Four-channel self-focus computer-generated hologram

Ren ChaoHong, Zhou Jin, and Gao WenQi

A new type of computer-generated hologram ~CGH! is described in this research. Upon the base of atwo-channel CGH, it can generate four independent images in four different directions with the additionof positive or negative quadratic phase factors on the object spectrum; it has the character of self-focus.Results of the experiment are provided. © 1997 Optical Society of America

Key words: Four-channel self-focus computer-generated hologram, quadratic phase factors, discreteFourier spectrum.

1. Introduction

As a branch of modern optics, the computer-generated hologram ~CGH! is applied broadly, suchas in matching filters, wave-mode transformation,and holography scanners. In the past few decades,several encoding methods have been introduced.1–3

With these methods only one object can be encoded ina CGH. Recently, Mendlovic and Kiryuschev4 sug-gested a two-channel CGH that can encode two ob-jects simultaneously and reconstructed their imagesin orthogonal diffraction directions as normal FourierCGH; their conjugate images are also reconstructed.

We multiply the object spectrum by a quadraticphase factor and obtain a four-channel self-focusCGH that can reconstruct images of two other objectsat the positions of the two conjugate images. Withthis CGH, four objects can be encoded simulta-neously. Because of the quadratic phase factors,this CGH can reconstruct images without a lens andhas the property of self-focus. Therefore we canmatch four objects at one time and increase the chan-nels of the CGH from one to four.

2. Principle of Encoding

Figure 1 shows a sample unit of the CGH. The phys-ical extent of every unit is dd 3 dd, the coordinate ofthe ~m, n! unit is ~mdd, ndd!, and every unit containsfour square apertures. The size of the rectangle ofwhich the four vertices are the four centers of the foursquare apertures is ~Wmn

mdd 3 Wmnndd!. ~Pmn

mdd,Pmn

ndd! is the displacement from the center of the

The authors are with the Department of Physics, Nanjing Uni-versity, Nanjing 210093, China.

Received 1 April 1997; revised manuscript received 29 July 1997.0003-6935y97y348844-04$10.00y0© 1997 Optical Society of America

8844 APPLIED OPTICS y Vol. 36, No. 34 y 1 December 1997

rectangle to the center of unit. Thus every unit hasfour variables ~Pmn

m, Wmnm, Pmn

n, Wmnn!. Another

four variables are defined as

hmnm 5 Pmn

m 1 Wmnmy2, jmn

m 5 Pmnm 2 Wmn

my2,

hmnn 5 Pmn

n 1 Wmnny2, jmn

n 5 Pmnn 2 Wmn

ny2, (1)

The transmissivity H~m, n! of the whole CGH can bewritten as

H~m, n! 5 (m

(nHrectFm 2 ~m 1 hmn

m!dda G

3 rectFn 2 ~n 1 hmnn!dd

a G1 rectFn 2 ~m 1 hmn

m!dd

a GrectFn 2 ~n 1 jmnn!dd

a G1 rectFn 2 ~m 1 jmn

m!dd

a GrectFn 2 ~n 1 hmnn!dd

a G1 rectFn 2 ~m 1 jmn

m!dd

a GrectFn 2 ~n 1 jmnn!dd

a GJ .

(2)The field distribution in the output plane is readilywritten as

h~x, y! 5 F21@H~m, n!#

5 (m

(nHexpF j

2pddlf

~xhmnm 1 yhmn

n!G1 expF j

2pddlf

~xhmnm 1 yjmn

n!G

1 expF j2pdd

lf~xjmn

m 1 yhmnn!G

1 expF j2pdd

lf~xjmn

m 1 yjmnn!GJ

3 sinc~ax! sinc~ay!

3 expF j2pdd

lf~xm 1 yn!G, (3)

where F21 represents the inverse Fourier transform.The coordinates of ~11, 0!, ~21, 0!, ~0, 11!, and ~0. 21!orders are, respectively, equal to ~x1, 0!, ~2x1, 0!, ~0,y1!, and ~0, 2y1!, where

x1ddlf

5 1,y1ddlf

5 1.

Taking a reasonable approximation and neglectingthe constant factor, we can obtain the field distribu-tion of h11,0, h21,0, h0,11, and h0,11 near ~x1, 0!, ~2x1,0!, ~0, y1!, and ~0, 2y1!:

h11,0~x9, y! 5 (m

(n

cos pWmnm exp~ j2pPmn

m!

3 expF j2pdd

lf~x9m 1 yn!G ~x9 5 x 2 x1!,

h21,0~x0, y! 5 (m

(n

cos pWmnm exp~ 2 j2pPmn

m!

3 expF j2pdd

lf~x0m 1 yn!G ~x0 5 x 1 x1!,

h0,11~x, y9! 5 (m

(n

cos pWmnn exp~ j2pPmn

m!

3 expF j2pdd

lf~xm 1 y9n!G ~y9 5 y 2 y1!,

h0,21~x, y0! 5 (m

(n

cos pWmnnexp~ 2 j2pPmn

n!

3 expF j2pdd

lf~xm 1 y0n!G ~y0 5 y 1 y1!,

(4)

Fig. 1. Sample unit of CGH.

From Eq. ~4!, evidently h11,0 and h21,0 are relatedonly to ~Pmn

m, Wmnm!, and h0,21 and h0,11 are related

only to ~Pmnn, Wmn

n!. ~Pmnm, Wmn

m! and ~Pmnn,

Wmnn! are independent of each other and can respec-

tively encode different object functions. To define:

H11,0 5 (m

(n

cos pWmnm exp~ j2pPmn

m!,

H21,0 5 (m

(n

cos pWmnm exp~ 2 j2pPmn

m!,

H0,11 5 (m

(n

cos pWmnn exp~ j2pPmn

n!,

H0,21 5 (m

(n

cos pWmnn exp~ 2 j2pPmn

n!.

We have H11,0 5 H*21,0, H0,11 5 H*0,21. Now Eq.~4! can be rewritten as

h11.0 5 F21(H11.0), h21.0 5 F21(H21.0),

h0,11 5 F21(H0,11), h0.1 5 F21(H0,21).

If we assume that the zero order of the CGH is H0,0,and if we compare Eq. 4 ~as rewritten! with Eq. ~3!and neglect the higher-order diffraction, we mayagain regard H~m, n! as being approximated to thesuperposition of H0,0, H11,0, H21,0, H0,11, and H0,21.

We define Fx and Fy as two complex discrete func-tions:

Fx~mdd, ndd! 5 (m

(n

Amnx exp~ jFmn

X!,

Fy~mdd, ndd! 5 (m

(n

Amny exp~ jFmn

y!,

where Amnx and Amn

y are two normalized real num-bers. Fx and Fy are encoded with the followingmethod:

Amnx 5 cos pWmn

m, Fmnx 5 2pPmn

m (5)

Amny 5 cos pWmn

n, Fmny 5 2pPmn

n.

We have Fx 5 H11,0 5 H*21,0, Fy~mdd, ndd! 5 H0,115 H*0,21. If Fx and Fy are two normalized discreteFourier spectra of two objects, with Eq. 5 we can findthat the two images of two objects are reconstructedat ~x1, 0! and ~0, y1! and their conjugate images are at~2x1, 0! and ~0, 2y1!. This is two-channel CGH.

Now we need to encode four objects in a CGH simul-taneously; Fx or Fy cannot simply be equal to an ob-ject’s discrete spectrum but to a somewhat syntheticdiscrete spectrum. X11, X21, Y11, and Y21 are fourobjects to be encoded; their object functions aregX11

~x, y!, gX21~x, y!, gY11

~x, y!, and gY21~x, y!, and their

Fourier discrete spectra are

(m

(n

AmnX11 exp~ jFmn

X11!, (m

(n

AmnX21 exp~ jFmn

X21!,

(m

(n

AmnY11 exp~ jFmn

Y11!, (m

(n

AmnY21 exp~ jFmn

Y21!.

Let us consider Fx first. We multiply the conjugatespectrum of X11 by a positive quadratic phase factorand the spectrum of X21 by a negative quadraticphase factor. Fx is equal to their sum. Fx is en-

1 December 1997 y Vol. 36, No. 34 y APPLIED OPTICS 8845

coded according to Eq. ~5!. Put CGH in the Fig. 2optical path, where the input illumination is coherentand collimated.

From the above discussion, the reconstructed im-ages of Fx are related only with H11,0 and H21,0. Weconsider H11,0 first. In the condition of the Fresneldiffraction approximation, the ~11, 0! order, corre-sponding to H11,0 in the output plane, is readily writ-ten as

Fx~mdd, ndd! 5 (m

(nHAmn

X11 exp~ jFmnX11)

3 expF2jplf

~m2 1 n2!dd2G1 Amn

X21 exp~2jFmnX21!

3 expFjplf

~m2 1 n2!dd2GJ .

The first part of the equation immediately above is anobject function of X11 multiplied by a phase factorthat has no effect on image intensity distribution; thesecond part is a speckle of X21 ~i.e., a defousing im-age!. Thus the image of X11 and the speckle of X21are reconstructed at the ~11, 0! order of CGH. Be-cause H21,0 5 H*11,0, it is readily known that theimage of X21 and the speckle of X11 are reconstructedat the ~21, 0! order of CGH.

We use the same method on Fy. Multiply thespectrum of Y11 by a negative quadratic phase factorand the conjugate spectrum of Y21 by a positive qua-dratic phase factor; Fy is equal to their sum:

Fy~mdd, ndd! 5 (m

(nHAmn

Y11 exp~ jFmnY11!

3 expF2jplf

~m2 1 n2!dd2G1 Amn

Y21 exp~2jFmnY21!

3 expFjplf

~m2 1 n2!dd2GJ .

In a similar fashion, we can find that the image ofY11 and the speckle of Y21 are reconstructed at the

Fig. 2. Setup for reconstructing four-channel self-focus CGH.Input illumination is coherent and collimated.

8846 APPLIED OPTICS y Vol. 36, No. 34 y 1 December 1997

~0, 11! order, and the image of Y21 and the speckle ofY11 are reconstructed at the ~0, 21! order.

3. Experiment and Discussion

In the experiment the four objects are four Englishcharacters: E, S, N, and W. Every object sampled is64 3 64 units. Fx is the sum of the E discrete Fou-rier spectrum multiplied by a negative quadraticphase factor and the W discrete conjugate Fourierspectrum multiplied by a positive quadratic phasefactor. Fy is the sum of the S discrete Fourier spec-trum multiplied by a negative quadratic phase factorand the N discrete conjugate Fourier spectrum mul-tiplied by a positive quadratic phase factor. Thevalue of f in the quadratic phase factors is 60 cm.The encoded result is output by a plotter and reduced80 times; we then obtain the CGH. Its physical sizeis 4 mm 3 5 mm. Figure 3 is a part of the CGHmagnified 80 times. From the experiment result inFig. 4, we can see that the reconstructed image isaffected by the speckles. The spectra of objects are

Fig. 3. Part of CGH, magnified 80 times.

Fig. 4. Result of experiment. Images of four objects are recon-structed in four different directions.

multiplied by different displacement phase factors,thus the distances from two images to the originalong the same axis are different.

4. Conclusion

Four-channel self-focus CGH can reconstruct im-ages of four objects in four directions. The positionof the output plane is related to f in the quadraticphase factors. In the experiment and the discus-sion, the value of f in the four quadratic phasefactors that multiply the four object spectra are thesame. If the four f values are not the same, fourimages will reconstruct in different planes; there-

fore the depths of field of the reconstructed imageswill be different.

References1. B. R. Brown and A. W. Lohman, “Complex spatial filter with

binary mask,” Appl. Opt. 5, 967–969 ~1966!.2. A. W. Lohman and D. P. Paris, “Binary Fraunhofer holograms

generated by computer,” Appl. Opt. 6, 1739–1749 ~1967!.3. W.-H. Lee, “Binary computer-generated holograms,” Appl.

Opt. 18, 3661–3669 ~1979!.4. D. Mendlovic and I. Kiryuschev, “Two-channel computer-

generated hologram and its application for optical correlation,”Opt. Commun. 116, 322–325 ~1995!.

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