Abstract

Now that ultrashort laser pulses can be used in holography, the temporal and spatial resolution approach the same order of magnitude. In that case the limited speed of light sometimes causes large measuring errors if correction methods are not introduced. Therefore, we want to revive the Minkowski diagram, which was invented in 1908 to visualize relativistic relations between time and space. We show how this diagram in a modified form can be used to derive both the static holodiagram, used for conventional holography, including ultrahigh-speed recordings of wavefronts, and a dynamic holodiagram used for studying the apparent distortions of objects recorded at relativistic speeds.

© 1988 Optical Society of America

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References

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  1. N. Abramson, The Making and Evaluation of Holograms (Academic, London, 1981), pp. 28, 110, 144, and 197.
  2. N. Abramson, “Light-in-Flight Recording. 3: Compensation for Optical Relativistic Effects,” Appl Opt. 23, 4007 (1984).
    [CrossRef] [PubMed]
  3. N. Abramson, “Light-in-Flight Recording. 4: Visualizing Optical Relativistic Phenomena,” Appl. Opt. 24, 3323 (1985).
    [CrossRef] [PubMed]
  4. A. Einstein, The Meaning of Relativity (Methuer & Co LTD, 1950), p. 36.
  5. A. Einstein et al., The Principle of Relativity (Dover, New York, 1952), p. 73.
  6. N. Abramson, “Light-in-Flight Recording: High-Speed Holographic Motion Picture of Ultrafast Phenomena,” Appl. Opt. 22, 215 (1983).
    [CrossRef] [PubMed]
  7. R. Bhandari, “Visual Appearance of a Moving Vertical Line,” Am. J. Phys. 38, 1200 (1970).
    [CrossRef]
  8. J. Terell, “Invisibility of the Lorentz Contraction,” Phys. Rev. 116, 1041 (1959).
    [CrossRef]
  9. P. M. Mathews, M. Lakshmanan, “On the Apparent Visual Forms of Relativistically Moving Objects,” I1 Nuovo Cimento 12, 168 (1972).
  10. G. D. Scott, H. J. van Driel, “Geometrical Appearances at Relativistic Speeds,” Am. J. Phys. 38, 971 (1970).
    [CrossRef]

1985 (1)

1984 (1)

N. Abramson, “Light-in-Flight Recording. 3: Compensation for Optical Relativistic Effects,” Appl Opt. 23, 4007 (1984).
[CrossRef] [PubMed]

1983 (1)

1972 (1)

P. M. Mathews, M. Lakshmanan, “On the Apparent Visual Forms of Relativistically Moving Objects,” I1 Nuovo Cimento 12, 168 (1972).

1970 (2)

G. D. Scott, H. J. van Driel, “Geometrical Appearances at Relativistic Speeds,” Am. J. Phys. 38, 971 (1970).
[CrossRef]

R. Bhandari, “Visual Appearance of a Moving Vertical Line,” Am. J. Phys. 38, 1200 (1970).
[CrossRef]

1959 (1)

J. Terell, “Invisibility of the Lorentz Contraction,” Phys. Rev. 116, 1041 (1959).
[CrossRef]

Abramson, N.

N. Abramson, “Light-in-Flight Recording. 4: Visualizing Optical Relativistic Phenomena,” Appl. Opt. 24, 3323 (1985).
[CrossRef] [PubMed]

N. Abramson, “Light-in-Flight Recording. 3: Compensation for Optical Relativistic Effects,” Appl Opt. 23, 4007 (1984).
[CrossRef] [PubMed]

N. Abramson, “Light-in-Flight Recording: High-Speed Holographic Motion Picture of Ultrafast Phenomena,” Appl. Opt. 22, 215 (1983).
[CrossRef] [PubMed]

N. Abramson, The Making and Evaluation of Holograms (Academic, London, 1981), pp. 28, 110, 144, and 197.

Bhandari, R.

R. Bhandari, “Visual Appearance of a Moving Vertical Line,” Am. J. Phys. 38, 1200 (1970).
[CrossRef]

Einstein, A.

A. Einstein, The Meaning of Relativity (Methuer & Co LTD, 1950), p. 36.

A. Einstein et al., The Principle of Relativity (Dover, New York, 1952), p. 73.

Lakshmanan, M.

P. M. Mathews, M. Lakshmanan, “On the Apparent Visual Forms of Relativistically Moving Objects,” I1 Nuovo Cimento 12, 168 (1972).

Mathews, P. M.

P. M. Mathews, M. Lakshmanan, “On the Apparent Visual Forms of Relativistically Moving Objects,” I1 Nuovo Cimento 12, 168 (1972).

Scott, G. D.

G. D. Scott, H. J. van Driel, “Geometrical Appearances at Relativistic Speeds,” Am. J. Phys. 38, 971 (1970).
[CrossRef]

Terell, J.

J. Terell, “Invisibility of the Lorentz Contraction,” Phys. Rev. 116, 1041 (1959).
[CrossRef]

van Driel, H. J.

G. D. Scott, H. J. van Driel, “Geometrical Appearances at Relativistic Speeds,” Am. J. Phys. 38, 971 (1970).
[CrossRef]

Am. J. Phys. (2)

R. Bhandari, “Visual Appearance of a Moving Vertical Line,” Am. J. Phys. 38, 1200 (1970).
[CrossRef]

G. D. Scott, H. J. van Driel, “Geometrical Appearances at Relativistic Speeds,” Am. J. Phys. 38, 971 (1970).
[CrossRef]

Appl Opt. (1)

N. Abramson, “Light-in-Flight Recording. 3: Compensation for Optical Relativistic Effects,” Appl Opt. 23, 4007 (1984).
[CrossRef] [PubMed]

Appl. Opt. (2)

I1 Nuovo Cimento (1)

P. M. Mathews, M. Lakshmanan, “On the Apparent Visual Forms of Relativistically Moving Objects,” I1 Nuovo Cimento 12, 168 (1972).

Phys. Rev. (1)

J. Terell, “Invisibility of the Lorentz Contraction,” Phys. Rev. 116, 1041 (1959).
[CrossRef]

Other (3)

A. Einstein, The Meaning of Relativity (Methuer & Co LTD, 1950), p. 36.

A. Einstein et al., The Principle of Relativity (Dover, New York, 1952), p. 73.

N. Abramson, The Making and Evaluation of Holograms (Academic, London, 1981), pp. 28, 110, 144, and 197.

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Figures (12)

Fig. 1
Fig. 1

Minkowski diagram. The x and y axes represent two dimensions of our ordinary world, while the vertical z axis represents time multiplied by the speed of light. Any constant velocity is represented by an inclined line inside the cone representing the speed of light. Thus, the expansion of a spherical wavefront is represented by a cone with its apex at the light source A. A converging spherical wavefront is represented by another, inverted cone with its apex at the point of observation B′, separated from A by x1. The two cones are out of contact because x1 is larger than ct1. This situation corresponds to a picosecond pulse of light emitted by A and a picosecond observation is, after a time delay of t1, made at B′. No light is seen because the observation was made before the light arrived.

Fig. 2
Fig. 2

Illumination and observation cones are separated in time but not in space. Two picosecond pulses are emitted at times t0 and t1, whereafter finally an observation is made at t2. Light can be transmitted from illumination to observation cones only by scattering objects situated at the circular intersections of the cones. These circles have half of the radius of the cones at the time of observation (t2). In our ordinary world this indicates that scattering or reflecting objects are seen only at a spherical surface that has half of the radius of the true spherical wavefront. Examples of this situation are radar and conventional interferometry where one fringe represents half of the wavelength.

Fig. 3
Fig. 3

Illumination and observation cones are separated both in time and space. One picosecond pulse is emitted at (O,O,O), and one picosecond observation is made at (x1,O,t1). At the intersection ellipse light is scattered by objects into the observation cone. When projected to the x-y plane, this ellipse produces one of the ellipses of the ordinary static holodiagram. If, however, the separation of emittance and observation is not static (x1) but dynamic (υt1), caused by a traveling observer, the projected intersection ellipse represents a relativistic effect. The desensitizing factor, halfway between A and B, is the same in the two cases and produces identical results, but in holography it is referred to as a trigonometric factor, while in relativity it is referred to as a transverse Doppler shift.

Fig. 4
Fig. 4

Differences between the static and the dynamic holodiagram are revealed when more than one picosecond pulse is studied. In this figure the separation of A and B is static and therefore the intersection ellipses, when projected down to the x-y plane, represent the total ordinary static holodiagram. They all have the two common focal points A (illumination) and B (observation). The smallest ellipses have the largest eccentricity and their corresponding intersection ellipses have the largest tilt.

Fig. 5
Fig. 5

Static holodiagram as derived from Fig. 4. When used in holographic interferometry, A is the point from which the divergent laser beam originates, while B is the point of observation behind the hologram plate. Light from A scattered to B by the object at C will not change its path length if C is displaced along an ellipse, while the difference in path lengths to adjacent ellipses is a constant number of wavelengths. The displacement perpendicular to the ellipses needed to cause one fringe is k 0.5λ, where k is constant along arcs of circles, each representing a different spacing of the ellipses.

Fig. 6
Fig. 6

To visualize the ellipses of the static holodiagram we have drawn a number of closely spaced ellipses with the common focal points A and B. Every second elliptic area between adjacent ellipses was painted black, so that the thickness of black-and-white areas represents the k value and thus the interferometric sensitivity to displacement. If two transparencies are made of this figure and one is displaced in relation to the other, moire fringes are formed that correspond to the interference fringes caused by that displacement.

Fig. 7
Fig. 7

Dynamic holodiagram. In contrast to that of Fig. 5 the distances along the x axis are solely caused by a constant velocity in the x direction by the person who makes the observation. He emits the first picosecond pulse at (O,O,O), the second at (υt1,O,t1) and finally he makes the single picosecond observation at (υt2,O,t2). When the intersection ellipses are projected down to the x-y plane, it is found that they all have one common focal point at B, but the other focal points (A1, A2, etc.) are different for each ellipse. In contrast to the static holodiagram of Figs. 57 it is also found that all the ellipses have the same eccentricity and that all the intersection ellipses have the same inclination. The k lines of the static holodiagram are in this dynamic holodiagram substituted by q lines that are constant, no along circles through B, but along straight lines through B.

Fig. 8
Fig. 8

Traveling observer moves at constant speed (0.6c) to the right and emits picosecond pulses at A1, A2, etc. separated by υdt. His lines of sight are aberrated from angle γ to the angles drawn in this dynamic holodiagram (the q lines). Along each q line the separation of the intersections by the ellipsoids of observation have a constant value, the q value. Doppler shift, apparent speed of time, and apparent longitudinal magnification are all functions of q, while transversal Doppler shift, time dilation, and Lorentz contraction depend only on the q line representing γ = 90°.

Fig. 9
Fig. 9

To visualize the ellipses of the dynamic holodiagram we have drawn a number of closely spaced ellipses all derived from the projected intersection ellipses as described in Fig. 7. Just as in the static diagram of Fig. 6 every second elliptic area between adjacent ellipses was painted black. The separation of these ellipsoids of observation divided by the separation of the ordinary spheres of observation corresponding to zero velocity represents the q value.

Fig. 10
Fig. 10

To the traveling observer an arbitrary point G of the stationary world appears to exist at K which is found by drawing a line of constant Y value from G to the sphere. Light rays emitted by the traveler at angle γ are aberrated by his velocity to angle α, while his lines of sight are aberrated from angle γ to angle β. This is the method used to produce both Figs. 8 and 11.

Fig. 11
Fig. 11

Flat surface (ss) perpendicular to the direction of travel appears distorted to the traveling observer (B) into the curved surface s′–s′, which is constructed in the following way: Draw lines parallel to the x axis from the points where ss intersects the ellipsoids of observation until they reach corresponding points on the spheres of observation. The same can be done going from the intersections with the aberrated lines of sight (q lines) to the original lines of sight (at angle γ). Connecting these points results in the traveler’s impression s′–s′ of the stationary surface ss. (a) Ellipsoids of observation and aberrated lines of sight; (b) spheres of observation and original lines of sight.

Fig. 12
Fig. 12

Orthogonal coordinate system of the stationary world (a) appears to the traveling observer transformed into that of (b). The traveler exists at the small circle (i,O) and is moving to the right at a speed of 0.6c. This situation is identical to that when the observer is stationary while the coordinate system is moving to the left. From the diagram we find that flat surfaces are transformed into hyperboloids. The plane (ii) through the observer is transformed into a cone. The back side can be seen on all objects that have passed this cone. The separation of advancing hyperboloids is increased, while that of those moving away is decreased.

Equations (12)

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k = 1 cos α = g h = 0 . 5 D x 0 . 5 D y = e .
q = 1 cos α = g h = 0 . 5 D x 0 . 5 D y = e = g g 2 f 2 = 1 1 ( υ c ) 2 .
k = 1 cos α ,
q = the distance B G divided by the distance B K ,
q = 1 υ c cos γ 1 ( υ c ) 2 .
s = k 0 . 5 c t .
t t r u e = L a p p / k 0 . 5 c .
υ a p p = k 0 . 5 c .
L t r u e = L a p p 0 . 5 k .
L a p p = 1 q L t r u e .
λ a p p = q λ t r u e .
L a p p = 1 q L t r u e .

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