Abstract

In the case of total reflection at a boundary surface between two different optical media, the ray reflected at the boundary is spatially shifted with respect to the point where the incident ray intersects the boundary. The light penetrates into the second medium, and the evanescent electromagnetic wave propagates along the boundary. The described effect is called the Goos–Hänchen effect. Our work describes the influence of the Goos–Hänchen effect on the imaging properties of planar optical systems, and a differential equation of a wave-front meridian that corresponds to a reflected bundle of rays is derived. It is shown that the wave front can be described by the d’Alambert differential equation. This equation makes it possible to determine the coordinates of individual points on the wave-front meridian. The influence of total reflection on the value of the Strehl definition of the reflected ray bundle, is also investigated.

© 2005 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1968).
  2. R. H. Renard, “Total reflection: a new evaluation of the Goos–Hänchen shift,” J. Opt. Soc. Am. 54, 1190–1197 (1964).
    [CrossRef]
  3. A. Miks, Applied Optics (Czech Technical University Press, Prague, 2000).
  4. H. K. V. Lotsch, “Reflection and refraction of a beam of light at a plane interface,” J. Opt. Soc. Am. 58, 551–561 (1968).
    [CrossRef]
  5. O. Bryndhal, “Evanescent waves in optical imaging,” in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, Amsterdam, 1973), pp. 167–221.
  6. F. Pilon, H. Gilles, S. Girard, “Experimental observation of the Imbert–Fedorov transverse displacement after a single total reflection,” Appl. Opt. 43, 1863–1869 (2004).
    [CrossRef]
  7. H. Maecker, “Quantitativer Nachweis von Grenzschichtwellen in der Optik,” Ann. Phys. (Leipzig) 4, 409–431 (1949).
    [CrossRef]
  8. J. Picht, “Neue Untersuchungen zur Totalreflexion,” Optik 12, 41–55 (1955).
  9. A. Miks, “Dependence of the wave-front aberration on the radius of the reference sphere,” J. Opt. Soc. Am. A 19, 1187–1190 (2002).
    [CrossRef]

2004 (1)

2002 (1)

1968 (1)

1964 (1)

1955 (1)

J. Picht, “Neue Untersuchungen zur Totalreflexion,” Optik 12, 41–55 (1955).

1949 (1)

H. Maecker, “Quantitativer Nachweis von Grenzschichtwellen in der Optik,” Ann. Phys. (Leipzig) 4, 409–431 (1949).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1968).

Bryndhal, O.

O. Bryndhal, “Evanescent waves in optical imaging,” in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, Amsterdam, 1973), pp. 167–221.

Gilles, H.

Girard, S.

Lotsch, H. K. V.

Maecker, H.

H. Maecker, “Quantitativer Nachweis von Grenzschichtwellen in der Optik,” Ann. Phys. (Leipzig) 4, 409–431 (1949).
[CrossRef]

Miks, A.

Picht, J.

J. Picht, “Neue Untersuchungen zur Totalreflexion,” Optik 12, 41–55 (1955).

Pilon, F.

Renard, R. H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1968).

Ann. Phys. (Leipzig) (1)

H. Maecker, “Quantitativer Nachweis von Grenzschichtwellen in der Optik,” Ann. Phys. (Leipzig) 4, 409–431 (1949).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Optik (1)

J. Picht, “Neue Untersuchungen zur Totalreflexion,” Optik 12, 41–55 (1955).

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1968).

A. Miks, Applied Optics (Czech Technical University Press, Prague, 2000).

O. Bryndhal, “Evanescent waves in optical imaging,” in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, Amsterdam, 1973), pp. 167–221.

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Figures (6)

Fig. 1
Fig. 1

Total reflection and the Goos–Hänchen shift.

Fig. 2
Fig. 2

Dependence of the Goos–Hänchen shift on angle of incidence (n1 = 1.5 , n2 = 1, λ = 633 nm).

Fig. 3
Fig. 3

Total reflection of a homocentric bundle of rays at the boundary between different optical media (n1>n2).

Fig. 4
Fig. 4

Wave aberration corresponding to the Goos–Hänchen shift x and x for λ=633 nm, n1=1.6, n2=1.

Fig. 5
Fig. 5

Reflection of a bundle of rays at the reflecting surface of a prism.

Fig. 6
Fig. 6

Wave aberration due to the Goos–Hänchen shift.

Tables (2)

Tables Icon

Table 1 Aberration Coefficients with Respect to the Ideal Reference Sphere: s Polarization

Tables Icon

Table 2 Aberration Coefficients with Respect to the Ideal Reference Sphere: p Polarization

Equations (42)

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dλ1=sin ε cos2 επ(1-n2)(sin2 ε-n2)1/2=xλ1cos ε,
dλ1=n2 sinε cos2 επ(n4 cos2 ε+sin2 ε-n2)(sin2 ε-n2)1/2=xλ1cos ε,
y=1tan ε (x-x), y=1tan ε (x-x),
x=sin ε cosεπ(1-n2)(sin2 ε-n2)1/2 λ1 x=n2 sin ε cos επ(n4 cos2 ε+sin2 ε-n2)(sin2 ε-n2)1/2 λ1.
sin ε=tan ε/(1+tan2 ε)1/2, cos ε=1/(1+tan2ε)1/2, sin ε cos ε=tan ε/(1+tan2 ε),
x=C(1+C2)[C2/(1+C2)-n2]1/2 L, x=Cn2(C2-n2)[C2/(1+C2)-n2]1/2 L,
C=tan ε,  L=λ1π(1-n2).
F(x, y, C)=F(x, y, C)=y-xC+1(1+C2)[C2/(1+C2)-n2]1/2 L=0, F(x, y, C)=F(x, y, C)=y-xC+n2(C2-n2)[C2/(1+C2)-n2]1/2 L=0.
F(x, y, C)=0,  Fx+Fy y=0,
G(x, y, y)=0.
G(x, y, -1/y)=0,
Fx=-1C,  Fy=1,
Fx+Fy y=-1C+y=0.
C=1/y.
G(x, y, y)=G(x, y, y)=y-xy+L(1+1/y2)[(1/y2)/(1+1/y2)-n2]1/2=0,G(x, y, y)=G(x, y, y)=y-xy+Ln2(1/y2-n2)[(1/y2)/(1+1/y2)-n2]1/2=0.
y+xy+L(1+y2)[y2/(1+y2)-n2]1/2=0, y+xy+Ln2(y2-n2)[y2/(1+y2)-n2]1/2=0.
y=xA(y)+B(y).
dx/dp+P(p)x+Q(p)=0,
P(p)=[dA(p)]/dpA(p)-p,  Q(p)=[dB(p)]/dpA(p)-p.
A(p)=-1p, B(p)=-L(1+p2)[p2/(1+p2)-n2]1/2, A(p)=-1p, B(p)=-Ln2(p2-n2)[p2/(1+p2)-n2]1/2.
P(p)=P(p)=-1p(1+p2),
Q(p)=-p2L(1+p2)3[p2/(1+p2)-n2]1/2×2+1p2-n2(1+p2), Q(p)=-p2n2L(p2-n2)2(1+p2)[p2/(1+p2)-n2]1/2×2+p2-n2(1+p2)[p2-n2(1+p2)].
dx/dp+P(p)x+Q(p)=0.
y=xA(p)+B(p),
p=-tan ε=-sin ε(1-sin2 ε)1/2.
sin εc=n2n1=n,  pc=-n(1-n2)1/2.
sin εn,  p-n(1-n2)1/2.
sin εc=n2/n1.
α<45°-εc.
αmax=45°-εc.
45°-αmax<ε<45°+αmax.
38.68°<ε<51.32°,  0.8<|p|<1.25.
W=W11r+W20r2+W31r3+W40r4+W51r5+W60r6,
ES=W20212+W20W406+4W40245+3W20W6020+W40W606+9W602112,
EK=W1124+W11W313+W3128+W11W514+W31W515+W51212,
E0=ES+EK,
SD=1-(2π/λ)2E0.
W11=-23W31-12W51,  W20=-W40-910W60.
Wmin=W31(r2-23)r+W40(r2-1)r2+W51(r4-12)r+W60(r4-910)r2.
(ES)min=W402180+W40W6060+9W602700, (EK)min=W31272+W31W5130+W51248, (E0)min=(ES)min+(EK)min, SDmax=1-2πλ2(E0)min.
SD=0.8623,  SD=0.6456.
W11=-0.0089,  W20=-0.0228, SD=0.9985, W11=-0.0344,  W20=-0.0134, SD=0.9904,

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