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Achromatized holographic phase shifter and modulator
Volume 67, number 3
OPTICS COMMUNICATIONS
1 July 1988
A C H R O M A T I Z E D H O L O G R A P H I C P H A S E S H I F T E R AND M O D U L A T O R ~ Nicholas G E O R G E and Thomas STONE The Institute of Optics, University of Rochester, Rochester, NY 14627, USA Received 11 January 1988; revised manuscript received 25 April 1988
Novel achromatic optical systems forming phase shifters and phase modulators are presented and shown to produce phaseangle shifts in incoming waves which are independent of wavelength over broad spectral ranges. Three achromatic phase shifters are described including the triangular, close cascade, and imaged grating configurations. Applications are discussed and experimental results are given in support of the theory.
1. Introduction
Gratings have been used as beamsplitters in interferometers [1,2 ] and as precision phase shifters in optical information processing [ 3 ]. In these applications limited to monochromatic illumination, their usefulness as phase shifters has been pointed out. Recent other descriptions of the use of gratings in white light interferometers as beamsplitters and color encoders are found in the literature [4,5]. However no achromatic phase shifter applications to broadband illumination have been found in the literature. Recently we have reported that simple hologram configurations exist which will provide controllable phase shifts even where a very broad spectrum of colors is used [ 6 ]. And most interestingly, one can obtain a phase shift that is independent of wavelength, approximating a holographic analog to a Fresnel rhomb. In the following sections of this paper, we describe the theory of operation and present a few basic configurations, and we conclude with experimental results that are in excellent agreement with the theory.
2. Theory for the holographic phase shifter Consider the triangular configuration of the holographic phase shifter shown in fig. 1 consisting of two gratings (I, III) of spatial period A spaced by 2H along the z-axis, as well as a vertex grating (II) of period A v = A / 2 with its centerpoint C2 offset by a distance X2o from the optical axis. A monochromatic beam normally incident on grating (I) is diffracted by an angle 0(2) so that it impinges on the vertex grating. Thereafter, due to the choice for the spatial period of Av, the beam exiting (II) is diffracted at an angle - 0 ( 2 ) with respect to z. Finally, this beam is diffracted by (III) so that the output beam is directed along the z-axis. One should note that we are considering only one diffracted beam from each t X2
1 A _xol~ -Av= -2
/
A_
....... Research supported by the U.S. Army Night Vision and Electro-Optics Laboratory, the U.S. Research Office, and the corporate sponsors under the New York State Center for Advanced Optical Technology.
P-~" I
0(x)
i
,,
\
-oT . . . . . . . .
'-I"
' %'-7
H
~p III
Fig. l. The triangular configuration of the achromatic phase shifter.
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
185
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grating such as is readily attainable by volume holographic gratings or that the direct beam along the z-axis has been blocked. The basic principle of this phase shifter depends upon the fact that the phase shift of the collimated beam exiting (III) depends directly on X2o, the offset of the centerpoint of grating (II), and not upon the wavelength 2. This is clearly seen from the theory presented below. In order to find the amplitude transmission function, t~3, for the three gratings we write the separate idealized transmission functions tl, t2, t3 for each of the gratings as
1 July 1988
(second member with H ) that is unimportant in many applications. Moreover, the imaged grating configuration described later does not contain such a term. For the sake of brevity, we have analyzed the triangular configuration assuming t~ = t3 and a vertex period Av=A/2. Formulas are readily obtained for more general choices using the same procedure. Herein, we would like to include as well the formulas for the case in which grating offset parameters X~o, X2o and X3o are considered. The resulting transmission function t'13 is given by t ' l a ( X l , Yl; X3, Y3)
tl (x~, Yl ) = e x p ( - i 2 ~ z x l / A ) ,
(1)
tz(x2, Y2) =exp{ +i2rc[2 (x2 -X2o) ]/A},
(2)
t3 (x3, Y3) = exp ( - i2zrx3/A) .
(3)
Note that we are using an exp (iogt) notation and that the vertex grating (II) has a period A/2, a rotation of the diffracted beam that is clockwise (hence the positive exponential), and a variable offset parameter x2o. Gratings (I) and (III) are identical imparting a counter clockwise rotation to an input plane wave. The diffraction angle 0(2) is of course given by solving exp{ - i 2 n [ x sin0(2) ] / 2 = e x p ( - i 2 ~ z x / A ) ,
(4)
i.e., by sin0(2) = 2 / A .
(5)
The overall transmission function t~3 is found from the equation
=exp{ + i(2rc/A) (X~o -2X2o +x30) - i(4~rH/;t) [ 1 - (2/A) 2 ] i/z}.
(8)
From eq. (8), we see that phase shift devices can be built in which any of the three gratings are translated relatively. Consider the use of gratings that have multiple orders. If the grating of period A is used in the order q, then the equation appropriately replacing eq. (8) is found by transliteration of A to A/q. Hence the phase shift Aq~ ~ch becomes A I~ _ qach = - (2zrq/A) (X~o-2X2o +X3o) •
(9)
Accordingly, light propagating in higher diffracted orders receives integer multiples of the achromatic phase shifts introduced into the first order, and the zero order light is not shifted in phase at all.
2.1. Theory for the holographic phase modulator t13(X3, Y3; Xt, Yl)
=tl t2t3 exp[ - i ( 2 n / 2 ) 2 H cos0(2) ] ,
(6)
in which total phase retardation terms for the continuous-wave case are included. Substitution of eqs. (1), (2), (3), (5) into eq. (6) gives tt 3 (x3,Y3 ; Xl, Yl ) = exp{ -
i4nx2o/A
- i(4~zH/,~) [ 1 - (2/d)2] 1/2}.
(7)
This interesting result shows that one can obtain a variation in phase shift of Aqb a¢h= 41tXEo/A rad that is strictly independent of wavelength 2, simply by varying the offset distance X2o. In eq. (7) the total phase shift does contain a wavelength dependent term 186
The triangular configuration just described also forms an achromatic phase modulator, in which phase modulation is achieved by varying the position of the vertex grating with time. This time-varying displacement function may be realized by coupling the vertex grating to a piezo-electric drive or a translation stage. One can also use the traveling waves in an acousto-optic modulator in order to form the moving vertex grating. Thus in the above analysis the grating shift x20 is replaced with the function of time x20(t). Accordingly the output waves are modulated with an achromatic temporal phase function described by
Volume 67, number 3 Aci~ach ( t )
OPTICS COMMUNICATIONS
=2ztX2o(t)/Av,
(10)
where Av=A/2 as defined earlier. Hence in eq. (7) we substitute
X2o(t)=L +vm(t) ,
(11)
in which the translation velocity of the modulation element is Vm. For unit amplitude input, the exiting scalar field component, v3=h3, is given by
v3(x3,Y3)=exp[-i2n(vm/Av)t+i0)t] ,
(12)
where the vertex spacing Av=A/2, 0)=2nc/2, and only time-varying phases are retained. Thus, the output signal has a final frequency 0)3 given by
0)3=0)-27~vm/Av.
(13)
This frequency shift of A0)ach=-2ZWrJAv is independent of wavelength, and it is just the Doppler shift for an electromagnetic wave incident and reflected at an angle of incidence of n / 2 - 0 ( 2 ) by a moving platform traveling at a velocity Vm. The Bragg planes of the volume vertex grating-hologram form the moving platforms. Achromatic frequency modulation can be obtained by sinusoidally varying Xo, i~e., by letting
Xo(t) = L + x m COS(0)mt) ,
1 July 1988
planes of the vertex element would also give achromatic phase and frequency shifts. The reason for this is that the achromatization in the overall system results from the wavelength variation in 0 caused by tt and t 3.
2.2. Close cascade configuration The close cascade configuration is illustrated in fig. 2 and consists in this case of two cascaded gratings Gt and G2 of equal period. Broadband or white light incident on the first grating is angularly dispersed and then incident on the second grating which restores the waves to their original propagating direction. Undiffracted or spurious-order light may be blocked by the introduction of a venetian-blind type material F, as illustrated. Achromatic phase and frequency shifts are introduced in the output waves by movement of one or both of the component gratings as described for the base gratings of the triangular configuration. The gratings are fabricated in a volume phase recording material with parameters such as refractive index modulation, element thickness, spatial period, etc. chosen to maximize diffraction efficiency over broad spectral and angular bandwidths [7 ].
(14)
in which Xm and com are the amplitude and temporal frequency, respectively, of the motion of the grating. Again, substitution of eq. (14) into eq. (7) gives us an output v3 as follows,
2. 3. Imaged-grating phase modulator The third configuration which is useful as a broadband achromatic phase/frequency shifter is illus-
V3(X3, Y3)
=exp[-i2~(xm/Av) COS(0)mt)+io)t]
~
N,.- pz T
.
(15)
In eq. (15) we recognize the standard frequencymodulation formula (non time-varying phases have been dropped) and notice that the phase modulation term is independent of wavelength. As a contrast to eq. (14) we note that simple modulation of an interferometer mirror would result in a phase term given by exp[--i4~(Xm/2) COS(0)mt)+io)t]. Either a constant frequency offset, eq. (13), or an FM-spectra, eq. ( 15 ), can be obtained by appropriate spatial modulation of the vertex grating. Additionally from the discussion above, it is clear that the same motion by a single mirror placed in the triangular configuration and aligned with the Bragg
F
It
OUTPUT INPUT
G1
G2
Fig. 2. Close cascade phase modulator with holographic gratings (G~) and (G2) separated by light control film (F). The translator (PZT) provides motion of (G2) for phase shifting or modulation. 187
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OPTICS COMMUNICATIONS
PZT
°'
PZT
LI.,A-_.~ RED~
L
V
V
Fig. 3. Imaged-gratingachromatic phase modulator.
1 July 1988
configuration devices. Further if input light is restricted to collimated sources lying in a plane containing the optical axis or the input light is axially collimated ( a common case), then only cylindrical lenses are required in the imaging configuration. In the latter case, only partial apertures are required for these cylindrical lenses.
3. P h a s e s h i f t e r for w h i t e l i g h t i n t e r f e r o m e t r y
3.1. Mach-Zehnder configuration trated in fig. 3, and consists of a pair of gratings which are imaged onto each other by a two lens unity magnification optical processor. This processor successively Fourier transforms the field exiting the first grating twice. In this system the lenses L1 and L2 perform a similar function to the vertex grating of fig. 1 in returning the varied spectral components diffracted by the first grating symmetrically to the second grating. This symmetry results in the spatial recombination by the second grating of the waves dispersed by the first grating as in the triangular cascade. The phase and frequency shifts are obtained in this configuration by the movement of one or both of the gratings in similar fashion to that described for the base gratings of the triangular configuration or in the close-cascade shifter. Other techniques may be used to image the two gratings onto each other including lens and mirror systems. However the two lens Fourier process imaging system shown may be superior in some applications due to its accurate phase imaging without the quadratic phase factors across the output which are present in other imaging systems. The imaged grating phase/frequency shifter/modulator configuration has many desirable properties. For example, the imaging relationship between the two gratings results in a very useful property when the device is used with light input throughout a broad field angle (e.g., with light of limited spatial coherence). For this case it is noted that broad spectrum light incident on an input-grating point from all field angles exits the system at the corresponding imaged point on the exit grating. In some applications this is much more convenient than the field-angle dependent spatial separation of each spectral component which exits in the output of the triangular 188
Modified Mach-Zehnder interferometers consisting of gratings as beamsplitters instead of mirrors are described in the literature together with recent applications to white-light interferometry [4,8,9]. In prior interferometric work for tiny phase shifts, it is common to rotate a single glass plate by a small angle. An improvement to this was first described by Raleigh who used two flats of slightly differing thicknesses mounted in a fixed symmetric tent on the same turntable. However we feel that with the high quality available in modern holographic gratings that achromatic phase shifters provide important new capabilities both in monochromatic and broadband applications. Consider the holographic Mach-Zehnder interferometer in fig. 4, with equal periods'A for all gratings G1-G4. This is a particularly convenient configuration for broadband illumination when samples and reference objects (S and R as shown) are to be inserted since the illumination in two arms travels parX2
f
R
t
/I r., /I
/,'1 It
,NPOTI/ I L V
G1
I
S
i i
~'
! r
.t
i i
/
l
-
I
~
J' OUTPUT D
>'1
11/ V
G3
Fig. 4. The Mach-Zehnder interferometer with holographic gratings (Gt-G4). Achromatic differential phase shifts between paths
1 and 2 can be obtained by motion of any of the gratings along x.
Volume 67, number 3
OPTICS COMMUNICATIONS
allel to a fixed optical axis even as the wavelength is varied. The achromatic phase shift is accomplished simply by providing translation of grating G2 along the direction x2. Additionally it is pointed out that any of the gratings can be moved to provide the phase shift since, following eq. (9) the direct or 0-order beams do not experience a phase shift.
3.2. Symmetrical modulator A broad variety of phase shifting white light or multi-wavelength interferometers may be constructed with the basic achromatic phase shifters described in sec. 2. Consider the symmetrical grating interferometer shown in fig. 5 consisting of an input splitter G~, a single vertex grating G2, and a combiner G3. The first base grating diffracts the incident white light beam into symmetric+ 1 and - 1 orders, which then illuminate the vertex grating. Both diffracted orders symmetrically exit the vertex grating and are recombined by the final grating. The optical path length is identical for a given wavelength traversing either the + 1 or - 1 diffracted order path. The analysis for this configuration follows directly from that of the triangular configuration. Hence the final output of grating G3, denoted by v3, is given by
V3(X3,Y3) = e x p ( --iAtib~ch) +exp(--iAti~ch), (16) in which A@~ch and A ~ ~h are the achromatic phase shifts in paths 1 and 2, respectively. Explicit values for these phases shifts are obtained from eq. (9) from the triangular configuration. It is also obvious that
A ~ ~ch = - A ~
1 July 1988 ch .
(17)
This device can be readily applied as a phase shifter in the spatial cosinusoidal transform system described in the literature [ 10]. The details of these configurations are omitted in the interest of brevity; however, the essence of these arrangements is that the output beam is separated into two beams using a pair of mirrors but with the same grating G3. This gives two separate outputs which have differential achromatic phases. Interferometers of this type could also play a role in optical experiments to demonstrate the validity of redshifts and blueshifts of spectral lines caused by source correlations [ 11 ].
4. Achromatic phase conjugation Phase-shifting white light interferometers, as discussed in sec. 3, may be readily applied to achromatize the real time phase conjugation process (real time holography and some aspects of four wave mixing), forming white light phase conjugators. The key point to the achromatization of phase conjugators is to form stationary achromatic (white light) fringe patterns in the nonlinear medium. Since from the description of the holographic Mach-Zehnder interferometer in sec. 3 it is clear that collimated input for all wavelengths forms fringes with period A at G4, one merely needs to replace G4 with a non-linear crystal. Fig. 6 shows a system with a "picture" wave S that passes through a mild diffusing medium (M). R" -G5
RED" 1 \ '
/.aJ A
PAT.
//
| ~ ~
G1 / /
'~xxx ~ G3
JE
Xl
)
OUTP;r
INPUT ~
G1 I
R X3 [A]
A
G3
0(~)
"--, -.
x\ PATH2x,
xx
RED"
/111 / / 1/11 / / /// /
G2
Fig. 5. Symmetricaltwin-triangular interferometer.
A
A
Fig. 6. White light phase conjugationusing the holographicMachZehnder intefferometer configuration. A nonlinear medium (labeled X3) is substituted for grating G4 (see fig. 4), in the region of white light fringes. 189
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OPTICS COMMUNICATIONS
The reference wave R is derived from the other arm of the Mach-Zehnder interferometer. The spectral components of this reference beam propagate at angles O(2)=-sin-l(2/A) with respect to the optical axis. As illustrated in fig. 6, "playback" is accomplished using the conjugate to the reference wave, R, which is generated by passing a collimated white light beam through a holographic grating (G5) that is identical to the others used in the configuration. The spectral components of this reconstructing wave travel at angles - 0 ( 2 ) . The returning conjugated wavefront then propagates through the diffuser returning the original wavefront S. Similarly any of the achromatic interferometer configurations may be used to form white light phase conjugators, by locating the nonlinear medium in the region of whitelight fringes. -
5. Experiments Experimental verification of the theory presented above was obtained using the grating interferometer illustrated in fig. 7. The base gratings G~, G3, and the vertex grating G2 had spatial periods along x of 1.26 and 0.633 ~tm, respectively. Identical grating sets were fabricated both using dichromated gelatin [ 12 ] and Polaroid DMP-128 recording materials. The Polaroid DMP-128 material is particulary useful due to its advantages of high diffraction efficiency, ex ' lent optical quality, low loss, and extreme east processing [ 13,14 ]. The peak diffraction efficie of all gratings was > 98%, although in the exp ment the base gratings were angularly detuned provide the 0-order reference path in the interfi meter as illustrated in fig. 7. Helium-neon and argon-ion lasers were used
PZT I
Ol \ 514.5
-I
B
..1"... I....~.~,
G1
R
_L--"~ CRO
"
D ~u t/
G3
CRO U_ID
nm
Fig. 7. Experiments with the triangular configuration.
190
provide narrowband beams of wavelength 0.6328 and 0.5145 pm, respectively, thus spanning approximately 1200 ~,. These beams were combined using a beamsplitter and were collinearly incident on the first base grating Gl. The reference path (R) was formed by the undiffracted light from Gl and G3 as discussed above, and exited the grating interferometer from G3 as illustrated. The modulated path was swept out by the + 1 orders of the base gratings and the - 1 order of the vertex grating as explained in sec. 2. The red light was diffracted through the angle 0(0.6328) = 30 ° and the green light by the smaller angle 0(0.5145) =24 °, the trajectories of which are represented by the dashed and solid lines in the modulated path of fig. 7. The green and red interferometer signals were then spatially separated using a dichroic beamsplitter (B), and were subsequently incident on separate PIN photodiodes (D) as shown in fig. 7. The outputs of the photodiodes were amplified and input in separate vertical channels of a dual-trace oscilloscope. Motion of the vertex grating could be accomplished using either Burleigh piezo-electric-or Newport Research d.c. motor-translators. Fig. 8 is a photograph of the oscilloscope output (red signal on top) as the vertex grating was translated with a uniform velocity of 50 Ixm/s. It is seen that the two widely separated wavelengths experience equal phase shifts, as each oscillation represents a 2re phase dffference between the reference and modulated paths. Phase shifts are commonly introduced in interferometers by moving mirrors, and it is useful to compare this technique to the utilization of the above achromatic device. Consider the case of a Michelson
G2
|
632.8 nm
1 July 1988
0
1
TIME (mi||iseco~s) Fig. 8. Cathode-ray oscilloscope display of achromatized fringe count: upper trace 632.8 nm and lower 514.5 nm.
Volume 67, number 3
OPTICS COMMUNICATIONS
interferometer. As one o f the m i r r o r s is m o v e d a distance p r o v i d i n g a count o f 200 fringes in the red, the shorter green wavelength experiences 246 osciUations. This was experimentally c o m p a r e d to the above a c h r o m a t i c phase shifter where for a fringe count o f 200 in the red signal, 200_+ 0.5 oscillations were observed in the blue signal. Hence using the achromatic phase shifter no detectable difference in the fringe count was observed.
Acknowledgements The authors acknowledge discussion o f phase conj u g a t i o n with R.W. Boyd a n d K. M a c D o n a l d . D. Schertler assisted in the e x p e r i m e n t s on the achromatic phase shifter. This research was s u p p o r t e d in part b y the U.S. A r m y Night Vision a n d Electro-Optics Laboratory, the U.S. A r m y Research Office, a n d the corporate sponsors u n d e r the N e w Y o r k State Center for A d v a n c e d Optical Technology.
1 July 1988
References [I ] K. Iizuka, Engineering optics (Springer-Verlag, Berlin, 1987). [2] P. Hariharan, Optical interferometry (Academic Press, Sydney, 1985). [ 3 ] G.L. Rogers, Noncoherent optical processing (John Wiley, New York, 1977). [4] E.N. Leith and G.J. Swanson, Appl. Optics 19 (1980) 638. [5] F.T.S. Yu, Appl. Optics (1980) 2457. [6] N. George and T. Stone, J. Opt. Soc. Am. A 4 (1987) 79. [ 7 ] T. Stone and N. George, Appl. Optics 24 ( 1985 ) 3797. [8] G.J. Swanson, J. Opt. Soc. Am. A 1 (1984) 1147. [9] F.J. Weinberg and N.B. Wood, J. Sci. Instr. 36 (1959) 227. [ 10 ] N. George and S. Wang, Appl. Optics 23 (1984) 787. [ l 1] E. Wolf, Optics Comm. 62 (1987) 12. [ 12] C.D. Leonard and B.D. Guenther, Technical Report T-7917 (U.S. Army Missile Research and Development Command, Redstone Arsenal, 1979). [ 13 ] R.T. Ingwall and H,L. Fielding, Opt. Eng. 24 ( 1985 ) 808. [ 14 ] R.T. Ingwall,A. Stuck and W.T. Vetterling, Proc. Soc. PhotoOpt. Instr. Eng. 615 (1986) 81.
191
OPTICS COMMUNICATIONS
1 July 1988
A C H R O M A T I Z E D H O L O G R A P H I C P H A S E S H I F T E R AND M O D U L A T O R ~ Nicholas G E O R G E and Thomas STONE The Institute of Optics, University of Rochester, Rochester, NY 14627, USA Received 11 January 1988; revised manuscript received 25 April 1988
Novel achromatic optical systems forming phase shifters and phase modulators are presented and shown to produce phaseangle shifts in incoming waves which are independent of wavelength over broad spectral ranges. Three achromatic phase shifters are described including the triangular, close cascade, and imaged grating configurations. Applications are discussed and experimental results are given in support of the theory.
1. Introduction
Gratings have been used as beamsplitters in interferometers [1,2 ] and as precision phase shifters in optical information processing [ 3 ]. In these applications limited to monochromatic illumination, their usefulness as phase shifters has been pointed out. Recent other descriptions of the use of gratings in white light interferometers as beamsplitters and color encoders are found in the literature [4,5]. However no achromatic phase shifter applications to broadband illumination have been found in the literature. Recently we have reported that simple hologram configurations exist which will provide controllable phase shifts even where a very broad spectrum of colors is used [ 6 ]. And most interestingly, one can obtain a phase shift that is independent of wavelength, approximating a holographic analog to a Fresnel rhomb. In the following sections of this paper, we describe the theory of operation and present a few basic configurations, and we conclude with experimental results that are in excellent agreement with the theory.
2. Theory for the holographic phase shifter Consider the triangular configuration of the holographic phase shifter shown in fig. 1 consisting of two gratings (I, III) of spatial period A spaced by 2H along the z-axis, as well as a vertex grating (II) of period A v = A / 2 with its centerpoint C2 offset by a distance X2o from the optical axis. A monochromatic beam normally incident on grating (I) is diffracted by an angle 0(2) so that it impinges on the vertex grating. Thereafter, due to the choice for the spatial period of Av, the beam exiting (II) is diffracted at an angle - 0 ( 2 ) with respect to z. Finally, this beam is diffracted by (III) so that the output beam is directed along the z-axis. One should note that we are considering only one diffracted beam from each t X2
1 A _xol~ -Av= -2
/
A_
....... Research supported by the U.S. Army Night Vision and Electro-Optics Laboratory, the U.S. Research Office, and the corporate sponsors under the New York State Center for Advanced Optical Technology.
P-~" I
0(x)
i
,,
\
-oT . . . . . . . .
'-I"
' %'-7
H
~p III
Fig. l. The triangular configuration of the achromatic phase shifter.
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
185
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OPTICS COMMUNICATIONS
grating such as is readily attainable by volume holographic gratings or that the direct beam along the z-axis has been blocked. The basic principle of this phase shifter depends upon the fact that the phase shift of the collimated beam exiting (III) depends directly on X2o, the offset of the centerpoint of grating (II), and not upon the wavelength 2. This is clearly seen from the theory presented below. In order to find the amplitude transmission function, t~3, for the three gratings we write the separate idealized transmission functions tl, t2, t3 for each of the gratings as
1 July 1988
(second member with H ) that is unimportant in many applications. Moreover, the imaged grating configuration described later does not contain such a term. For the sake of brevity, we have analyzed the triangular configuration assuming t~ = t3 and a vertex period Av=A/2. Formulas are readily obtained for more general choices using the same procedure. Herein, we would like to include as well the formulas for the case in which grating offset parameters X~o, X2o and X3o are considered. The resulting transmission function t'13 is given by t ' l a ( X l , Yl; X3, Y3)
tl (x~, Yl ) = e x p ( - i 2 ~ z x l / A ) ,
(1)
tz(x2, Y2) =exp{ +i2rc[2 (x2 -X2o) ]/A},
(2)
t3 (x3, Y3) = exp ( - i2zrx3/A) .
(3)
Note that we are using an exp (iogt) notation and that the vertex grating (II) has a period A/2, a rotation of the diffracted beam that is clockwise (hence the positive exponential), and a variable offset parameter x2o. Gratings (I) and (III) are identical imparting a counter clockwise rotation to an input plane wave. The diffraction angle 0(2) is of course given by solving exp{ - i 2 n [ x sin0(2) ] / 2 = e x p ( - i 2 ~ z x / A ) ,
(4)
i.e., by sin0(2) = 2 / A .
(5)
The overall transmission function t~3 is found from the equation
=exp{ + i(2rc/A) (X~o -2X2o +x30) - i(4~rH/;t) [ 1 - (2/A) 2 ] i/z}.
(8)
From eq. (8), we see that phase shift devices can be built in which any of the three gratings are translated relatively. Consider the use of gratings that have multiple orders. If the grating of period A is used in the order q, then the equation appropriately replacing eq. (8) is found by transliteration of A to A/q. Hence the phase shift Aq~ ~ch becomes A I~ _ qach = - (2zrq/A) (X~o-2X2o +X3o) •
(9)
Accordingly, light propagating in higher diffracted orders receives integer multiples of the achromatic phase shifts introduced into the first order, and the zero order light is not shifted in phase at all.
2.1. Theory for the holographic phase modulator t13(X3, Y3; Xt, Yl)
=tl t2t3 exp[ - i ( 2 n / 2 ) 2 H cos0(2) ] ,
(6)
in which total phase retardation terms for the continuous-wave case are included. Substitution of eqs. (1), (2), (3), (5) into eq. (6) gives tt 3 (x3,Y3 ; Xl, Yl ) = exp{ -
i4nx2o/A
- i(4~zH/,~) [ 1 - (2/d)2] 1/2}.
(7)
This interesting result shows that one can obtain a variation in phase shift of Aqb a¢h= 41tXEo/A rad that is strictly independent of wavelength 2, simply by varying the offset distance X2o. In eq. (7) the total phase shift does contain a wavelength dependent term 186
The triangular configuration just described also forms an achromatic phase modulator, in which phase modulation is achieved by varying the position of the vertex grating with time. This time-varying displacement function may be realized by coupling the vertex grating to a piezo-electric drive or a translation stage. One can also use the traveling waves in an acousto-optic modulator in order to form the moving vertex grating. Thus in the above analysis the grating shift x20 is replaced with the function of time x20(t). Accordingly the output waves are modulated with an achromatic temporal phase function described by
Volume 67, number 3 Aci~ach ( t )
OPTICS COMMUNICATIONS
=2ztX2o(t)/Av,
(10)
where Av=A/2 as defined earlier. Hence in eq. (7) we substitute
X2o(t)=L +vm(t) ,
(11)
in which the translation velocity of the modulation element is Vm. For unit amplitude input, the exiting scalar field component, v3=h3, is given by
v3(x3,Y3)=exp[-i2n(vm/Av)t+i0)t] ,
(12)
where the vertex spacing Av=A/2, 0)=2nc/2, and only time-varying phases are retained. Thus, the output signal has a final frequency 0)3 given by
0)3=0)-27~vm/Av.
(13)
This frequency shift of A0)ach=-2ZWrJAv is independent of wavelength, and it is just the Doppler shift for an electromagnetic wave incident and reflected at an angle of incidence of n / 2 - 0 ( 2 ) by a moving platform traveling at a velocity Vm. The Bragg planes of the volume vertex grating-hologram form the moving platforms. Achromatic frequency modulation can be obtained by sinusoidally varying Xo, i~e., by letting
Xo(t) = L + x m COS(0)mt) ,
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planes of the vertex element would also give achromatic phase and frequency shifts. The reason for this is that the achromatization in the overall system results from the wavelength variation in 0 caused by tt and t 3.
2.2. Close cascade configuration The close cascade configuration is illustrated in fig. 2 and consists in this case of two cascaded gratings Gt and G2 of equal period. Broadband or white light incident on the first grating is angularly dispersed and then incident on the second grating which restores the waves to their original propagating direction. Undiffracted or spurious-order light may be blocked by the introduction of a venetian-blind type material F, as illustrated. Achromatic phase and frequency shifts are introduced in the output waves by movement of one or both of the component gratings as described for the base gratings of the triangular configuration. The gratings are fabricated in a volume phase recording material with parameters such as refractive index modulation, element thickness, spatial period, etc. chosen to maximize diffraction efficiency over broad spectral and angular bandwidths [7 ].
(14)
in which Xm and com are the amplitude and temporal frequency, respectively, of the motion of the grating. Again, substitution of eq. (14) into eq. (7) gives us an output v3 as follows,
2. 3. Imaged-grating phase modulator The third configuration which is useful as a broadband achromatic phase/frequency shifter is illus-
V3(X3, Y3)
=exp[-i2~(xm/Av) COS(0)mt)+io)t]
~
N,.- pz T
.
(15)
In eq. (15) we recognize the standard frequencymodulation formula (non time-varying phases have been dropped) and notice that the phase modulation term is independent of wavelength. As a contrast to eq. (14) we note that simple modulation of an interferometer mirror would result in a phase term given by exp[--i4~(Xm/2) COS(0)mt)+io)t]. Either a constant frequency offset, eq. (13), or an FM-spectra, eq. ( 15 ), can be obtained by appropriate spatial modulation of the vertex grating. Additionally from the discussion above, it is clear that the same motion by a single mirror placed in the triangular configuration and aligned with the Bragg
F
It
OUTPUT INPUT
G1
G2
Fig. 2. Close cascade phase modulator with holographic gratings (G~) and (G2) separated by light control film (F). The translator (PZT) provides motion of (G2) for phase shifting or modulation. 187
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PZT
°'
PZT
LI.,A-_.~ RED~
L
V
V
Fig. 3. Imaged-gratingachromatic phase modulator.
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configuration devices. Further if input light is restricted to collimated sources lying in a plane containing the optical axis or the input light is axially collimated ( a common case), then only cylindrical lenses are required in the imaging configuration. In the latter case, only partial apertures are required for these cylindrical lenses.
3. P h a s e s h i f t e r for w h i t e l i g h t i n t e r f e r o m e t r y
3.1. Mach-Zehnder configuration trated in fig. 3, and consists of a pair of gratings which are imaged onto each other by a two lens unity magnification optical processor. This processor successively Fourier transforms the field exiting the first grating twice. In this system the lenses L1 and L2 perform a similar function to the vertex grating of fig. 1 in returning the varied spectral components diffracted by the first grating symmetrically to the second grating. This symmetry results in the spatial recombination by the second grating of the waves dispersed by the first grating as in the triangular cascade. The phase and frequency shifts are obtained in this configuration by the movement of one or both of the gratings in similar fashion to that described for the base gratings of the triangular configuration or in the close-cascade shifter. Other techniques may be used to image the two gratings onto each other including lens and mirror systems. However the two lens Fourier process imaging system shown may be superior in some applications due to its accurate phase imaging without the quadratic phase factors across the output which are present in other imaging systems. The imaged grating phase/frequency shifter/modulator configuration has many desirable properties. For example, the imaging relationship between the two gratings results in a very useful property when the device is used with light input throughout a broad field angle (e.g., with light of limited spatial coherence). For this case it is noted that broad spectrum light incident on an input-grating point from all field angles exits the system at the corresponding imaged point on the exit grating. In some applications this is much more convenient than the field-angle dependent spatial separation of each spectral component which exits in the output of the triangular 188
Modified Mach-Zehnder interferometers consisting of gratings as beamsplitters instead of mirrors are described in the literature together with recent applications to white-light interferometry [4,8,9]. In prior interferometric work for tiny phase shifts, it is common to rotate a single glass plate by a small angle. An improvement to this was first described by Raleigh who used two flats of slightly differing thicknesses mounted in a fixed symmetric tent on the same turntable. However we feel that with the high quality available in modern holographic gratings that achromatic phase shifters provide important new capabilities both in monochromatic and broadband applications. Consider the holographic Mach-Zehnder interferometer in fig. 4, with equal periods'A for all gratings G1-G4. This is a particularly convenient configuration for broadband illumination when samples and reference objects (S and R as shown) are to be inserted since the illumination in two arms travels parX2
f
R
t
/I r., /I
/,'1 It
,NPOTI/ I L V
G1
I
S
i i
~'
! r
.t
i i
/
l
-
I
~
J' OUTPUT D
>'1
11/ V
G3
Fig. 4. The Mach-Zehnder interferometer with holographic gratings (Gt-G4). Achromatic differential phase shifts between paths
1 and 2 can be obtained by motion of any of the gratings along x.
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allel to a fixed optical axis even as the wavelength is varied. The achromatic phase shift is accomplished simply by providing translation of grating G2 along the direction x2. Additionally it is pointed out that any of the gratings can be moved to provide the phase shift since, following eq. (9) the direct or 0-order beams do not experience a phase shift.
3.2. Symmetrical modulator A broad variety of phase shifting white light or multi-wavelength interferometers may be constructed with the basic achromatic phase shifters described in sec. 2. Consider the symmetrical grating interferometer shown in fig. 5 consisting of an input splitter G~, a single vertex grating G2, and a combiner G3. The first base grating diffracts the incident white light beam into symmetric+ 1 and - 1 orders, which then illuminate the vertex grating. Both diffracted orders symmetrically exit the vertex grating and are recombined by the final grating. The optical path length is identical for a given wavelength traversing either the + 1 or - 1 diffracted order path. The analysis for this configuration follows directly from that of the triangular configuration. Hence the final output of grating G3, denoted by v3, is given by
V3(X3,Y3) = e x p ( --iAtib~ch) +exp(--iAti~ch), (16) in which A@~ch and A ~ ~h are the achromatic phase shifts in paths 1 and 2, respectively. Explicit values for these phases shifts are obtained from eq. (9) from the triangular configuration. It is also obvious that
A ~ ~ch = - A ~
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(17)
This device can be readily applied as a phase shifter in the spatial cosinusoidal transform system described in the literature [ 10]. The details of these configurations are omitted in the interest of brevity; however, the essence of these arrangements is that the output beam is separated into two beams using a pair of mirrors but with the same grating G3. This gives two separate outputs which have differential achromatic phases. Interferometers of this type could also play a role in optical experiments to demonstrate the validity of redshifts and blueshifts of spectral lines caused by source correlations [ 11 ].
4. Achromatic phase conjugation Phase-shifting white light interferometers, as discussed in sec. 3, may be readily applied to achromatize the real time phase conjugation process (real time holography and some aspects of four wave mixing), forming white light phase conjugators. The key point to the achromatization of phase conjugators is to form stationary achromatic (white light) fringe patterns in the nonlinear medium. Since from the description of the holographic Mach-Zehnder interferometer in sec. 3 it is clear that collimated input for all wavelengths forms fringes with period A at G4, one merely needs to replace G4 with a non-linear crystal. Fig. 6 shows a system with a "picture" wave S that passes through a mild diffusing medium (M). R" -G5
RED" 1 \ '
/.aJ A
PAT.
//
| ~ ~
G1 / /
'~xxx ~ G3
JE
Xl
)
OUTP;r
INPUT ~
G1 I
R X3 [A]
A
G3
0(~)
"--, -.
x\ PATH2x,
xx
RED"
/111 / / 1/11 / / /// /
G2
Fig. 5. Symmetricaltwin-triangular interferometer.
A
A
Fig. 6. White light phase conjugationusing the holographicMachZehnder intefferometer configuration. A nonlinear medium (labeled X3) is substituted for grating G4 (see fig. 4), in the region of white light fringes. 189
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The reference wave R is derived from the other arm of the Mach-Zehnder interferometer. The spectral components of this reference beam propagate at angles O(2)=-sin-l(2/A) with respect to the optical axis. As illustrated in fig. 6, "playback" is accomplished using the conjugate to the reference wave, R, which is generated by passing a collimated white light beam through a holographic grating (G5) that is identical to the others used in the configuration. The spectral components of this reconstructing wave travel at angles - 0 ( 2 ) . The returning conjugated wavefront then propagates through the diffuser returning the original wavefront S. Similarly any of the achromatic interferometer configurations may be used to form white light phase conjugators, by locating the nonlinear medium in the region of whitelight fringes. -
5. Experiments Experimental verification of the theory presented above was obtained using the grating interferometer illustrated in fig. 7. The base gratings G~, G3, and the vertex grating G2 had spatial periods along x of 1.26 and 0.633 ~tm, respectively. Identical grating sets were fabricated both using dichromated gelatin [ 12 ] and Polaroid DMP-128 recording materials. The Polaroid DMP-128 material is particulary useful due to its advantages of high diffraction efficiency, ex ' lent optical quality, low loss, and extreme east processing [ 13,14 ]. The peak diffraction efficie of all gratings was > 98%, although in the exp ment the base gratings were angularly detuned provide the 0-order reference path in the interfi meter as illustrated in fig. 7. Helium-neon and argon-ion lasers were used
PZT I
Ol \ 514.5
-I
B
..1"... I....~.~,
G1
R
_L--"~ CRO
"
D ~u t/
G3
CRO U_ID
nm
Fig. 7. Experiments with the triangular configuration.
190
provide narrowband beams of wavelength 0.6328 and 0.5145 pm, respectively, thus spanning approximately 1200 ~,. These beams were combined using a beamsplitter and were collinearly incident on the first base grating Gl. The reference path (R) was formed by the undiffracted light from Gl and G3 as discussed above, and exited the grating interferometer from G3 as illustrated. The modulated path was swept out by the + 1 orders of the base gratings and the - 1 order of the vertex grating as explained in sec. 2. The red light was diffracted through the angle 0(0.6328) = 30 ° and the green light by the smaller angle 0(0.5145) =24 °, the trajectories of which are represented by the dashed and solid lines in the modulated path of fig. 7. The green and red interferometer signals were then spatially separated using a dichroic beamsplitter (B), and were subsequently incident on separate PIN photodiodes (D) as shown in fig. 7. The outputs of the photodiodes were amplified and input in separate vertical channels of a dual-trace oscilloscope. Motion of the vertex grating could be accomplished using either Burleigh piezo-electric-or Newport Research d.c. motor-translators. Fig. 8 is a photograph of the oscilloscope output (red signal on top) as the vertex grating was translated with a uniform velocity of 50 Ixm/s. It is seen that the two widely separated wavelengths experience equal phase shifts, as each oscillation represents a 2re phase dffference between the reference and modulated paths. Phase shifts are commonly introduced in interferometers by moving mirrors, and it is useful to compare this technique to the utilization of the above achromatic device. Consider the case of a Michelson
G2
|
632.8 nm
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0
1
TIME (mi||iseco~s) Fig. 8. Cathode-ray oscilloscope display of achromatized fringe count: upper trace 632.8 nm and lower 514.5 nm.
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interferometer. As one o f the m i r r o r s is m o v e d a distance p r o v i d i n g a count o f 200 fringes in the red, the shorter green wavelength experiences 246 osciUations. This was experimentally c o m p a r e d to the above a c h r o m a t i c phase shifter where for a fringe count o f 200 in the red signal, 200_+ 0.5 oscillations were observed in the blue signal. Hence using the achromatic phase shifter no detectable difference in the fringe count was observed.
Acknowledgements The authors acknowledge discussion o f phase conj u g a t i o n with R.W. Boyd a n d K. M a c D o n a l d . D. Schertler assisted in the e x p e r i m e n t s on the achromatic phase shifter. This research was s u p p o r t e d in part b y the U.S. A r m y Night Vision a n d Electro-Optics Laboratory, the U.S. A r m y Research Office, a n d the corporate sponsors u n d e r the N e w Y o r k State Center for A d v a n c e d Optical Technology.
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References [I ] K. Iizuka, Engineering optics (Springer-Verlag, Berlin, 1987). [2] P. Hariharan, Optical interferometry (Academic Press, Sydney, 1985). [ 3 ] G.L. Rogers, Noncoherent optical processing (John Wiley, New York, 1977). [4] E.N. Leith and G.J. Swanson, Appl. Optics 19 (1980) 638. [5] F.T.S. Yu, Appl. Optics (1980) 2457. [6] N. George and T. Stone, J. Opt. Soc. Am. A 4 (1987) 79. [ 7 ] T. Stone and N. George, Appl. Optics 24 ( 1985 ) 3797. [8] G.J. Swanson, J. Opt. Soc. Am. A 1 (1984) 1147. [9] F.J. Weinberg and N.B. Wood, J. Sci. Instr. 36 (1959) 227. [ 10 ] N. George and S. Wang, Appl. Optics 23 (1984) 787. [ l 1] E. Wolf, Optics Comm. 62 (1987) 12. [ 12] C.D. Leonard and B.D. Guenther, Technical Report T-7917 (U.S. Army Missile Research and Development Command, Redstone Arsenal, 1979). [ 13 ] R.T. Ingwall and H,L. Fielding, Opt. Eng. 24 ( 1985 ) 808. [ 14 ] R.T. Ingwall,A. Stuck and W.T. Vetterling, Proc. Soc. PhotoOpt. Instr. Eng. 615 (1986) 81.
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