Design of polarization-selective diffractive phase ... - OSA Publishing

Design of polarization-selective diffractive phase elements in a laser cavity Juan Liu, Ben-Yuan Gu, and Guo-Zhen Yang

We report a new design for a polarization-selective laser cavity with birefringent diffractive phase elements. This laser cavity can create two modes with different polarizations and profiles launched separately from two end mirrors. The numerical simulation results show that the constructed laser cavity can successfully generate two orthogonal polarization modes with a uniform circular shaped pattern output from one end mirror and a uniform square-shaped pattern output from another end mirror. © 1999 Optical Society of America OCIS codes: 140.0140, 230.5440, 320.5540, 050.1970, 260.1440.

1. Introduction

Conventional Fabry–Perot resonators are widely used in many commercial lasers. Characteristics of the gain medium determine the shapes of the spherical mirrors in laser resonators as well as eigenmode loss.1 A low fundamental-mode loss is required for a low-gain system. However, low-loss resonators require long resonator lengths and small-diameter scale modes so that the power output remains a lower value, and discrimination between lowest-order modes is small. Although unstable resonators can support a large-diameter scale fundamental mode as well as preserve adequate higher-order mode discrimination, they have an obstructed output aperture that produces an undesirable near-field pattern, i.e., an inherently lossy fundamental mode. Therefore unstable resonators are not suitable for low- and medium-gain laser systems. Unconventional mirrors have been used to generate specific modes. Variable-reflectivity mirrors2,3 and phase-step mirrors4 have been used to reduce the diffraction loss of the hard edges of an unstable resonator and to increase the uniformity of the near-field intensity. Recently, Fourier spatial filters5 have been used to produce a uniphase beam with a flat spatial profile and minimal beam divergence. Graded phase mir-

The authors are with the Institute of Physics, Academia Sinica, P.O. Box 603, Beijing 100080, China. The e-mail address for J. Liu is [email protected]. Received 19 March 1999; revised manuscript received 18 June 1999. 0003-6935兾99兾336887-05$15.00兾0 © 1999 Optical Society of America

rors6 have been applied to tailor the shape of the desired fundamental mode. Diffractive mode-selecting mirrors have been fabricated and operated in a solidstate laser system, most notably those presented by Leger and co-workers.7–9 The advantages of these customized cavities demonstrate the superior discrimination against higher-order modes, permitting singlemode operation and flexible construction of a fundamental-mode pattern with the desired profile. The output profile of the real mode is completely governed by a single diffractive phase element, and an arbitrary mode profile can be produced. Here we extend the Leger et al. approach to the design of a novel laser cavity with polarization-selective diffractive phase elements 共PDPE’s兲 as its two end mirrors. This laser can generate two arbitrary real modes with different patterns and orthogonal polarization states with each launched from two end mirrors. We chose PDPE’s rather than diffractive phase elements as the cavity mirrors because PDPE’s can easily be used to modulate different polarization beams separately, which makes them the best candidates for designing a novel laser cavity with the output of two separate polarization beams from two end mirrors. This implies that a polarization state can provide an additional degree of freedom to achieve the desired result. We used the model design of a novel laser cavity based on the theory of angular spectrum. To provide a better understanding of the design approach, we rederive a set of eigenequations for determination of the surface-relief depth distributions of the PDPE’s to generate two real polarization modes with different shapes in a stable resonator. By use of the simulated annealing algorithm,10 these equations can be solved numerically. We present numerical simulation re20 November 1999 兾 Vol. 38, No. 33 兾 APPLIED OPTICS

6887

Fig. 1. Schematic view of the laser cavity containing two polarization-selective diffractive phase elements.

sults of one model design example for which we designed a particular laser cavity that can be used to generate a uniform circular pattern with ordinary-ray polarization and a uniform square pattern with extraordinary-ray polarization, separately launched from two end mirrors in this laser system. We describe the relevant formulas and the calculation method in Section 2. Simulation results are presented in Section 3, and Section 4 provides a brief summary. 2. Formulas and Calculation Method Used in the Design

The optical cavity system is composed of two PDPE’s made of uniaxial birefringent material, M1 and M2, separated by a distance l, as shown in Fig. 1. The outer surface of each PDPE is coated with highreflective film for one polarization state, partial reflection for the other polarization state, and its inner surface is antireflection coated. This implies that the outer surface of the PDPE1 共PDPE2兲 should be coated for high reflection for polarization beam II 共I兲, whereas partial reflection for polarization beam I共II兲 and the inner surface of PDPE1 共PDPE2兲 should be antireflection coated for both polarization modes to avoid parasitic reflections. Both optical axes of the two birefringent crystal wafers lie on the surface of the substrate wafers. The surface-relief depths of the PDPE’s are denoted by h1共x, y兲 and h2共x⬘, y⬘兲. We must note that the same depth distribution, h1共2兲共x, y兲, on the end mirror will give rise to different phase retardation when the light beams with different polarization states transmit the same PDPE because of the different refractive index n␣ of the PDPE with respect to the polarization state. The phase is related to n␣ by ␾i,␣ ⫽ 2␲hi 共n␣ ⫺ 1兲兾␭, where subscript i ⫽ 1 or 2 denotes the PDPE1 or PDPE2, respectively, and subscript ␣ ⫽ I or II represents different orthogonal polarization modes. The designed PDPE’s must be capable of generating special modes with specified profiles as eigenmodes of this resonator system. As is well known, when two lin6888

APPLIED OPTICS 兾 Vol. 38, No. 33 兾 20 November 1999

ear polarization beams transmit uniaxial birefringent crystal substrates, different phase retardations are acquired for different polarization states. Therefore, it is possible to separate these two linear polarization beams. For example, we assume that one linear polarization beam 共ordinary ray兲 with the polarization direction perpendicular to the optical axis of the uniaxial substrate material is launched from the first mirror 共M1兲 while another orthogonally linear polarization beam 共extraordinary ray兲 is emitted from the second mirror 共M2兲 when the beams are normally incident on PDPE’s. It is assumed that proper intracavity loss or gain distribution exists to prevent the other modes from being excited. The complex function ui␣ describes a complex optical field, in which subscript i and superscript ␣ denote, respectively, two mirrors and two polarization states. We now consider linear polarization light beam I for which ␳10共x, y兲 denotes its real mode distribution, located at the back 共outer兲 surface of the first mirror 共M1兲 but just after reflection by the back surface of this mirror. When this light beam transmits the PDPE1 and arrives at the front 共inner兲 surface of mirror M1, the corresponding wave front can be expressed by a complex function u1I共x, y兲 in the form u1I共x, y兲 ⫽ ␳10共x, y兲exp关i2␲h1共x, y兲共n1 ⫺ 1兲兾␭兴, (1) where n1 and ␭ represent the refractive index of the PDPE for this polarization state and the wavelength of the beam, respectively. Now, h1共x, y兲 can be regarded as an arbitrary function. We expand this field in terms of its angular plane-wave spectrum U1I共 fx, fy兲 as u1I共x, y兲 ⫽

兰兰 ⬁

⬁

⫺⬁

⫺⬁

U1I共 fx, fy兲exp关i2␲共xfx ⫹ yfy兲兴dfxdfy, (2)

where fx and fy are spatial frequencies in the x and y directions, respectively. The u1I wave propagates a cavity length l in free space and arrives at the front

共inner兲 surface of the second PDPE 共M2兲. The corresponding field distribution is then given by u2 共x⬘, y⬘兲 ⫽ I

兰兰 ⬁

⬁

⫺⬁

⫺⬁

u2II共x⬘, y⬘兲 ⫽ ␳20共x⬘, y⬘兲exp关i2␲h2共x⬘, y⬘兲共n2 ⫺ 1兲兾␭兴,

U1 共 fx, fy兲exp关i2␲共x⬘fx ⫹ y⬘fy兲兴 I

⫻ exp兵ikl关1 ⫺ 共␭fx兲2 ⫺ 共␭fy兲2兴1兾2其dfxdfy.

of this mirror can be described by the complex function u2II共x⬘, y⬘兲:

(7) (3)

If the complex reflection coefficient r2I共x⬘, y⬘兲 of the second mirror for polarization state I is

where n2 represents the refractive index of PDPE for this polarization state. If one chooses the complex reflection coefficient of the first mirror M1 as r1II共x, y兲 ⫽

I

r2I共x⬘, y⬘兲 ⫽

u2 *共x⬘, y⬘兲 ⫽ H2共␳10, h1兲 u2I共x⬘, y⬘兲

⫽ exp关i4␲h1共n2 ⫺ 1兲兾␭兴,

⫽ exp关i4␲h2共n1 ⫺ 1兲兾␭兴,

(4a)

where the asterisk represents the complex conjugate, we can obtain h2共x⬘, y⬘兲 as h2共x⬘, y⬘兲 ⫽ ⫽

冋

that is, h1共x, y兲 ⫽

册

␭ u2I*共x⬘, y⬘兲 arg 4␲共n1 ⫺ 1兲 u2I共x⬘, y⬘兲

␭ arg关H2共␳10, h1兲兴. 4␲共n1 ⫺ 1兲

⫽ (4b)

Thus the optical field of the returning wave located at the front 共inner兲 surface of mirror M2 after reflection with this mirror should be given by r2Iu2I共x⬘, y⬘兲 ⫽ u2I*共x⬘, y⬘兲.

(5)

When this wave further propagates through the cavity length l distance in free space and arrives at the front 共inner兲 surface of the first mirror, the corresponding wave front becomes u1I*共x, y兲 ⫽

兰兰 ⬁

⬁

⫺⬁

⫺⬁

U1I*共 fx, fy兲

⫻ exp关⫺i2␲共xfx ⫹ yfy兲兴dfxdfy. With reference to Eq. 共1兲, after transmitting mirror M1 and being successively reflected by its back 共outer兲 surface with high-reflective film, one can rewrite this returning wave field as ␳10⬘共x, y兲 ⫽ r10 exp关i2␲h1共n1 ⫺ 1兲兾␭兴u1I*共x, y兲 ⫽ r10␳10共x, y兲 ⫽ ␳10共x, y兲,

u1II*共x, y兲 ⫽ H1共␳20, h2兲 u1II共x, y兲

(6)

where we have assumed that the back surface of this mirror possesses complete reflection for polarization state I, that is, r1 0 ⫽ 1. The initial distribution ␳10共x, y兲 automatically duplicates itself after one round trip through the laser cavity, so ␳10共x, y兲 becomes an eigenmode of the system only if Eq. 共4兲 is satisfied and h1共x, y兲 is free at this step. Similarly, we can follow the same procedure to derive corresponding equations for the second orthogonal linear polarization beam II. It is assumed that the desired mode of the second polarization beam is also real and launched from the second mirror. We denote it as ␳20共x⬘, y⬘兲. This beam transmits the second mirror, and then its wave front on the front 共inner兲 surface

(8a)

冋

册

u1II*共x, y兲 ␭ arg 4␲共n2 ⫺ 1兲 u1II共x, y兲

␭ arg关H1共␳20, h2兲兴, 4␲共n2 ⫺ 1兲

(8b)

the second real polarization mode automatically duplicates itself in this cavity. Here we have u1II共x, y兲 ⫽

兰兰 ⬁

⬁

⫺⬁

⫺⬁

U2II共u, v兲exp关i2␲共xu ⫹ yv兲兴

⫻ exp兵ikl关1 ⫺ 共␭u兲2 ⫺ 共␭v兲2兴1兾2其dudv, where U2II共u, v兲 is the angular plane-wave spectrum of u2II共x⬘, y⬘兲. Since the complex reflection coefficients of the two mirrors 关Eqs. 共4兲 and 共8兲兴 are phase only, they can be fabricated as PDPE’s. We emphasize that the phases of eigenmodes are constant when the designed phase distributions of PDPE’s are used in the cavity. The phases of both modes can be set to zero, whereas the phase distributions of both PDPE’s are much more complex. To guarantee two orthogonal polarization modes serving as eigenmodes of this resonator, it is necessary to have both functional equations, 共4兲 and 共8兲, satisfied at the same time. One can see from Eqs. 共4兲 and 共8兲 that, when ␳10 and ␳20 are given, h1 and h2 depend on each other. These functional equations can be solved numerically with the simulated annealing 共SA兲 algorithm.10 The optimization procedure can be performed initially in Eq. 共4兲 to obtain h2共x⬘, y⬘兲 from an arbitrary function h1共x, y兲 共h1 is a 64-level depth in SA兲 and the known ␳10共x, y兲, so that one can create the first linear polarization mode. Then we can calculate the second linear polarization mode ␳20⬘共x⬘, y⬘兲 by substituting h1共x, y兲, the calculated h2共x⬘, y⬘兲, the desired ␳10共x, y兲, and ␳20共x⬘, y⬘兲 into the expression ␳20⬘共x⬘, y⬘兲 as

兰兰

␳20⬘共 x⬘, y⬘兲 ⫽ r20 exp关i2␲h2共n2 ⫺ 1兲兾␭兴

⬁

⬁

⫺⬁

⫺⬁

⫻ exp兵ikl关1 ⫺ 共␭u兲 ⫺ 共␭v兲 兴 其r1II 2

2 1兾2

⫻ exp兵ikl关1 ⫺ 共␭u兲2 ⫺ 共␭v兲2兴1兾2其U2II共u, v兲 ⫻ exp关i2␲共x⬘u ⫹ y⬘v兲兴dudv. 20 November 1999 兾 Vol. 38, No. 33 兾 APPLIED OPTICS

(9) 6889

Equation 共9兲 calculates the field distribution of the second polarization mode on the second end mirror after this mode completes one round trip in the laser cavity. The reflective index of the back surface of the second mirror is assumed to be r20 ⫽ 1. As one requires ␳20⬘共x⬘, y⬘兲 3 ␳20共x⬘, y⬘兲, the objective function D in SA is

兰兰

D ⫽ 储␳20⬘2 ⫺ ␳202储2 ⫽

⬁

⬁

⫺⬁

⫺⬁

共␳20⬘2 ⫺ ␳202兲2dx⬘dy⬘. (10)

In this procedure, h2 is regarded as a function argument, that is, as soon as h1 is determined, h2 can be evaluated by Eq. 共4兲.11 The surface-relief depths h1 and h2 are determined when D 3 0. After h1 and h2 are decided, the surface-relief PDPE’s can be fabricated by lithography, for example, the calcite substrate can be etched to produce functions h1共x, y兲 and h2共x⬘, y⬘兲. The calcite can also be deposited upon an ordinary material or upon a calcite substrate with a specified depth of h0 ⫽ ⫾␭兾共n1 ⫺ n2兲. 3. Numerical Simulation Results

Inasmuch as the uniform beam profile is useful in many practical applications, we prefer to present one modal design of a laser cavity with PDPE’s that can generate two real modes with orthogonal linear polarizations: one with an amplitude of uniform circular shape emitted from one mirror 共M1兲 and a second with an amplitude of a uniform square shape launched from the other mirror 共M2兲. The two real modes, for example, read ␳10共x, y兲 ⫽ ␳20共x, y兲 ⫽

再 再

1 for 共x2 ⫹ y2兲 ⱕ a , 0 otherwise Fig. 2. Distribution of the surface-relief depth of 共a兲 the first PDPE 共M1兲 plotted as a gray-level representation and 共b兲 the second PDPE 共M2兲 plotted as a gray-level representation.

1 for 兩x兩 ⱕ b and 兩y兩 ⱕ b . 0 otherwise

To assess the uniformity of the entire intensity distribution over a given region, we define the measurement of the degree of nonuniformity as nonuniformity ⫽

共具Ireal2典 ⫺ 具Ireal典2兲1兾2 , 具Iideal典

(11)

where 具. . .典, Ireal, and Iideal represent the average values of the calculated eigenmode intensity and the ideal 共specified兲 wave field intensity, respectively. The signal-to-noise ratio 共SNR兲 is defined as SNR ⫽

具I1典 , 具I0典

(12)

where 具I1典 and 具I0典 represent the average intensity over the signal domain and that outside the signal domain, respectively. For our design, we chose the large size PDPE’s to maintain the minimal diffractive loss and the negligible fundamental mode loss. We set a ⫽ b ⫽ 0.5 mm. The birefringent material used for the PDPE’s is calcite crystal with refractive indices of no ⫽ 1.658 for the ordinary ray and ne ⫽ 1.486 for the extraordinary ray 6890

APPLIED OPTICS 兾 Vol. 38, No. 33 兾 20 November 1999

when the laser wavelength was 0.6328 ␮m. We assumed a cavity length of 1.1 m. Two PDPE’s have the same size of 9.4 mm, which is larger than that of the fundamental mode. This condition is necessary to establish the desired mode by favorable constructive and destructive interferences of the waves in the cavity. The Fresnel number of the resonator is approximately 32. The surface-relief depth distributions of the PDPE’s at mirrors M1 and M2 are displayed in Figs. 2共a兲 and 2共b兲 as gray-level representations, respectively. Because both real polarization modes are symmetrical in our design, it can easily be seen that the distributions of surface-relief depths of two mirrors are exactly symmetrical. The corresponding eigenmode profiles of the laser cavity with PDPE’s are shown in Figs. 3 and 4, where both mode phases are flat. It is obvious that the output beams have a good uniform circular shape and a good uniform square shape for two orthogonal polarization modes. The degree of nonuniformity of the extraordinary ray is as small as 5.28 ⫻ 10⫺3 in the signal domain and the SNR is 3.73 ⫻ 104. The ordi-

continuous relief structures 共kinoforms兲 can be constructed with photoreduced gray-scale patterns as lithographic masks, and its procedures include mask exposure, development, and etch. Since continuous relief phase structures have been fabricated commercially by use of specially designed systems for direct laser beam writing lithography,12 it is better for us to use the second method. 4. Summary

Fig. 3. Three-dimensional plot of the amplitude distribution of the linear polarization eigenmode 共ordinary ray兲 with the uniform circular shaped output from M1.

nary ray mode is even better with a degree of nonuniformity of 3.13 ⫻ 10⫺4 and a SNR as high as 3.77 ⫻ 1012. Furthermore, the numerical results show that the loss per transit for these two modes is negligibly small, that is, 5.7 ⫻ 10⫺4 for the extraordinary ray and 1.06 ⫻ 10⫺7 for the ordinary ray. Inasmuch as polarization-selective diffractive mirrors can be fabricated accurately by lithography, the profile of the cavity modes can be arbitrarily preset. With regard to practical fabrication of PDPE’s, there are some different methods of fabricating efficient PDPE’s. The earliest and best-developed technique of PDPE fabrication is a two-step process that uses binary masks for multiple lithographic cycles. The multiple lithographic procedures include mask’s exposure, development, etch, and alignment. The most useful technique is a single-step process that uses a variable-intensity exposure with a direct-write system with an electron beam or a laser beam, in which the procedures include variable exposure, development, and etch. Another technique is a single-step process that uses a gray-scale mask so that multiple-level or

Fig. 4. Three-dimensional plot of the amplitude distribution of the linear polarization eigenmode 共extraordinary ray兲 with the uniform square shaped output from M2.

In summary, we have presented a new approach, based on the theory of angular spectrum, for designing a novel laser cavity with two PDPE’s. This laser cavity can produce two orthogonally real polarization modes with any specified patterns. A set of equations is rederived for determination of the surfacerelief depth of the PDPE’s of the laser cavity. The SA algorithm was used to solve these equations. The simulation results show that the He–Ne laser cavity can satisfactorily create two polarization beams with specified output patterns from two end mirrors. It is worth noting that the eigenmode phases are constants and the phase distributions of PDPE’s are complex. We believe that the present approach will provide a useful and effective tool for designing a multifunctional laser cavity that can satisfy different demands in practical applications. This research was supported by the National Natural Science Foundation of China. References 1. A. E. Siegman, Lasers 共University Science, Mill Valley, Calif., 1986兲, Chap. 23. 2. N. McCarthy and P. Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 共1985兲. 3. N. McCarthy and P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: misalignment sensitivity,” Appl. Opt. 22, 2704 –2708 共1983兲. 4. M. Piche and D. Cantin, “Enhancement of modal feedback in unstable resonators using mirrors with a phase step,” Opt. Lett. 16, 1135–1137 共1991兲. 5. V. Kermene, A. Saviot, M. Vampouille, B. Colombeau, and C. Froehly, “Flattening of the spatial laser beam profile with low losses and minimal beam divergence,” Opt. Lett. 17, 859–961 共1992兲. 6. P. A. Belanger and C. Pare, “Optical resonators using gradedphase mirrors,” Opt. Lett. 16, 1057–1059 共1991兲. 7. J. R. Leger, D. Chen, and Z. Wang, “Diffractive optical element for mode shaping of a Nd:YAG laser,” Opt. Lett. 19, 108–110 共1994兲. 8. J. R. Leger, D. Chen, and G. Mowry, “Design and performance of diffractive optics for custom laser resonators,” Appl. Opt. 34, 2498 –2509 共1995兲. 9. J. R. Leger, D. Chen, and K. Dai, “High modal discrimination in a Nd:YAG laser resonator with internal phase gratings,” Opt. Lett. 19, 1976 –1978 共1994兲. 10. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671– 680 共1983兲. 11. I. Richter, P. C. Sun, F. Xu, and Y. Fainman, “Design considerations of form-birefringent microstructures,” Appl. Opt. 34, 2421–2429 共1995兲. 12. T. J. Suleski and D. C. O’Shea, “Gray-scale masks for diffractiveoptics fabrication. I. Commercial slide imagers,” Appl. Opt. 34, 7507–7517 共1995兲.

20 November 1999 兾 Vol. 38, No. 33 兾 APPLIED OPTICS

6891