Abstract

Multilayer interference filters having various transmission characteristics are described and compared. The emphasis is on filters with narrow stop bands such as might be used to eliminate the hazard from a laser beam. Four types of filters are considered: (1) quarter-wave stacks of two dielectric materials having matching layers one-eighth wavelength thick; (2) quarter-wave stacks of two dielectric materials having all layers of the same optical thickness (including the end layers); (3) quarter-wave stacks wherein all layers are of the same optical thickness, but the refractive indices of the layers may all be different to achieve equal reflection ripples in the passband; and (4) multilayer stacks of two dielectric materials wherein each layer may be of a different optical thickness to achieve nearly equal reflection ripples in the passband. The new formulas presented give the bandwidths between nulls of all the various filters as well as the bandwidths between equal-ripple points of the equal-ripple filters. Explicit formulas are stated for the ripple envelopes of filter types (1) and (2), and for the ripple heights of equal-ripple filters of types (3) and (4). A first-order design procedure based on the theory of linear arrays is given and evaluated by working numerical examples; general design criteria are presented to establish the validity of the first-order theory.

© 1967 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955).
  2. A. Vačicek, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960).
  3. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964) Sec. 1.6.
  4. P. W. Baumeister, Handbook of Optical Design, Chap. 20, MIL-Handbook No. 141 (5October1962), obtainable from Control Center 550, Frankford Arsenal, Philadelphia, Pa.
  5. A. F. Turner, P. W. Baumeister, Appl. Opt. 5, 69 (1966).
    [CrossRef] [PubMed]
  6. J. S. Seeley, S. D. Smith, Appl. Opt. 5, 81 (1966).
    [CrossRef] [PubMed]
  7. L. Young, E. G. Cristal, Appl. Opt. 5, 77 (1966).
    [CrossRef] [PubMed]
  8. J. A. Dobrowolski, Appl. Opt. 4, 937 (1965).
    [CrossRef]
  9. J. S. Seeley, J. Opt. Soc. Am. 54, 342 (1964).
    [CrossRef]
  10. G. Tricoles, J. Phys. 25, 262 (1964).
    [CrossRef]
  11. A. Thelen, J. Opt. Soc. Am. 53, 1266 (1963).
    [CrossRef]
  12. L. Young, J. Opt. Soc. Am. 52, 753 (1962).
    [CrossRef]
  13. L. Young, J. Opt. Soc. Am. 51, 967 (1961).
    [CrossRef]
  14. R. J. Pegis, J. Opt. Soc. Am. 51, 1255 (1961).
    [CrossRef]
  15. A. Herpin, Compt, Rend. 225, 182 (1947).
  16. L. I. Epstein, J. Opt. Soc. Am. 42, 806 (1952).
    [CrossRef]
  17. L. Young, E. G. Cristal, IEEE Trans. MTT-14, 75 (1966).
  18. P. H. Berning, J. Opt. Soc. Am. 52, 431 (1962).
    [CrossRef]
  19. M. Iwata, S. Katsube, T. Fukuda, Sci. Light Tokyo 7, No. 2, 33 (1958).
  20. M. Iwata, Sci. Light Tokyo 2, 116 (1953).
  21. F. Abelès, Compt. Rend. 226, 1872 (1948).
  22. A. Herpin, Compt. Rend. 17 (1947).
  23. F. Abelès, Ann. Phys. 5,596, 706 (1950).
  24. K. D. Mielenz, J. Res. Natl. Bur. Std. 63A, 297 (1959).
    [CrossRef]
  25. K. D. Mielenz, J. Opt. Soc. Am. 50, 1014 (1960).
    [CrossRef]
  26. C. D. Herpin, A. Herpin, Rev. Opt. 32, 321 (1953).
  27. P. G. Kard, Opt. Spectry. 11, 49 (1960).
  28. P. G. Kard, Opt. Spectry. 14, 121 (1963).
  29. L. Young, IEEE Trans. MTT-10, 339 (1962).
  30. R. Levy, IEEE Trans. MTT-13, 514 (1965).
  31. L. Young, IEEE Trans. MTT-13, 488 (1965).
  32. G. L. Matthaei, L. Young, E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures (McGraw-Hill Book Co., Inc., New York, 1964).
  33. C. L. Dolph, Proc. Inst. Radio Engrs. 34, 335 (1946). See also H. J. Riblet, C. L. Dolph, Proc. Inst. Radio Engrs. 35, 489 (1957).
  34. L. B. Brown, G. A. Sharp, “Tchebyscheff Antenna Distribution, Beamwidth, and Gain Tables”, NAYORD Report 4629, NOLO Report 383, Naval Ordnance Laboratory, Corona, Calif. (February1958).
  35. M. L. Reuss, “Some Design Considerations Concerning Linear Arrays Having Dolph-Tchebycheff Amplitude Distributions”, NLR Report 5240, Project NE 120-000–27, U. S. Naval Research Laboratory, Washington, D. C. (12February1959) (AD No. 212 621).
  36. G. J. Van der Maas, J. Appl. Phys. 25, 121 (1954).
    [CrossRef]
  37. G. J. Van der Maas, J. Appl. Phys. 24, 1250 (1953).
    [CrossRef]
  38. C. J. Drane, Proc. IEEE, 110, 1755 (1963).
  39. R. S. Elliott, IEEE Trans. AP-11, 707 (1963).
    [CrossRef]
  40. R. S. Elliott, IEEE Trans. AP-11, 378 (1963).
    [CrossRef]
  41. R. J. Stegen, Proc. Inst. Radio Engrs. 41, 1671 (1953).
  42. R. J. Stegen, Inst. Radio Engrs. Trans. AP-8, 629 (1980).
  43. D. Barbiere, Proc. Inst. Radio Engrs. 40, 78 (1952).
  44. L. L. Bailin, M. J. Ehrlich, Proc. Inst. Radio Engrs. 41, 235 (1953).
  45. H. E. Salzer, “The Use of Poisson’s Formula in Pattern Synthesis”, DOFL Report TR-28, Project No. 4104-106448, Diamond Ordnance Fuze Laboratories, Washington D.C. (19January1954).

1980 (1)

R. J. Stegen, Inst. Radio Engrs. Trans. AP-8, 629 (1980).

1966 (4)

1965 (3)

J. A. Dobrowolski, Appl. Opt. 4, 937 (1965).
[CrossRef]

R. Levy, IEEE Trans. MTT-13, 514 (1965).

L. Young, IEEE Trans. MTT-13, 488 (1965).

1964 (2)

1963 (5)

A. Thelen, J. Opt. Soc. Am. 53, 1266 (1963).
[CrossRef]

C. J. Drane, Proc. IEEE, 110, 1755 (1963).

R. S. Elliott, IEEE Trans. AP-11, 707 (1963).
[CrossRef]

R. S. Elliott, IEEE Trans. AP-11, 378 (1963).
[CrossRef]

P. G. Kard, Opt. Spectry. 14, 121 (1963).

1962 (3)

1961 (2)

1960 (2)

P. G. Kard, Opt. Spectry. 11, 49 (1960).

K. D. Mielenz, J. Opt. Soc. Am. 50, 1014 (1960).
[CrossRef]

1959 (1)

K. D. Mielenz, J. Res. Natl. Bur. Std. 63A, 297 (1959).
[CrossRef]

1958 (1)

M. Iwata, S. Katsube, T. Fukuda, Sci. Light Tokyo 7, No. 2, 33 (1958).

1954 (1)

G. J. Van der Maas, J. Appl. Phys. 25, 121 (1954).
[CrossRef]

1953 (5)

G. J. Van der Maas, J. Appl. Phys. 24, 1250 (1953).
[CrossRef]

M. Iwata, Sci. Light Tokyo 2, 116 (1953).

C. D. Herpin, A. Herpin, Rev. Opt. 32, 321 (1953).

R. J. Stegen, Proc. Inst. Radio Engrs. 41, 1671 (1953).

L. L. Bailin, M. J. Ehrlich, Proc. Inst. Radio Engrs. 41, 235 (1953).

1952 (2)

D. Barbiere, Proc. Inst. Radio Engrs. 40, 78 (1952).

L. I. Epstein, J. Opt. Soc. Am. 42, 806 (1952).
[CrossRef]

1950 (1)

F. Abelès, Ann. Phys. 5,596, 706 (1950).

1948 (1)

F. Abelès, Compt. Rend. 226, 1872 (1948).

1947 (2)

A. Herpin, Compt. Rend. 17 (1947).

A. Herpin, Compt, Rend. 225, 182 (1947).

1946 (1)

C. L. Dolph, Proc. Inst. Radio Engrs. 34, 335 (1946). See also H. J. Riblet, C. L. Dolph, Proc. Inst. Radio Engrs. 35, 489 (1957).

Abelès, F.

F. Abelès, Ann. Phys. 5,596, 706 (1950).

F. Abelès, Compt. Rend. 226, 1872 (1948).

Bailin, L. L.

L. L. Bailin, M. J. Ehrlich, Proc. Inst. Radio Engrs. 41, 235 (1953).

Barbiere, D.

D. Barbiere, Proc. Inst. Radio Engrs. 40, 78 (1952).

Baumeister, P. W.

A. F. Turner, P. W. Baumeister, Appl. Opt. 5, 69 (1966).
[CrossRef] [PubMed]

P. W. Baumeister, Handbook of Optical Design, Chap. 20, MIL-Handbook No. 141 (5October1962), obtainable from Control Center 550, Frankford Arsenal, Philadelphia, Pa.

Berning, P. H.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964) Sec. 1.6.

Brown, L. B.

L. B. Brown, G. A. Sharp, “Tchebyscheff Antenna Distribution, Beamwidth, and Gain Tables”, NAYORD Report 4629, NOLO Report 383, Naval Ordnance Laboratory, Corona, Calif. (February1958).

Cristal, E. G.

L. Young, E. G. Cristal, IEEE Trans. MTT-14, 75 (1966).

L. Young, E. G. Cristal, Appl. Opt. 5, 77 (1966).
[CrossRef] [PubMed]

Dobrowolski, J. A.

Dolph, C. L.

C. L. Dolph, Proc. Inst. Radio Engrs. 34, 335 (1946). See also H. J. Riblet, C. L. Dolph, Proc. Inst. Radio Engrs. 35, 489 (1957).

Drane, C. J.

C. J. Drane, Proc. IEEE, 110, 1755 (1963).

Ehrlich, M. J.

L. L. Bailin, M. J. Ehrlich, Proc. Inst. Radio Engrs. 41, 235 (1953).

Elliott, R. S.

R. S. Elliott, IEEE Trans. AP-11, 378 (1963).
[CrossRef]

R. S. Elliott, IEEE Trans. AP-11, 707 (1963).
[CrossRef]

Epstein, L. I.

Fukuda, T.

M. Iwata, S. Katsube, T. Fukuda, Sci. Light Tokyo 7, No. 2, 33 (1958).

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955).

Herpin, A.

C. D. Herpin, A. Herpin, Rev. Opt. 32, 321 (1953).

A. Herpin, Compt, Rend. 225, 182 (1947).

A. Herpin, Compt. Rend. 17 (1947).

Herpin, C. D.

C. D. Herpin, A. Herpin, Rev. Opt. 32, 321 (1953).

Iwata, M.

M. Iwata, S. Katsube, T. Fukuda, Sci. Light Tokyo 7, No. 2, 33 (1958).

M. Iwata, Sci. Light Tokyo 2, 116 (1953).

Jones, E. M. T.

G. L. Matthaei, L. Young, E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures (McGraw-Hill Book Co., Inc., New York, 1964).

Kard, P. G.

P. G. Kard, Opt. Spectry. 14, 121 (1963).

P. G. Kard, Opt. Spectry. 11, 49 (1960).

Katsube, S.

M. Iwata, S. Katsube, T. Fukuda, Sci. Light Tokyo 7, No. 2, 33 (1958).

Levy, R.

R. Levy, IEEE Trans. MTT-13, 514 (1965).

Matthaei, G. L.

G. L. Matthaei, L. Young, E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures (McGraw-Hill Book Co., Inc., New York, 1964).

Mielenz, K. D.

K. D. Mielenz, J. Opt. Soc. Am. 50, 1014 (1960).
[CrossRef]

K. D. Mielenz, J. Res. Natl. Bur. Std. 63A, 297 (1959).
[CrossRef]

Pegis, R. J.

Reuss, M. L.

M. L. Reuss, “Some Design Considerations Concerning Linear Arrays Having Dolph-Tchebycheff Amplitude Distributions”, NLR Report 5240, Project NE 120-000–27, U. S. Naval Research Laboratory, Washington, D. C. (12February1959) (AD No. 212 621).

Salzer, H. E.

H. E. Salzer, “The Use of Poisson’s Formula in Pattern Synthesis”, DOFL Report TR-28, Project No. 4104-106448, Diamond Ordnance Fuze Laboratories, Washington D.C. (19January1954).

Seeley, J. S.

Sharp, G. A.

L. B. Brown, G. A. Sharp, “Tchebyscheff Antenna Distribution, Beamwidth, and Gain Tables”, NAYORD Report 4629, NOLO Report 383, Naval Ordnance Laboratory, Corona, Calif. (February1958).

Smith, S. D.

Stegen, R. J.

R. J. Stegen, Inst. Radio Engrs. Trans. AP-8, 629 (1980).

R. J. Stegen, Proc. Inst. Radio Engrs. 41, 1671 (1953).

Thelen, A.

Tricoles, G.

G. Tricoles, J. Phys. 25, 262 (1964).
[CrossRef]

Turner, A. F.

Vacicek, A.

A. Vačicek, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960).

Van der Maas, G. J.

G. J. Van der Maas, J. Appl. Phys. 25, 121 (1954).
[CrossRef]

G. J. Van der Maas, J. Appl. Phys. 24, 1250 (1953).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964) Sec. 1.6.

Young, L.

L. Young, E. G. Cristal, Appl. Opt. 5, 77 (1966).
[CrossRef] [PubMed]

L. Young, E. G. Cristal, IEEE Trans. MTT-14, 75 (1966).

L. Young, IEEE Trans. MTT-13, 488 (1965).

L. Young, IEEE Trans. MTT-10, 339 (1962).

L. Young, J. Opt. Soc. Am. 52, 753 (1962).
[CrossRef]

L. Young, J. Opt. Soc. Am. 51, 967 (1961).
[CrossRef]

G. L. Matthaei, L. Young, E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures (McGraw-Hill Book Co., Inc., New York, 1964).

Ann. Phys. (1)

F. Abelès, Ann. Phys. 5,596, 706 (1950).

Appl. Opt. (4)

Compt, Rend. (1)

A. Herpin, Compt, Rend. 225, 182 (1947).

Compt. Rend. (2)

F. Abelès, Compt. Rend. 226, 1872 (1948).

A. Herpin, Compt. Rend. 17 (1947).

IEEE Trans. (6)

L. Young, E. G. Cristal, IEEE Trans. MTT-14, 75 (1966).

L. Young, IEEE Trans. MTT-10, 339 (1962).

R. Levy, IEEE Trans. MTT-13, 514 (1965).

L. Young, IEEE Trans. MTT-13, 488 (1965).

R. S. Elliott, IEEE Trans. AP-11, 707 (1963).
[CrossRef]

R. S. Elliott, IEEE Trans. AP-11, 378 (1963).
[CrossRef]

Inst. Radio Engrs. Trans. (1)

R. J. Stegen, Inst. Radio Engrs. Trans. AP-8, 629 (1980).

J. Appl. Phys. (2)

G. J. Van der Maas, J. Appl. Phys. 25, 121 (1954).
[CrossRef]

G. J. Van der Maas, J. Appl. Phys. 24, 1250 (1953).
[CrossRef]

J. Opt. Soc. Am. (8)

J. Phys. (1)

G. Tricoles, J. Phys. 25, 262 (1964).
[CrossRef]

J. Res. Natl. Bur. Std. (1)

K. D. Mielenz, J. Res. Natl. Bur. Std. 63A, 297 (1959).
[CrossRef]

Opt. Spectry. (2)

P. G. Kard, Opt. Spectry. 11, 49 (1960).

P. G. Kard, Opt. Spectry. 14, 121 (1963).

Proc. IEEE (1)

C. J. Drane, Proc. IEEE, 110, 1755 (1963).

Proc. Inst. Radio Engrs. (4)

C. L. Dolph, Proc. Inst. Radio Engrs. 34, 335 (1946). See also H. J. Riblet, C. L. Dolph, Proc. Inst. Radio Engrs. 35, 489 (1957).

D. Barbiere, Proc. Inst. Radio Engrs. 40, 78 (1952).

L. L. Bailin, M. J. Ehrlich, Proc. Inst. Radio Engrs. 41, 235 (1953).

R. J. Stegen, Proc. Inst. Radio Engrs. 41, 1671 (1953).

Rev. Opt. (1)

C. D. Herpin, A. Herpin, Rev. Opt. 32, 321 (1953).

Sci. Light Tokyo (2)

M. Iwata, S. Katsube, T. Fukuda, Sci. Light Tokyo 7, No. 2, 33 (1958).

M. Iwata, Sci. Light Tokyo 2, 116 (1953).

Other (8)

H. E. Salzer, “The Use of Poisson’s Formula in Pattern Synthesis”, DOFL Report TR-28, Project No. 4104-106448, Diamond Ordnance Fuze Laboratories, Washington D.C. (19January1954).

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955).

A. Vačicek, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960).

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964) Sec. 1.6.

P. W. Baumeister, Handbook of Optical Design, Chap. 20, MIL-Handbook No. 141 (5October1962), obtainable from Control Center 550, Frankford Arsenal, Philadelphia, Pa.

L. B. Brown, G. A. Sharp, “Tchebyscheff Antenna Distribution, Beamwidth, and Gain Tables”, NAYORD Report 4629, NOLO Report 383, Naval Ordnance Laboratory, Corona, Calif. (February1958).

M. L. Reuss, “Some Design Considerations Concerning Linear Arrays Having Dolph-Tchebycheff Amplitude Distributions”, NLR Report 5240, Project NE 120-000–27, U. S. Naval Research Laboratory, Washington, D. C. (12February1959) (AD No. 212 621).

G. L. Matthaei, L. Young, E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures (McGraw-Hill Book Co., Inc., New York, 1964).

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Figures (17)

Fig. 1
Fig. 1

A dielectric multilayer stack with end layers having half of the optical thickness of interior layers. The interior layers have twice the optical thickness of the end layers.

Fig. 2
Fig. 2

Typical envelope of a large number of reflectance curves of various stacks with different numbers of layers, showing one typical curve.

Fig. 3
Fig. 3

Fractional bandwidth between nulls (w1) as a function of refractive index ratio for m = 9, m = 19, m = 29, and m = infinity (w1 = w H ).

Fig. 4
Fig. 4

Symmetrical three-layer sandwiches forming the basic period of many filters.

Fig. 5
Fig. 5

Quarter-wave stack consisting of many layers of identical optical thickness. Each layer has the same optical thickness.

Fig. 6
Fig. 6

Multilayer filter in which the refractive indices of the layers may all be different.

Fig. 7
Fig. 7

Possible equal-ripple response curve obtainable with filter shown in Fig. 6.

Fig. 8
Fig. 8

Schematic of linear array and its radiation pattern.

Fig. 9
Fig. 9

Response curve for the filter Case A.

Fig. 10
Fig. 10

Response curve for the filter Case B.

Fig. 11
Fig. 11

Response curve for the filter Case C.

Fig. 12
Fig. 12

Response curve for the filter Case D.

Fig. 13
Fig. 13

Response curve for the filter Case E.

Fig. 14
Fig. 14

Multilayer filter consisting of alternate thin and thick layers.

Fig. 15
Fig. 15

Response curve of filter Case F.

Fig. 16
Fig. 16

Response curve of filter Case G.

Fig. 17
Fig. 17

Dielectric layer of refractive index n2 between two semiinfinite media of refractive index n1.

Tables (3)

Tables Icon

Table I Current Distributions on Array Elements for 40-dB Sidelobe Ratio (Normalized with Respect to the Center Element)

Tables Icon

Table II Summary of Numerical Results and Comparisons for the Five Filter Cases A, B, C, D, E (All Designs are Based on a 40-dB Sidelobe Ratio, as in Table I)

Tables Icon

Table A-I Excess Loss Ratio in Decibels, 10 log10 (S/S r ) = 10 log10 [T m 2 (x)] a

Equations (58)

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n 0 = n 1 ½ n 2 ½ .
n 0 = n 1 ³ / n 2 - ½ .
w H = 4 π [ sin - 1 ( n 1 - n 2 ) / ( n 1 + n 2 ) rad ] = 1 45 [ sin - 1 ( n 1 - n 2 ) / ( n 1 + n 2 ) deg ] ,
θ i = cos - 1 { [ ( n 1 - n 2 ) 2 + 4 n 1 n 2 sin 2 ( π i / m ) ] ½ n 1 + n 2 }
= cos - 1 [ Γ 12 2 + ( 1 - Γ 12 2 ) sin 2 ( π i / m ) ] ,
θ 1 = cos - 1 { [ ( n 1 - n 2 ) 2 + 4 n 1 n 2 sin 2 ( π / m ) ] ½ n 1 + n 2 }
= cos - 1 [ Γ 12 2 + ( 1 - Γ 12 2 ) sin 2 ( π / m ) ] .
w 1 = 2 - [ θ 1 ( deg ) / 45 ] .
V env = [ ( n 1 / n 2 ) ( u + v ) / ( u - v ) ] ± 1 > 1 , where u = ( n 1 + n 2 ) cos θ v = n 1 - n 2 } .
R env = [ ( n 1 2 - n 2 2 ) ( 1 - cos θ ) ( n 1 + n 2 ) 2 cos θ - ( n 1 - n 2 ) 2 ] 2 .
V env = [ ( n 1 / n 0 ) 2 ( u + v ) / ( u - v ) ] ± 1 > 1
R env = [ ( n 1 2 - n 0 2 ) u + ( n 1 2 + n 0 2 ) v ( n 1 2 + n 0 2 ) u + ( n 1 2 - n 0 2 ) v ] 2 .
V max ( n 1 / n 2 ) ± ( m + 1 ) > 1.
Γ = ( V - 1 ) / ( v + 1 )
R = Γ 2 T = 1 - R } .
Γ env ( n 1 - n 2 ) / ( n 1 + n 2 ) cos θ
= Γ 12 / cos θ .
ɛ ( θ ) = ɛ r T m 2 ( sin θ / sin θ 0 ) ,
T m 2 ( x ) = cos 2 ( m cos - 1 x ) , x 1
= cosh 2 ( m cosh - 1 x ) , x 1 ,
θ 0 = sin - 1 { cosh [ ( 1 / m ) cosh - 1 ( ɛ max / ɛ r ) ½ ] } - 1 ,
L P = ɛ + 1.
ɛ = R / ( 1 - R )
= ( V - 1 ) 2 / 4 V
L P = ( 1 - R ) - 1
= ( V + 1 ) 2 / 4 V .
L A = - 10 log 10 ( L P )             dB .
ɛ max = R max / ( 1 - R max )
= ( V max - 1 ) 2 / 4 V max .
ɛ r = R r / ( 1 - R r )
= ( V r - 1 ) 2 / 4 V r .
x = sin θ / sin θ 0 .
w 0 = 2 - θ 0 ( deg ) 45 .
10 log 10 ( ɛ max / ɛ r ) = 10 log 10 ( 1000 - 1 ) / ( 1.259 - 1 ) = 35.86 dB .
x = sin 90° / sin θ 0 = sin 90° / sin 81° = 1.0125.
10 log 10 ( ɛ / ɛ r ) = 10 log 10 ( 100 - 1 ) / ( 1.259 - 1 ) = 25.82 dB .
( 180 d sin ψ ) / λ ( θ - 90 ) ,
bandwidth = 4 / m ,
V 12 = n 1 / n 2 or n 2 / n 1 , whichever is > 1.
bandwidth = ( 4 / π ) ( V 12 - 1 ) / ( V 12 + 1 ) ,
V 12 = ( m + π ) / ( m - π ) .
V max = V 12 m + 1 ( 1 + 2 π m ) m + 1 e 2 π .
bandwidth , w 1 a = 2 sin ( beam width / 2 ) ,
w 0 a = 2 - 1 45 { sin - 1 { cosh [ ( 1 / m ) cosh - 1 ( SLR ) ] } - 1 deg } .
sidelobe ratio , SLR log ( V max ) / log ( V r ) ,
V r = 1 + [ log e ( V max ) ] / SLR .
s = 20 log 10 ( SLR ) .
w 0 a 2 - 1 45 { sin - 1 [ 1 + 1 2 ( 0.69 + 0.115 s ) 2 / m 2 ] - 1 deg } ,
m > s > 1
w ( w 0 2 + w 1 2 ) ½ ,
n i / n 0 = 1 - x k = 1 i ( - 1 ) k I k             ( i = 1 to m ) .
θ 1 = y I 1 θ 3 = y I 2 θ 5 = y I 3 ........ θ i = y I ( i + 1 ) / 2 , i = 1 , 3 , 5 , , 2 m + 1 }
θ 2 = 180° - 1 2 ( θ 1 + θ 3 ) ........ θ i = 180° - 1 2 ( θ i - 1 + θ i + 1 ) , i = 2 , 4 , , 2 m .
a = 1 2 ( n 1 + 1 n 1 ) b = 1 2 ( n 1 n 2 2 + n 2 2 n 1 ) } ,
R = a cos 2 θ + b sin 2 θ - 1 a cos 2 θ + b sin 2 θ - 1 ,
V = [ 1 + ( R ) ½ ] / [ 1 - ( R ) ½ ] .
θ = sin - 1 [ ( c - a ) / ( c - b ) ] ,
c = 1 2 ( V + 1 / V ) = ( 1 + R ) / ( 1 - R ) .

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