϵ(z)V(z)d,
dVdz(z)=ikϵ(z)U(z)d.
U(0)=1,
V(0)=pl,
pl=ϵlϵ1sin2ϕ
pl=ϵlϵlϵ1sin2ϕ
rs(p)=p1U(1)V(1)p1U(1)+V(1),
ts(p)=2p1p1U(1)+V(1).
p1=ϵ1cosϕ
p1=ϵ1cosϕ
Rs(p)=|rs(p)|2
Ts(p)=plp1|ts(p)|2.
R(λ,ϕ)=F(Rs(p)(n,d,λ,ϕ),Ts(p)(n,d,λ,ϕ))=F(n,d,λ,ϕ).
n=n(z,λ)=f(λ,q(z)).
0q1.
R(λ,ϕ)=F(q(z),d,λ,ϕ).
J(q,d)=F(q,d,λ,ϕ)R(λ,ϕ)L22,
F(q,d)R(λ,ϕ)L22=Λ×Φ|F(q,d)R(λ,ϕ)|2dλdϕ.
(q,d)=argminJ(q,d);
Λ={λ1,,λnl},Φ={ϕ1,,ϕnp}
R={R(λ,ϕ)}λΛ,ϕΦ.
J(q,d)=i=1,,nl1j=1,,np1ΔλiΔϕj|F(q,d,λi,ϕj)R(λi,ϕj)|2,
J(q,d)=i=1,,nl1j=1,,np1ΔλiΔϕjΔRij|F(q,d,λi,ϕj)R(λi,ϕj)|2.
Jα(q,d)=F(q,d)RL22+αP(q,d),
(qα,dα)=argminJα(q,d);
F(q,d)=FRs(p)Rs(p)q+FTs(p)Ts(p)q+FRs(p)Rs(p)d+FTs(p)Ts(p)d.
dHdz(z)=ikG(z)d,
dGdz(z)=ik(ϵ(z)ϵ1sin2ϕ)H(z)d.
dHdz(z)=ikϵ(z)G(z)d,
dGdz(z)=ikϵ(z)ϵ1sin2ϕϵ(z)V(z)d.
H(1)=ts(p)(1rs(p))2,
G(1)=ts(p)(1+rs(p))2p1.
δrs=ikd01UGdϵdqδqdz
δrp=ikd01(UG+VHϵ1sin2ϕϵ)dϵdqδqdz.
rsd=ikd01[VH+UG(ϵϵ1sin2ϕ)]dz
rpd=ikd01[UGϵ+VH(ϵϵ1sin2ϕ)ϵ]dz.
H(1)=ts(p)22,
G(1)=ts(p)22p1.
{q:[0,1]R|0q(z)1z[0,1]},
J(q,d)=F(q,d)R2
mk=qJ(qk,d)=2Fq(qk,d)*(RF(qk,d)),
Fq(q,d)g,h=g,Fq(q,d)*h.
qk+1=qk+βkmk,k=0,1,2,
βk=β,kN,
F(q,d)F(q¯,d)Fq(q¯,d)(qq¯)ηF(q,d)F(q¯,d),η<12,
βk=argminβRJ(qk+βmk).
0q(z)1,z[0,1],
π(q):[0,1][0,1],π(q)(z)={0ifq(z)<0q(z)if0q(z)11otherwise.
qk+1=π(qk+βkmk),k=0,1,2,
F(q,d)Rδ,
Jα(q,d)=F(q,d)R2+αP(q),
Jα(q,d)=F(q,d)R2+αq2.
mk=qJ(qk,d)=2Fq(qk,d)*(RF(qk,d))+2qk.
Jαs(q,a,d)=Jα(q,d)+Cqa2F(q,d)F(a,d)2,
qk+1=argminqJαs(q,qk,d),k=0,1,2,
qJα(q,a,d)=2(α+C)q+2Fq(q,d)*(F(a,d)R)2Ca=0,
q=1α+C[Ca+Fq(q,d)*(RF(a,d))].
qk+1=1α+C[Cqk+Fq(qk+1,d)*(RF(qk,d))],
qkn+1=1α+C[Cqk+Fq(qkn,d)*(RF(qk,d))].
F(qk)(qk+1qk)=RF(qk),
qk+q=qk+F(qk)1[RF(qk)].
qk+q=qk+(F(qk)*F(qk)+αI)1[F(qk)*(RF(qk)+αqk)].
Jα=RF(qk)F(qk)(qqk)2+αqk2.
qk+q=qk+(F(qk)*F(qk)+αI)1(F(qk)*(RF(qk)).
qpwc(z)i=1Nqiχ[zi1,zi)(z),
χI(z){1zI,0zI.
F(qpwc(z))R.
Qpwc{q|q(z)=i=1Nqiχ[zi1,zi)(z)}
Jα(q,d)min.
ziiNf¨uri=0,,N.
N=dc,
qi1|Ii|Iiq(z)dzf¨uri=1,,N.
ψ(x){10x<12,112x<1,0else.
ψjk(·)2j/2ψ(2j·k),j,kZ
suppψ=[0,1]andsuppψjk=[2j,2j(k+1)].
φχ[0,1).
ψ(x)=φ(2x)φ(2x1).
f(x)=kZf,φ(·k)φ(xk)+kZ,j>0f,ψjkψjk(x).
q˜(x)kZq˜0kφ(xk)+kZ,j˜>j>0q˜jkψjk(x).
givenRand a partitionzifindq⃗such thatF(qpwc)R.
q⃗=(q1,,qN)withJ(qpwc)F(qpwc)R2min.
J(q)=(Jqi(q))i=1,,N=0.
Jqi(q)=2F(q)*(F(q)R),χIi=0,i=1,N.
q⃗m+1=q⃗mβJ(q⃗m).
qiPqimitPqiargminqi˜Qadm|qiqi˜|.
qϵHϵeffϵH+2ϵeff+(1q)ϵLϵeffϵL+2ϵeff=0.
J=i[Rs(q,d,λi)ΔRi]2,
Rm=iR(λi)M,
Rm=0.1029(nd)0.9758.
F1(Rs(p),Ts(p))=Ts,F2(Rs(p),Ts(p))=Tp.

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