f[x0_,y0_,y_,d_]=ArcSin[Sin[x0]*(Sinh[d]-Sinh[y0]+Sinh[y0-d])/(Sinh[d]-Sinh[y]+Sinh[y-d])]

HxT=(Sin[xT]/Sinh[dT])*(Cosh[yT]-Cosh[yT-dT])

HyT=(Cos[xT]/Sinh[dT])*(Sinh[dT]-Sinh[yT]+Sinh[yT-dT])

LoadVectorFieldPlots`

dT=2

plot1=PlotVectorField[{HxT,HyT},{xT,0,2*Pi},{yT,0,dT}]

plot2={GrayLevel[.8],Rectangle[{0,0},{2*Pi,dT}]}

Show[plot1,Frame->True,FrameLabel->{ax,ay}]

Show[Graphics[plot2],plot1,Frame->True,FrameLabel->{ax,ay}]

d=2

p1=ParametricPlot[{{ArcSin[f[Pi/2,.1,y,2]],y},{Pi-ArcSin[f[Pi/2,.1,y,2]],y},{Pi+ArcSin[f[Pi/2,.1,y,2]],y},{2*Pi-ArcSin[f[Pi/2,.1,y,2]],y}},{y,0.1,1.9}, PlotRange->{{0,2*Pi},{0,2}}]

p2=ParametricPlot[{{ArcSin[f[Pi/2,.2,y,2]],y},{Pi-ArcSin[f[Pi/2,.2,y,2]],y},{Pi+ArcSin[f[Pi/2,.2,y,2]],y},{2*Pi-ArcSin[f[Pi/2,.2,y,2]],y}},{y,0.2,1.8}, PlotRange->{{0,2*Pi},{0,2}}]

p3=ParametricPlot[{{ArcSin[f[Pi/2,.3,y,2]],y},{Pi-ArcSin[f[Pi/2,.3,y,2]],y},{Pi+ArcSin[f[Pi/2,.3,y,2]],y},{2*Pi-ArcSin[f[Pi/2,.3,y,2]],y}},{y,0.3,1.7}, PlotRange->{{0,2*Pi},{0,2}}]

p4=ParametricPlot[{{ArcSin[f[Pi/2,.4,y,2]],y},{Pi-ArcSin[f[Pi/2,.4,y,2]],y},{Pi+ArcSin[f[Pi/2,.4,y,2]],y},{2*Pi-ArcSin[f[Pi/2,.4,y,2]],y}},{y,0.4,1.6}, PlotRange->{{0,2*Pi},{0,2}}]

p5=ParametricPlot[{{ArcSin[f[Pi/2,.5,y,2]],y},{Pi-ArcSin[f[Pi/2,.5,y,2]],y},{Pi+ArcSin[f[Pi/2,.5,y,2]],y},{2*Pi-ArcSin[f[Pi/2,.5,y,2]],y}},{y,0.5,1.5}, PlotRange->{{0,2*Pi},{0,2}}]

p6=ParametricPlot[{{ArcSin[f[Pi/2,.58,y,2]],y},{Pi-ArcSin[f[Pi/2,.58,y,2]],y},{Pi+ArcSin[f[Pi/2,.58,y,2]],y},{2*Pi-ArcSin[f[Pi/2,.58,y,2]],y}},{y,.58,1.42}, PlotRange->{{0,2*Pi},{0,2}}]

p7=ParametricPlot[{{ArcSin[f[Pi/2,.02,y,2]],y},{Pi-ArcSin[f[Pi/2,.02,y,2]],y},{Pi+ArcSin[f[Pi/2,.02,y,2]],y},{2*Pi-ArcSin[f[Pi/2,.02,y,2]],y}},{y,.01,1.99}, PlotRange->{{0,2*Pi},{0,2}}]

Show[p1,p2,p3,p4,p5,p6,p7]

g1=Graphics[{Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {Pi/2,1.99}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {Pi/2,1.89}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {Pi/2,1.77}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {Pi/2,1.64}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {Pi/2,1.49}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {Pi/2,1.32}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi/2,.7}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi/2,.53}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi/2,.38}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi/2,.26}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi/2,.14}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi/2,0.05}]}]

g2=Graphics[{Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {3*Pi/2,1.99}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {3*Pi/2,1.89}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {3*Pi/2,1.77}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {3*Pi/2,1.64}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {3*Pi/2,1.49}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {3*Pi/2,1.32}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {3*Pi/2,.7}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {3*Pi/2,.53}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {3*Pi/2,.38}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {3*Pi/2,.26}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {3*Pi/2,.14}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {3*Pi/2,0.05}]}]

g3=Graphics[{Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {.04,1.},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {.2,1},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {.39,1.},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {.57,1},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {.78,1.},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {1.05,1.},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {1.4,1.},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi-1.45,1.},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi-1.07,1.},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi-.82,1.},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi-.62,1.},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi-.42,1.},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi-.23,1.},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi-.06,1.},{0,0},{0,1}]}]

g4=Graphics[{Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi+.04,1.},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi+.2,1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi+.39,1.},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi+.57,1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi+.78,1.},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi+1.05,1.},{0,0},{0,1}],Text[StyleForm["<",FontSize->14,FontWeight->"Bold"], {Pi+1.4,1.},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {2*Pi-1.45,1.},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {2*Pi-1.09,1.},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {2*Pi-.83,1.},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {2*Pi-.62,1.},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {2*Pi-.42,1.},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {2*Pi-.23,1.},{0,0},{0,1}],Text[StyleForm[">",FontSize->14,FontWeight->"Bold"], {2*Pi-.06,1.},{0,0},{0,1}]}]

p8=Plot[{0,2},{x,0,2*Pi}]

Line1=Graphics[Line[{{0,0},{0,2}}]]

Line2=Graphics[Line[{{2*Pi,0},{2*Pi,2}}]]

Show[p1,p2,p3,p4,p5,p6,p7,p8,g1,g2,g3,g4,Line1,Line2,PlotRange->{-.1,2.1},Frame->True,FrameLabel->{ax,ay}]