Abstract

A new layer-by-layer multilayer design method is presented. The method is demonstrated mathematically and makes possible the optimization of the multilayer for the highest possible reflectance either at normal incidence or at nonnormal incidence for s- or p-polarized radiation. With the current method multilayers can be designed regardless of the number of different materials used. The optimum layer thickness is determined by means of functions suitable for implementation in a computer code. The new multilayer design method is fast and accurate.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. K. Carniglia, J. H. Apfel, “Maximum reflectance of multilayer dielectric mirrors in the presence of slight absorption,” J. Opt. Soc. Am. 70, 523–534 (1980).
    [CrossRef]
  2. E. Spiller, Soft X-Ray Optics (SPIE–The International Society for Optical Engineering, Bellingham, Wash., 1994), p. 143.
  3. M. Yamamoto, T. Namioka, “Layer-by-layer design method for soft-x-ray multilayers,” Appl. Opt. 31, 1622–1630 (1992).
    [CrossRef] [PubMed]
  4. See, for instance, W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).
  5. J. F. Meekins, R. G. Cruddace, H. Gursky, “Optimization of layered synthetic microstructures for narrowband reflectivity at soft x-ray and EUV wavelengths,” Appl. Opt. 25, 2757–2763 (1986).
    [CrossRef] [PubMed]
  6. J. I. Larruquert, “Reflectance enhancement through sub-quarterwave multilayers with more than two materials,” J. Opt. Soc. Am. A 19, 391–397 (2001).
    [CrossRef]
  7. M. Singh, J. J. M. Braat, “Design of multilayer extreme-ultraviolet mirrors for enhanced reflectivity,” Appl. Opt. 39, 2189–2197 (2000).
    [CrossRef]
  8. J. I. Larruquert, “Reflectance enhancement with sub-quarterwave multilayers of highly absorbing materials,” J. Opt. Soc. Am. A 18, 1406–1414 (2001).
    [CrossRef]
  9. J. I. Larruquert, “General theory of sub-quarterwave multilayers with highly absorbing materials,” J. Opt. Soc. Am. A 18, 2617–2627 (2001).
    [CrossRef]

2001 (3)

2000 (1)

1992 (1)

1986 (1)

1980 (1)

Apfel, J. H.

Braat, J. J. M.

Carniglia, C. K.

Cruddace, R. G.

Flannery, B. P.

See, for instance, W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Gursky, H.

Larruquert, J. I.

Meekins, J. F.

Namioka, T.

Press, W. H.

See, for instance, W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Singh, M.

Spiller, E.

E. Spiller, Soft X-Ray Optics (SPIE–The International Society for Optical Engineering, Bellingham, Wash., 1994), p. 143.

Teukolsky, S. A.

See, for instance, W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Vetterling, W. T.

See, for instance, W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Yamamoto, M.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Other (2)

E. Spiller, Soft X-Ray Optics (SPIE–The International Society for Optical Engineering, Bellingham, Wash., 1994), p. 143.

See, for instance, W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Optimum layer thicknesses of a multilayer optimized for highest reflectance at 20.0 nm. The data of this paper are compared with those obtained by Yamamoto and Namioka.3 The latter were shifted horizontally to make the comparison easier.

Tables (1)

Tables Icon

Table 1 Comparison of a Multilayera Designed with the Current Method and Designed with the Methods of Meekins et al. b and Yamamoto and Namiokac

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

ri-1=fi-1+riexp(βi)1+fi-1riexp(βi),
rm=fm.
βi=4πjNicos θixiλ,
Nicos θi=(Ni2-N02sin2 θ0)1/2.
R0=r0*r0.
Rximax=0,i=1tom,
(-1)i det1,i2Rxjxkmax>0,i=1tom,
Rx1max=2 Rer0*r0x1=0,
Rx2max=2 Rer0*r0x2=2 Rer0*r0x1×r0/x2r0/x1=-2 Imr0*r0x1Im(r0/r1) (r1/x2)r0/x1=0,
Rxmmax=2 Rer0*r0xm=-2 Imr0*r0x1×Im(rm-2/rm-1) (rm-1/xm)rm-2/xm-1×i=2m-1Re(ri-2/ri-1) (ri-1/xi)ri-2/xi-1=0,
Rer0*r0x1=0,i=1,
Im(ri-2/ri-1) (ri-1/xi)ri-2/xi-1=0,i=2tom.
Rer0*r0x1=Rej4πλR0N0cos θ0u1=-4πλR0N0cos θ0Im(u1),
Im(ri-2/ri-1) (ri-1/xi)ri-2/xi-1
=Im(ui),i=2tom,
ui=Nicos θiNi-1cos θi-1(1-fi-12)riexp βi(fi-1+riexp βi)(1+fi-1riexp βi),
i=1tom.
Im(ui)=0,i=1tom.
2Rxixjmax=-24πλR0N0cos θ0×l=1iRe(ul)×k1=1jIm(uk1/xk1)Re(uk1)k2=k1+1jRe(uk2),
i,j=1tom,ji.
Imuixik=1i-1Re(uk)>0,i=1tom.
Re(wi)k=1i-1Re(uk)>0,i=1tom,
wi=Ni-1cos θi-1fi-1[1-ri2exp(2βi)](1-fi-12)riexp βi,
i=1tom.
Im(ui)=0Re(wi)>0,i=1tom  Re(ui)>0,i=1tom.
Im(ui)=0Re(wi)>0,i=1tom.
Rxi=2 Rer0*r0xi=2 Rer0*r0x1×(r0/r1) (r1/x2)r0/x1××(ri-2/ri-1) (ri-1/xi)ri-2/xi-1.
Rxi=-24πλR0N0cos θ0Im(u1u2ui).
2Rxixjmax=-24πλR0N0cos θ0×Imu1xjl=2iRe(ul)+Re(u1)Imu2xjl=3iRe(ul)++k=1j-1Re(uk)Imujxjl=j+1iRe(ul),i,j=1tom,ji.
2Rxjximax=-24πλR0N0cos θ0×Imu1xil=2jRe(ul)+Re(u1)Imu2xil=3jRe(ul)++k=1j-1Re(uk)Imujxi,i,j=1tom,ji.
Imujxi=Imujxjl=j+1iRe(ul),j<i.
2Rxixjmax=-24πλR0N0cos θ0l=1iRe(ul)×k1=1jIm(uk1/xk1)Re(uk1)k2=k1+1jRe(uk2),i,j=1tom,ji,

Metrics