Accumulated Gouy phase shift in Gaussian beam propagation through first-order optical systems

Transcription

1 90 J. Opt. Soc. Am. A/Vol. 4, No. 9/September 997 M. F. Erden and H. M. Ozaktas Accumulated Gouy phase shift Gaussian beam propagation through first-order optical systems M. Fatih Erden and Haldun M. Ozaktas Department of Electrical Engeerg, ilkent University, ilkent, Ankara, Turkey Received August 6, 996; revised manuscript received January 4, 997; accepted February 8, 997 We defe the accumulated Gouy phase shift as the on-axis phase accumulated by a Gaussian beam passg through an optical system, excess of the phase accumulated by a plane wave. We give an expression for the accumulated Gouy phase shift terms of the parameters of the system through which the beam propagates. This quantity complements the beam diameter and the wave-front radius of curvature to constitute three parameters that uniquely characterize the beam with respect to a reference pot the system. Measurement of these parameters allows one to uniquely recover the parameters characterizg the first-order system through which the beam propagates. 997 Optical Society of America [S (97)0408-7]. INTRODUCTION A paraxial wave is a plane wave exp(ikz) modulated by a complex envelope A(x, y, z) that is assumed to be a slowly varyg function of position. In paraxial waves, order for the complex amplitude U(x, y, z), expressed as Ux, y, z Ax, y, zexpikz, () to satisfy Helmholtz equation, A(x, y, z) must satisfy the paraxial Helmholtz equation. One of the solutions of the paraxial Helmholtz equation is a Gaussian beam, for which we can express the complex envelope A(x, y, z) as Ax,y,z A qz expikx y /qz, qz z iz 0, () where z 0 is known as the Rayleigh range and A is an arbitrary complex constant. We can also express q(z) through qz rz i w z, (3) where w(z) and r(z) are the beam width and the wavefront radius of curvature, respectively, of Gaussian beams. With these newly defed parameters, the expression Eq. () becomes Ax, y, z A 0 wz exp x y w z exp i x y rz expiz, (4) where A 0 A /iz 0 is another complex constant and wz z /z 0, rz z z 0 /z, z tan z/z 0, z 0 /. (5) In these equations, and (z) are defed as the beamwaist diameter and the Gouy phase shift, respectively. These expressions assume that the beam waist is located at z 0, but they can be generalized to any waist location at z z w by simply replacg z with z z w. In this paper we consider Gaussian beams passg through centered, axially symmetric quadratic-phase optical systems under the standard approximations of Fourier optics. Th lenses, arbitrary sections of free space (under the Fresnel approximation), quadratic gradeddex media, and any combations of these belong to the class of quadratic-phase systems. We characterize the members of quadratic-phase systems through 6 p out x hx, xp xdx, hx, x K expix xx x, (6) where K is a complex constant and,, and are real constants. Thus, apart from the constant factor K, which has no effect on the resultg spatial distribution, a member of the class of quadratic-phase systems is completely specified by the three parameters,, and. Alternatively, such a system can also be completely specified by the transformation matrix 6 T A C D / / / /, (7) with unity determant (i.e., AD C ). If several systems each characterized by such a matrix are cascaded, the matrix characterizg the overall system can be found by multiplyg the matrices of the several systems.,4,5 The matrix defed above also corresponds to the well-known ray matrix employed ray optical analysis.,3 If we want to express the transformation Eq. (6) terms of the matrix elements ACD of the medium, it turns out that 5, /97/ $ Optical Society of America

2 M. F. Erden and H. M. Ozaktas Vol. 4, No. 9/September 997/J. Opt. Soc. Am. A 9 p out x expikl 0 h x, xp xdx, h x, x i exp i Dx xx Ax. (8) In this equation, L 0 is the on-axis optical length of the medium. 7 For two-dimensional, centered, and axially symmetric systems this transformation becomes p out x, y expikl 0 h x, xh y, y p x, ydxdy. (9) In this paper we defe the accumulated Gouy phase shift and give an expression for it terms of the transformation matrix elements ACD of the medium. We observe that this quantity complements the beam diameter and the wave-front radius of curvature to constitute three parameters that uniquely characterize the first-order optical system parameters through which the beam propagates.. ACCUMULATED GOUY PHASE SHIFT A. Defition The Gouy phase shift defed Eq. (5) is the on-axis phase of a Gaussian beam with respect to the beam waist excess of the phase of a plane wave exp(ikz). It is not dependent of beam diameter w and the wave-front radius of curvature r. That is, if we know w and r of a beam at a specific plane, we can also obta the Gouy phase shift at that plane. More specifically, from Eq. (5) we can fd z tan w z. (0) rz When a Gaussian beam propagates through an optical system composed of several lenses, there may be a number of beam waists. In this case it does not seem possible to terpret the Gouy phase shift as defed above a meangful manner. It may be possible to terpret as the on-axis phase of the beam excess of that of a plane wave with respect to the last (possibly virtual) waist. Of greater terest would be the phase shift accumulated by the beam as it passes through several lenses and sections of free space with respect to a sgle reference pot the system. Thus we defe the accumulated Gouy phase shift of a Gaussian beam passg through an optical system as the on-axis phase accumulated by the beam excess of the factor exp(ikl 0 ) Eq. (9) (this factor can also be thought as the on-axis phase that would be accumulated by a plane wave). Mathematically, arg p out 0, 0 arg p 0, 0 kl 0, () where p (x, y) and p out (x, y) denote the put and the output Gaussian beams, respectively, and arg denotes the argument (phase). In Ref. 8, transformation of the generalized beam-mode parameters is analyzed with operator algebra. The output parameters are related to the put parameters through transformation matrix elements ACD of the medium. When the matrix elements are real, as is the case here, the generalized beams correspond to conventional Hermite Gaussian beams, the lowest order of which is the Gaussian beam. Let us now consider a Gaussian beam with parameters and r put to a quadratic-phase system, and let the system be characterized by the transformation matrix with elements ACD. The output beam parameters are and r out. Let us defe the complex parameters q and q out associated with the put and the output Gaussian beams, respectively, through Eq. (3). We know that q out is related to q through matrix elements ACD as,7,8 q out Aq Cq D. () Usg this expression, we can relate the output parameters and r out to the put parameters and r through ACD as A r out r C D r A w, (3) r D 4 A. (4) r 4 In this paper we show that the accumulated Gouy phase shift can be similarly expressed as tan A r. (5) Two proofs of this expression are given Section 3. The first one is algebraically straightforward, but the second one although somewhat long, is more structive. As a result, if we consider a Gaussian beam with parameters and r put to a quadratic-phase system characterized by the matrix with elements ACD, the parameters and r out of the output beam as well as the accumulated Gouy phase shift from put to output can be found from Eqs. (3) (5). Notice that the accumulated Gouy phase shift depends on only two of the three dependent parameters that characterize the first-order system. Thus two systems that have the same values of A and but different values of C and D may have the same accumulated Gouy phase shift.. Interpretation As an Independent Parameter We may also look at the problem the other way around. Let us assume that we know the parameters and r of the Gaussian beam at the put and that we can measure and r out at the output of the quadratic-phase system, and let us assume that we also know the accumulated Gouy phase shift between the put and the output planes (if we know the on-axis optical length of the system, we can obta both the output wave-front radius of curvature and the accumulated Gouy phase shift by terferg the

3 9 J. Opt. Soc. Am. A/Vol. 4, No. 9/September 997 M. F. Erden and H. M. Ozaktas beam with a plane). Then we can recover the matrix elements ACD of the system by usg Eqs. (3) (5) and the identity ADC as A cos s r, (6) s, (7) C A 4 w r A r r out A, (8) r 4 C D. (9) A Notice that the ACD parameters of the system could not be recovered from measurements of a sgle Gaussian probe by employg the conventional Gouy phase shift [Eq. (0)]. This is because the conventional Gouy phase shift is not dependent of beam diameter w and the wave-front radius of curvature r. In contrast, the accumulated Gouy phase shift is an dependent parameter that complements the beam diameter and the wave-front radius of curvature to constitute three parameters that uniquely characterize the beam with respect to a reference pot the system. This means that knowledge of these three parameters at any sgle plane the system allows them to be calculated at any other plane the system. Measurement of these parameters allows one to uniquely recover the parameters that completely characterize the quadratic-phase system through which the beam propagates. Remember that such systems are characterized by three parameters (,,, or any three of ACD). Thus we would expect the effect of such a system on a beam passg through it also to be characterized by three parameters. The conventional Gouy phase shift, which is not dependent of w and r and is difficult to terpret meangfully multilens systems, cannot serve as a third parameter. This was the motivation behd defg the accumulated Gouy phase shift. The importance of this quantity is further supported by its direct relation to the fractional-fourier-transform parameter, 3,9,0 as discussed at length elsewhere. 3. PROOF OF THE ACCUMULATED GOUY PHASE-SHIFT EXPRESSION In this section we give two proofs for the expression Eq. (5). A. Proof A The output Gaussian beam p out (x, y) is related to the put Gaussian beam p (x, y) through Eq. (9) as p out x, y expikl 0 h x, xh y, y p x, ydxdy, (0) where h (, ) is the kernel Eq. (8) and p x, y A exp x y exp i x y r. () We know from Ref. 7 that expx r qxdx exp q. () r 4r Usg this identity, after some algebra we obta p out (x, y) as where p out x, y A expikl 0 exp x y r out tan exp i x y A r C D r A A r out expi, (3) w, (4) r D 4 A, r 4 r (5). (6). Proof Lemma : The expression Eq. (5) is consistent under concatenation [i.e., if two systems for which Eq. (5) holds are cascaded, the resultant accumulated Gouy phase shift is also the form of Eq. (5) and is expressed as the summation of the phase shifts associated with the dividual systems]. Proof: Suppose that we have two systems with transformation matrix elements A C D and A C D concatenated one after another. Let us call the Gaussian beam parameters at the put of the first system and r, at the output of the first system and r out r (which are also the put parameters of the second system), and at the output of the second system and r out. We also have the accumulated Gouy phase shifts and associated with the first and second systems, respectively. Let us also assume that both systems satisfy Eq. (5). Then tan tan A, (7) r A. (8) r out We can obta and r out used Eq. (8) terms of, r and the matrix elements of the first system

4 M. F. Erden and H. M. Ozaktas Vol. 4, No. 9/September 997/J. Opt. Soc. Am. A 93 A out x, y A 0 exp x y exp i x y r through Eqs. (3) and (4). We also have the overall system transformation matrix elements ACD through A C D A C D A C D. (9) We have all the necessary equations, so although the computations are somewhat long, it is just a matter of straightforward algebra to show that r tan A, (30) r where r is the accumulated Gouy phase shift associated with the cascaded system. Lemma : Equation (5) holds for systems with upper triangular matrices of the form T d. (3) 0 (The matrix associated with a section of free space of length d is of this form.) Proof: Let us consider a Gaussian beam with and r put to a system. With these parameters, the complex envelope of the put Gaussian beam defed Eq. () becomes A x, y A 0 exp x y exp i x y r. (3) After takg the Fresnel tegral of this expression, we fd the output Gaussian beam as A out x, y A 0 exp x y exp i x y r out expi, (33) where and r out Eq. (33) are the form of Eqs. (3) and (4) with ACD as the parameters of the matrix Eq. (3) and tan d d r. (34) We see that this expression is consistent with the one Eq. (5) with A and d, as is the case for the matrix Eq. (3). Lemma 3: Equation (5) holds for systems with lower triangular matrices of the form 0 T (35), f (The matrix associated with a th lens of focal length f is of this form.) Proof: Aga startg with the complex-envelope expression of the put Gaussian beam Eq. (3), we know that after passage through a lens of focal length f, this complex envelope becomes exp i f x. (36) Observg Eqs. (3) and (36), we see that when a Gaussian beam passes through a lens, its wave-front radius of curvature r out will be the only parameter that changes accordg to r out r f, whereas the beam diameter remas unchanged and there is no accumulated Gouy phase shift ( 0). This result is consistent with Eq. (5), as we can aga obta 0 from this equation together with the transformation matrix expressed Eq. (35). Proof of equation 5: Any matrix of unity determant can be decomposed as T A A /C C D 0 D /C. 0 C 0 (37) Thus any quadratic-phase system can be modeled as two sections of free space with a lens between. We have shown lemmas, 3 that the expression Eq. (5) holds for both lenses and free-space sections. Then with the help of lemma we can say that the accumulated Gouy phase-shift expression Eq. (5) also holds for the cascaded system Eq. (37) and hence for quadraticphase systems. 4. CONCLUSION In this paper we defed the accumulated Gouy phase shift, and gave an expression for it terms of the transformation matrix elements of the system through which the beam propagates. We observed that this quantity is dependent of beam diameter w and the wave-front radius of curvature r. Thus, measurement of the accumulated Gouy phase shift, the beam diameter, and the wavefront radius of curvature allows one to uniquely recover the parameters that completely characterize the quadratic-phase system through which the beam propagates. REFERENCES.. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 99).. M. J. astiaans, Wigner distribution function and its application to first-order optics, J. Opt. Soc. Am. 69, (979). 3. H. M. Ozaktas and D. Mendlovic, Fractional Fourier optics, J. Opt. Soc. Am. A, (995). 4. K.. Wolf, Integral Transforms Science and Engeerg (Plenum, New York, 979). 5. M. Nazarathy and J. Shamir, First-order optics a canonical operator representation: lossless systems, J. Opt. Soc. Am. 7, (98). 6. S. Abe and J. T. Sheridan, Optical operations on wave functions as the Abelian subgroups of the special affe Fourier transformation, Opt. Lett. 9, (994). 7. A. E. Siegman, Lasers (University Science ooks, Mill Valley, Calif., 986). 8. M. Nazarathy, A. Hardy, and J. Shamir, Generalized mode

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